Metric-measure boundary and geodesic flow on Alexandrov spaces

TL;DR

This paper explores the relationship between geodesic flow and volume growth in Alexandrov spaces, establishing conditions for the existence and measure-preserving properties of geodesic flow using analytic and integral geometry tools.

Contribution

It introduces a new analytic approach linking geodesic flow existence to volume growth, with implications for measure preservation in Alexandrov spaces.

Findings

Geodesic flow exists and preserves Liouville measure in key cases.

Volume growth of balls relates to the abundance of infinite geodesics.

Analytic tools connect integral geometry with geodesic flow analysis.

Abstract

We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry.

Equations

$$b_r(x):=\mu (B(x,r)).$$

$$v_r(x)=1-\frac{b_r(x)}{{\omega }_n\cdot r^n}$$

$${\mathcal{V}}_r=v_r\cdot \mu ,$$

$$\left\{{\mathcal{V}}_r/r;0<r\le 1\right\}$$

$$E(v)=(\pi (v),\pi ({\varphi }_1(v)).$$

$$d(\gamma (t),\gamma (s))=\lambda \cdot |t-s|.$$

$${\int }_A\mu (B(x,r))\cdot d\nu (x)\le {\int }_{B(A,r)}\nu (B(x,r))\cdot d\mu (x).$$

$$S=\left\{(y,x)\in X\times X;y\in A,d(y,x)<r\right\}$$

$$T=\left\{(y,x)\in X\times X;x\in B(A,r),d(y,x)<r\right\}$$

$${\exp }_x(v)={\gamma }_v(1).$$

$${\mathcal{V}}_r\ge -C\cdot r^2\cdot {\mathcal{H}}^n,$$

$$\frac{1}{1+\delta }\le \frac{|f(x)-f(y)|}{d(x,y)}\le 1+\delta$$

$$(1+\delta {)}^{-2\cdot n}\le \frac{b_r(x)}{{\omega }_n\cdot r^n}\le (1+\delta {)}^{2\cdot n}.$$

$$TX=\underset{x\in X}{⨆}{T}_{x}X.$$

$$\mathcal{M}(A)={\int }_X{\mathcal{H}}^n(A\cap T_{x}X)\cdot d{\mathcal{H}}^n(x).$$

$${\varphi }_t(\lambda \cdot v)=\lambda \cdot {\varphi }_{\lambda \cdot t}(v)\text{and}{\varphi }_{t+s}(v)={\varphi }_t({\varphi }_s(v)),$$

$$D=\bigcup _{x\in X}{D}_x\subset TX;$$

$$\pi ({\varphi }_1(\lambda \cdot v))={\exp }_{\pi (v)}(\lambda \cdot v).$$

$$(\pi (w),\exp (w))=(\exp (v),\pi (v))\in X\times X.$$

$$\epsilon \cdot r_m^{n+1}\le \mathcal{M}(A_m)-\mathcal{M}({\varphi }_1(A_m)).$$

$$\epsilon <\mathcal{M}(A)-\mathcal{M}({\varphi }_1(A)).$$

$${\varphi }_1(A)$$

$$=2\cdot {\varphi }_1\circ {\varphi }_1(\frac{1}{2}\cdot A),$$

$$\frac{\epsilon }{2^n}$$

$$\displaystyle\quad+\mathcal{M}\big{(}\phi_{1}(\tfrac{1}{2}\cdot A)\big{)}-\mathcal{M}\big{(}\phi_{1}\big{(}\phi_{1}(\tfrac{1}{2}\cdot A)\big{)}\big{)}.$$

$$\frac{\varepsilon}{2^{n+1}}<\mathcal{M}\big{(}A_{\frac{1}{2}}\big{)}-\mathcal{M}\big{(}\phi_{1}(A_{\frac{1}{2}})\big{)}.$$

$$\mathcal{M}(T^{r}K)={\omega }_n\cdot r^n\cdot {\mathcal{H}}^n(K).$$

$${\mathcal{H}}^{2\cdot n}(U^r(K))={\int }_K{b}_r(x)\cdot d{\mathcal{H}}^n(x).$$

$$\mathcal{V}_{r}(K)=\frac{1}{\omega_{n}\cdot r^{n}}\cdot\Big{(}\mathcal{M}(T^{r}K)-\mathcal{H}^{2\cdot n}(U^{r}(K))\Big{)}.$$

$$E(v)=(\pi (v),\pi (\exp (v))).$$

$$E(D^r)=U^r.$$

Full text

2010 Mathematics Subject classification. Primary 53C20, 52A15, 53C23. Keywords: Alexandrov spaces, convex hypersurfaces, geodesic flow, Liouville measure

1. Introduction

1.1. Motivation and application

The following question in the theory of Alexandrov spaces was formulated in a slightly different way in [PP96] and remains open.

• •

Are there “many” infinite geodesics on any Alexandrov space without boundary?

We address this question and obtain an affirmative answer in several cases. The main new tool is the investigation of the Taylor expansion of the average volume growth. The central results relate the first coefficient of this expansion to the geodesic flow and show how to control the Taylor expansion. This tool might be interesting in its own right, beyond the realm of Alexandrov geometry.

In particular, we prove the existence of such infinite geodesics in the most classical examples of non-smooth Alexandrov spaces:

THEOREM 1.1**.**

Let $X$ be the boundary of a convex body in $\mathbb{R}^{n+1}$. Then almost any direction in the tangent bundle $TX$ is the starting direction of a unique infinite geodesic on $X$. Moreover, the geodesic flow is defined almost everywhere and preserves the Liouville measure.

Apparently, the existence of a single infinite geodesic has not been known, even in the two-dimensional case [Zam92]. Our result might appear somewhat surprising since on most convex surfaces most points in the sense of Baire categories are not inner points of any geodesic; see [Zam82].

1.2. Metric-measure-boundary

On a smooth manifold with boundary the geodesic flow is not defined for all times. The amount of geodesics terminating at the boundary in a given time depends on the the size of this boundary, due to Santalo’s integral formula.

We are going to capture the size of the boundary by estimating the average volumes of small balls and their deviations from the corresponding volumes in the Euclidean space.

Let $(X,d)$ be a locally compact separable metric space, $\mu$ be a Radon measure on $X$ which takes finite values on the bounded subsets. For $x\in X$ and $r>0$ denote by $B(x,r)$ the open metric ball of radius $r$ around $x$. Consider the volume growth function $b_{r}\colon X\to[0,\infty)$,

[TABLE]

For a natural number $n>0$, let $\omega_{n}$ be the volume of the $n$-dimensional unit Euclidean ball. The deviation function

[TABLE]

measures in a very rough sense the deviation of the metric measure space $(X,d,\mu)$ from $\mathbb{R}^{n}$. Moreover, one can expect the behaviour of $v_{r}$ at the origin $r=0$ to reflect some curvature-like properties of the space $X$, as in the following fundamental example.

Example 1.2*.*

Let $X^{n}$ be a smooth Riemannian manifold with Riemannian volume $\mu$. Then, $v_{r}(x)=\frac{1}{6\cdot(n+2)}\cdot scal\cdot r^{2}$ up to terms of higher order in $r$. Here $scal$ denotes the scalar curvature of $X$.

In this paper we are interested in the order of vanishing of $v_{r}$ at $r=0$ and the first non-vanishing coefficient; in particular we assume that $v_{r}$ converges to zero in some integral sense. In most interesting metric spaces $(X,d)$, at least in the cases investigated here, the only reasonable choice of the measure $\mu$ for which $v_{r}$ is “sufficiently small” in $r$ is the $n$-dimensional Hausdorff measure $\mathcal{H}^{n}$:

Example 1.3*.*

Let $X$ be a countably $n$-rectifiable metric space. Assume that the Radon measure $\mu$ non-zero on open subsets of $X$. If $\mu=\mathcal{H}^{n}$ then the functions $v_{r}$ converge $\mathcal{H}^{n}$-almost everywhere to [math]. Moreover, $\mu=\mathcal{H}^{n}$ is the only measure with this property [AK00][Theorem 5.4].

Therefore, in the sequel, the number $n$ will always be the Hausdorff dimension of $X$ and $\mu$ will be the $n$-dimensional Hausdorff measure.

As seen in Example 1.3, most points in reasonably nice spaces are rather regular. It is conceivable, that by averaging the deviation functions $v_{r}$ we will smooth out the “wildest singularities”. The obtained objects will experience better behaviour at $r=0$ and tell us more about the regularity of the space.

Thus, instead of looking on the point-wise behaviour of $v_{r}$ at $r=0$ we define the deviation measure $\mathcal{V}_{r}$ of $X$ as a signed Radon measure

[TABLE]

absolutely continuous with respect to $\mu$.

The vector space $\mathrm{M}(X)$ of signed Radon measures on $X$ is dual to the topological vector space of compactly supported continuous functions $C_{c}(X)$. We consider the space $\mathrm{M}(X)$ with the topology of weak convergence. Recall that a subset $\mathcal{F}\subset\mathrm{M}(X)$ is relatively compact if and only if it is uniformly bounded; that is, if for any compact subset $K\subset X$ the values $\nu(K),\nu\in\mathcal{F}$ are uniformly bounded.

The next example, fundamental for this paper, can be obtained by computations in local coordinates. Since it is formally not needed in the sequel, we omit the details, however a rigorous proof can be extracted from the proof of Theorem 1.7 in Section 7 below.

Example 1.4*.*

Let $X$ be a smooth $n$-dimensional Riemannian manifold with boundary $\partial X$. Then, for $r\to 0$, the measures $\mathcal{V}_{r}/r$ converge in $\mathrm{M}(X)$ to $c_{n}\cdot\mathcal{H}^{n-1}_{\partial X}$, for some constant $c_{n}>0$ depending only on $n$.

This example suggests to view the first Taylor coefficient of $\mathcal{V}_{r}$ as the “boundary” of the metric-measure space $(X,d,\mu)$. It motivates the following definition.

Definition 1.5**.**

Let $(X,d,\mu)$ be a metric measure space as above. Let $\mathcal{V}_{r}$ be the deviation measure of $X$, as in (1.2). We say that $X$ has locally finite metric-measure boundary, abbreviated as mm-boundary, if the family of signed Radon measures

[TABLE]

is uniformly bounded. If $\lim_{r\to 0}\mathcal{V}_{r}/r=\nu$ in $\mathrm{M}(X)$, we call $\nu$ the mm-boundary of $X$. If $\nu=0$ we say that $X$ has vanishing mm-boundary.

We refer to Subsection 1.8 and Section 8 for a discussion of examples and questions, and state now our central result connecting mm-boundaries to the existence of infinite geodesics in Alexandrov spaces:

THEOREM 1.6**.**

Let $X$ be an Alexandrov space. If $X$ has vanishing mm-boundary, then almost each direction of the tangent bundle $TX$ is the starting direction of an infinite geodesic. Moreover, the geodesic flow preserves the Liouville measure on $TX$.

In different settings, geodesic flow on singular spaces have been investigated in [BB95] and [Bam16].

1.3. Size of the mm-boundary in Alexandrov spaces

The next theorem shows that, similarly to Example 1.4, the topological boundary is closely related to the mm-boundary in Alexandrov spaces.

THEOREM 1.7**.**

Let $X^{n}$ be an $n$-dimensional Alexandrov space. Then $X$ has locally finite mm-boundary. If $\nu=\lim\frac{\mathcal{V}_{s_{j}}}{s_{j}}$, for a sequence $s_{j}\to 0$, then $\nu$ is a Radon measure and the following holds true.

1. (1)

There is a Borel set $A_{0}$ with $\mathcal{H}^{n}(X\setminus A_{0})=\nu(A_{0})=0$. 2. (2)

If the topological boundary $\partial X$ is non-empty then $\nu\geq c\cdot\mathcal{H}^{n-1}_{\partial X}$, for a positive constant $c$ depending only on $n$. 3. (3)

If the topological boundary $\partial X$ is empty then $\nu(A)=0$, for any Borel subset $A\subset X$ with $\mathcal{H}^{n-1}(A)<\infty$.

We believe that an Alexandrov space with empty topological boundary $\partial X$ has vanishing mm-boundary, which would solve the question about the existence of infinite geodesics. This conjecture will be proved in two cases.

THEOREM 1.8**.**

Let $X^{n}$ be a convex hypersurface in $\mathbb{R}^{n+1}$ or let $X$ be a two-dimensional Alexandrov space without boundary. Then $X$ has vanishing mm-boundary.

In combination with Theorem 1.6 this proves Theorem 1.1. The two-dimensional case could be derived from the statement about convex hypersurfaces and Alexandrov’s embedding theorems. Another proof follows from a much stronger result discussed in the next subsection.

1.4. Metric-measure-curvature

Motivated by Example 1.2 one can naively hope that the second Taylor coefficient at [math] of the map $\mathcal{V}_{r}\colon r\mapsto\mathrm{M}(X)$ describes the scalar curvature of the space.

Definition 1.9**.**

Let $X,\mathcal{V}_{r}$ be as in Definition 1.5. If the family $\mathcal{V}_{r}/r^{2},r\leq 1$ is uniformly bounded then we say that $X$ has locally finite mm-curvature. If the measures $\mathcal{V}_{r}/r^{2}$ converge to a measure $\nu$, we call $\nu$ the mm-curvature of $X$.

Clearly, local finiteness of mm-curvature as defined above implies that the mm-boundary vanishes. Thus, the following result proves Theorem 1.8 in the $2$-dimensional case.

THEOREM 1.10**.**

Let $X$ be a 2-dimensional Alexandrov space without boundary. Then $X$ has locally finite mm-curvature.

This finiteness result holds true in the much greater generality of surfaces with bounded integral curvature in the sense of Alexandrov–Zallgaler–Reshetnyak [Res93], [AZ67], see Section 4.

Note, however, that the mm-curvature in Theorem 1.10 does not need to coincide with the “curvature measure” as defined in [AZ67], even in the case of a cone; compare to Example 1.14. In particular, this shows that the mm-curvatures in $2$-dimensional Alexandrov spaces are not stable under Gromov–Hausdorff convergence.

Remark 1.11*.*

Nina Lebedeva and the third named author have found in [LP17] a “scalar curvature measure” on all smoothable Alexandrov spaces. There is a hope, supported by our proof of Theorem 1.10, that a better understanding of this “stable curvature measure” will lead to some control of the mm-boundary and mm-curvature discussed here.

1.5. Relation to the Lipschitz–Killing curvatures

Let $M$ be a compact smooth submanifold in $\mathbb{R}^{n}$. Given $r>0$, consider the volume $w(r)=\mathcal{H}^{n}(B(M,r))$ of the distance tube $B(M,r)$ around $M$. The function $r\mapsto w(r)$ is a polynomial, at least for small positive $r$. The coefficients of $w(r)$, called the Lipschitz–Killing curvatures of $M$, are given as integrals of some intrinsically defined curvature terms. Moreover, these coefficients can be localized and considered as measures on $M$. We refer to [Ale16] for a short account of the theory, connection of the theory with [LP17] and further hypothetical relations with the theory of Alexandrov spaces.

To make the formal similarity with our approach to mm-boundary and mm-curvature more transparent, we observe that (at least for a smooth $n$-dimensional manifold $M$) the number $\int_{M}\mathcal{H}^{n}(B(x,r))\,\cdot d\mathcal{H}^{n}(x)$ can be interpreted as the $\mathcal{H}^{2\cdot n}$-measures of the distance tubes $B(\Delta,{\frac{r}{\sqrt{2}}})$ around the diagonal $\Delta$ in the Cartesian product $M\times M$.

1.6. Idea of the proof of Theorem 1.6

The interpretation of the tangent bundle of $M$ as the normal bundle of the diagonal $\Delta$ in $M\times M$ gives a connection between the measure theoretical properties of the tubes around $\Delta$ and the dynamical properties of the geodesic flow.

We clarify this abstract statement by explaining the main idea of our proof of Theorem 1.6 in the case of a complete smooth Riemannian manifold $X=M$. In this case the existence of geodesics is trivial. Thus, we just sketch a new proof of the classical fact that the geodesic flow $\phi$ preserves the Liouville measure $\mathcal{M}$ on $TM$. This proof is sufficiently stable to be transferred to the singular situation,

Denote by $\pi\colon TM\to M$ the tangent bundle of $M$. Let $\phi_{t}\colon TM\to TM$ be the geodesic flow for time $t$. Define $E\colon TM\to M\times M$ by

[TABLE]

By construction, $E(-\phi_{1}(v))=J(E(v))$, where $J$ is the involution of $M\times M$ which switches the coordinates. Since $J$ preserves the measure $\mathcal{H}^{2\cdot n}$ on $M\times M$ and $v\to-v$ preserves the Liouville measure $\mathcal{M}$ on $TM$, the statement that $\phi$ is measure preserving hinges upon the smallness of measure-distortion of the map $E\colon(TM,\mathcal{M})\to(M\times M,\mathcal{H}^{2\cdot n})$ close to the [math]-section.

In the present case of a Riemannian manifold, this property of $\phi_{1}$ is expressed by the fact that the differential of $E$ is the identity (after suitable identifications). Similarly, in the general case of Alexandrov spaces, we observe that the “infinitesimal” deviation (via the canonical map $E$) between $J$ being measure preserving (which we know) and $\phi_{1}$ being measure preserving (which is what we want to show) is expressed as the triviality of the mm-boundary.

1.7. Stability and relation with quasi-geodesics

Many Alexandrov spaces, for instance all convex hypersurfaces, appear naturally as Gromov–Hausdorff limits of smooth Riemannian manifolds. However, the properties of the geodesic flow, mm-boundaries and mm-curvature are unstable under limit operations; see also the discussion at the end of Subsection 1.4. Thus, there is no hope to deduce Theorem 1.10, Theorem 1.8 or Theorem 1.1 by a direct limiting argument.

For instance, being a geodesic is a local notion, not preserved under limits. However, any limit of geodesics in a non-collapsed limit of Alexandrov spaces is a curve sharing many properties with geodesics. These properties are used to define the so called quasi-geodesics; see [PP96], [Pet07] and the references therein. It was shown that any direction is the starting direction of an infinite quasi-geodesic. One motivation for the present paper was an attempt to prove Liouville’s theorem for the “quasi-geodesic flow”, see Subsection 3.6.

1.8. Examples

The estimates of the mm-boundary and mm-curvature are quite involved even in quite simple situations. The following examples are not needed in the sequel and we omit the somewhat tedious computations. Examples 1.14, 1.15 and 1.16 should be compared with [Ber03] and [Ber02] revealing further natural connections to the theory of Lipschitz–Killing curvature on singular subsets of the Euclidean space.

Example 1.12*.*

Let $X$ be a Riemannian manifold with a Lipschitz continuous metric. Then $X$ has vanishing mm-boundary.

Example 1.13*.*

If $X$ is a manifold with two-sided bounded curvature in the sense of Alexandrov then its mm-curvature is well-defined and absolutely continuous with respect to the Hausdorff measure.

Example 1.14*.*

Let $X$ be the Euclidean cone over the circle $S_{\rho}$ of length $\rho$. The curvature measure and the mm-curvature are Dirac measures concentrated at the tip of the cone. The mass of the curvature measure is $\alpha=2\pi-\rho$. From example 1.2 one would expect the mass of the mm-curvature to be $m(\alpha)=\frac{\alpha}{12}$. However, a straightforward calculation shows that $m(\alpha)=\frac{\alpha}{12}+f(\alpha)$, where $f(\alpha)=O(\alpha^{2})$ is a non-zero function.

Example 1.15*.*

Let $X$ be a finite $n$-dimensional simplicial complex with an intrinsic metric $d$. Assume that the restriction of $d$ to each simplex is given by a smooth Riemannian metric. Then $X$ has a finite mm-boundary $\nu$ with the support on the $(n-1)$-skeleton $X^{n-1}$.

Example 1.16*.*

Assume that $X$ as in the last example is a pseudo-manifold. Then $X$ has finite mm-curvature. If all simplices are flat then the mm-curvature is concentrated on the $(n-2)$-skeleton.

1.9. Structure of the paper

After preliminaries collected in Section 2, we prove Theorem 1.6 in Section 3 along the lines sketched above. In Sections 4, 5 and 7 we prove the remaining theorems 1.10, 1.8 and 1.7 respectively. The proofs of these theorems all rely on a decomposition of the space into a regular and a singular part, with a quantitative estimate of the size of the singular part. Finally, on the regular part we estimate the mm-curvature and mm-boundary by comparing it to other natural measures on these spaces.

In the case of surfaces, this comparison measure is the classical curvature measure, in the case of convex hypersurfaces, this comparison measure is the mean curvature. Finally, in the case of a general Alexandrov space, the comparison is given by the derivative of the metric tensor expressed in DC-coordinates, [Per95].

The needed control of the ball growth in terms of these measures is given by a theorem of Mario Bonk and Urs Lang in the case of surfaces and follows from classical convex geometry in the case of hypersurfaces. The analytical comparison result needed for Alexandrov spaces is established in Section 6.

In the final Section 8 we collect a number of comments and open questions which naturally arose during the work on this paper.

1.10. Acknowledgments

The authors are grateful for helpful conversations and comments to Semyon Alesker, Richard Bamler, Andreas Bernig, Ivan Izmestiev, Aaron Naber, Koichi Nagano.

The first author was supported in part by a Discovery grant from NSERC and by a Simons Fellowship from the Simons foundation (award 390117). The second author was supported in part by the DFG grants SFB TRR 191 and SPP 2026. The third author was partially supported by NSF grant DMS 1309340.

2. Preliminaries

2.1. Metric spaces

We refer to [BBI01] for basics on metric spaces. The distance between points $x,y$ in a metric space $X$ will be denoted by $d(x,y)$. By $B(x,r)$ we will denote the open metric ball of radius $r$ around a point $x$. For $A\subset X$ we denote by $B(A,r)$ the open $r$-neighborhood $B(A,r)=\cup_{x\in A}B(x,r)$.

A minimizing geodesic $\gamma$ in a metric space $X$ is a map $\gamma\colon\mathbb{I}\to X$ defined on an interval $\mathbb{I}$ such that for some number $\lambda\geq 0$ and all $t,s\in\mathbb{I}$

[TABLE]

In particular, we allow $\gamma$ to have any constant velocity $\lambda\geq 0$. A geodesic is a curve $\gamma\colon\mathbb{I}\to X$ such that its restriction to a small neighborhood of any point in $\mathbb{I}$ is a minimizing geodesic. Note that a geodesic is a curve of constant velocity.

2.2. Metric measure spaces

We refer to [Fed69] and [EG15] for basics on measure theory.

Let $X$ be a locally compact separable metric space. A Radon measure on $X$ is a measure on $X$ for which all compact subsets are measurable and have finite measure. Any Radon measure defines an element of $\mathrm{M}(X)$ the dual space to the topological vector space $\mathcal{C}_{c}(X)$ of compactly supported continuous functions on $X$. All elements in $\mathrm{M}(X)$ are called signed Radon measures. Any $\mu\in M(X)$ can be uniquely written as $\mu_{+}-\mu_{-}$, where $\mu_{\pm}$ are a Radon measures concentrated on disjoint subsets. The measure $|\mu|=\mu_{+}+\mu_{-}$ is called the total variation of $\mu$.

A family $\mathcal{F}$ of signed Radon measures on $X$ is uniformly bounded if for any compact subset $K\subset X$ there exists a constant $C(K)>0$ such that $|\mu|(K)\leq C(K)$ for any $\mu\in\mathcal{F}$. Any uniformly bounded sequence of signed measures $\mu_{i}$ has a convergent subsequence.

The following lemma will be repeatedly used.

Lemma 2.1**.**

Let $X$ be a metric space with two Radon measures $\mu$ and $\nu$. Let $r>0$ be arbitrary and let $A\subset X$ be a Borel subset. Then

[TABLE]

Proof.

Due to Fubini’s theorem the left hand side is the volume of

[TABLE]

with respect to the product measure $\nu\otimes\mu$. And on the right hand side of the inequality is the volume of the larger set

[TABLE]

with respect to the same measure.

Since $S\subset T$, the statement follows. ∎

2.3. Alexandrov spaces

We are assuming that the reader is familiar with basic theory of Alexandrov spaces and refer to [BGP92] as an introduction to the subject. In this paper, an Alexandrov space is a complete, locally compact, geodesic metric space of finite Hausdorff dimension and of curvature bounded from below by some $\kappa\in\mathbb{R}$. For Alexandrov spaces, an upper index will indicate the Hausdorff dimension; that is, $X^{n}$ denotes an $n$-dimensional Alexandrov space, equipped with the $n$-dimensional Hausdorff measure $\mathcal{H}^{n}$.

The set of starting directions of geodesics starting at a given point $x\in X$ carries a natural metric, whose completion is the tangent space $T_{x}=T_{x}X$ of $X$ at the point $x$. It is an $n$-dimensional Alexandrov space of non-negative curvature. Moreover, it is the Euclidean cone over the space $\Sigma_{x}$ of unit directions. The Euclidean cone structure defines multiplications by positive scalars $\lambda\geq 0$ on $T_{x}X$. The origin of the cone $T_{x}X$ is denoted by $0=0_{x}$. Elements of $T_{x}X$ are called tangent vectors at $x$, despite that $T_{x}X$ is not a vector space in general. For $v\in T_{x}X$ the norm $|v|$ of $v$ is the distance of $v$ from the origin $0_{x}$.

Geodesics in $X$ do not branch, moreover, any two geodesics with identical starting vectors coincide. For $x\in X$ the exponential map $\exp_{x}$ is defined as follows. Let $D_{x}$ denote the set of all vectors $v\in T_{x}X$ for which there exists an (always unique) minimizing geodesic $\gamma_{v}\colon[0,1]\to X$ with starting direction $v$. The exponential map is defined on $D_{x}$ as

[TABLE]

For any $r>0$, the map $\exp_{x}$ sends $D_{x}\cap B(0_{x},r)\subset T_{x}X$ surjectively onto $B(x,r)\subset X$. Moreover, for a constant $C=C(\kappa)\geq 0$ and all $r<\frac{1}{C}$ the map $\exp_{x}\colon D_{x}\cap B(0_{x},r)\to B(x,r)$ is $(1+C\cdot r^{2})$-Lipschitz continuous.

By the theorem of Bishop–Gromov, the volume $b_{r}(x)=\mathcal{H}^{n}(B(x,r))$ is bounded from above by the corresponding volume in the space of constant curvature $\kappa$. In particular, $b_{r}(x)\leq\omega_{n}\cdot r^{n}+C\cdot r^{n+2}$ for all $r\leq\frac{1}{C}$, where the constant $C$ can be chosen as before. Thus, the deviation measures $\mathcal{V}_{r}$ from (1.2) satisfy

[TABLE]

for all sufficiently small $r$. Here and in the previous paragraph, one can set $C=0$ if $\kappa\geq 0$.

Denote by $X_{reg}$ the set of all points $x\in X$ with $T_{x}X$ isometric to the Euclidean space. The set $X_{reg}$ has full $\mathcal{H}^{n}$-measure in $X$. Any inner point of any geodesic starting on $X_{reg}$ is contained in $X_{reg}$, [Pet98].

The topological boundary $\partial X$ of $X$ can be defined as the closure of the set of all points $x\in X$ with $T_{x}X$ isometric to a Euclidean half-space. Up to a subset of Hausdorff dimension $n-2$, $\partial X$ is an $(n-1)$-dimensional Lipschitz manifold.

2.4. Volume and bi-Lipschitz maps

Let $\mu=\mathcal{H}^{n}$ be a Radon measure on the metric space $X$. Let $U\subset X$ and $V\subset\mathbb{R}^{n}$ be open and assume that there is an $(1+\delta)$-bi-Lipschitz map $f\colon U\to V$; that is,

[TABLE]

for any pair of distinct points $x,y\in U$.

Let $A\subset U$ be given with $B(A,{(1+\delta)\cdot r})\subset U$ and $B(f(A),{(1+\delta)\cdot r})\subset V$. Then, for all $x\in A$,

[TABLE]

Therefore, if $\delta$ is sufficiently small, $|\mathcal{V}_{r}|(A)\leq 3\cdot n\cdot\delta\cdot\mathcal{H}^{n}(A).$

3. Liouville measure and geodesics

3.1. Tangent bundle and Liouville measure

Let $X$ be an $n$-dimensional Alexandrov space. Denote by $TX$ the disjoint union of the tangent spaces at all points,

[TABLE]

Let $\pi\colon TX\to X$ be the footpoint projection, so $\pi(T_{x}X)=\{x\}$ for any $x\in X$. For a subset $K\subset X$ denote by $TK$ the inverse image $\pi^{-1}(K)=\cup_{x\in K}T_{x}X$. Given $r>0$, denote by $T^{r}K$ the set of all vectors in $TK$ of norm smaller than $r$.

The Riemannian structure on the set of regular points discussed in [OS94] (see also [KMS01], [Per95]) provides $TX_{reg}$ with a structure of a Euclidean vector bundle over $X_{reg}$. In this topology, for any sequence of geodesics $\gamma_{i}$ in $X_{reg}$ converging to a geodesic $\gamma$, the starting directions of $\gamma_{i}$ converge to the starting direction of $\gamma$.

On the Euclidean vector bundle $TX_{reg}$ over $X_{reg}$ we have a natural choice of measure, which locally coincides with the product measure of $\mathcal{H}^{n}_{X}$ and the Lebesgue measures on the fibers. More precisely, it is the unique Borel measure $\mathcal{M}$ on $TX_{reg}$ such that for any Borel set $A\subset TX_{reg}$

[TABLE]

We extend $\mathcal{M}$ to a measure on $TX$ by setting $\mathcal{M}(TX\setminus TX_{reg})$ to be [math].

By the definition, a subset $A\subset TX$ is $\mathcal{M}$-measurable if and only if there exists a Borel subset $A^{\prime}\subset A\cap TX_{reg}$ such that for $\mathcal{H}^{n}$-almost all $x\in X$ the intersection $(A\setminus A^{\prime})\cap T_{x}X$ has $\mathcal{H}^{n}$-measure zero in $T_{x}X$.

For any $\lambda>0$, we have $\mathcal{M}(\lambda\cdot A)=\lambda^{n}\cdot\mathcal{M}(A)$ for any measurable set $A\subset TX$. The involution $I\colon TX_{reg}\to TX_{reg}$, defined by $I(v)=-v$, preserves $\mathcal{M}$ since it preserves the Lebesgue measure in each tangent space.

3.2. Geodesic flow

Let us define the geodesic flow $\phi$ on a maximal subset $\mathcal{F}$ of $TX\times\mathbb{R}$.

For any $v\in T_{x}X$ we set $\phi_{0}(v)=v$. If no geodesic starts in the direction of $v$, the value $\phi_{t}(v)$ will not be defined for $t\neq 0$. If such a geodesic $\gamma_{v}$ exists, then $\gamma_{v}$ can be uniquely extended to a maximal possible half-open interval $\gamma_{v}\colon[0,a)\to X$. For $t\geq a$ the value $\phi_{t}(v)$ will not be defined. For $0<t<a$ we set $\phi_{t}(v)$ to be $\gamma_{v}^{+}(t)\in T_{\gamma_{v}(t)}X$, the starting direction of $\gamma_{v}\colon[t,a)\to X$ at $\gamma_{v}(t)$.

If the geodesic $\gamma_{v}\colon[0,a)\to X$ extends to an (again uniquely defined, maximal) geodesic $\gamma_{v}\colon(b,a)\to X$ for some $b<0$ then we define $\phi_{t}(v)$ for $b<t<0$ to be $\gamma_{v}^{+}(t)$ as above.

We denote by $\mathcal{F}$ the set of all pairs $(v,t)\in TX\times\mathbb{R}$ for which $\phi_{t}(v)$ is defined.

For $\lambda>0$, for $t,s\in\mathbb{R}$ and $v\in T_{x}X$ we have

[TABLE]

whenever the right hand side is defined.

The partial flow $\phi$ preserves the norm of tangent vectors. Since inner points of geodesics starting in $X_{reg}$ are contained in $X_{reg}$, the set $TX_{reg}$ is invariant under the flow $\phi$.

By construction, the domain of the definition of the geodesic flow almost includes the domain of the definition of the exponential map. More precisely, consider the set

[TABLE]

that is, the set of all vectors $v\in TX$ for which $\exp_{\pi(v)}(v)$ is defined. Note that $\lambda\cdot D\subset D$ for any $0\leq\lambda\leq 1$. Moreover, for all $v\in D$ and all $0\leq\lambda<1$ the geodesic flow $\phi_{1}(\lambda\cdot v)$ is defined (equivalently $(\lambda\cdot v,1)\in\mathcal{F}$) and

[TABLE]

Thus, for $\mathcal{M}$-almost all $v\in D$ we have the following

• •

$v\in TX_{reg}$;

• •

$\phi_{1}(v)\in TX_{reg}$ is defined, hence $(v,1)\in\mathcal{F}$;

• •

$w=-\phi_{1}(v)\in D$ and

[TABLE]

3.3. Measurability

In order to use measure theoretic arguments we will need the following lemma, see also Subsection 3.6.

Lemma 3.1**.**

The set $\mathcal{F}\subset TX\times\mathbb{R}$ is measurable with respect to the product of the Liouville’s measure $\mathcal{M}$ on $TX$ and the Lebesgue measure on $\mathbb{R}$. Moreover the map $\phi\colon\mathcal{F}\to TX$ is measurable.

Proof.

Fix $(v,\tau)\in\mathcal{F}$ and set $\gamma(\tau\cdot t)=\phi_{t}(v)$, $t\in[0,1]$. Note that there exists some $k>0$ such that the restriction of $\gamma$ to any subinterval of length $\frac{1}{k}$ is a (minimizing) geodesic. We will call such $\gamma$ a $k$-geodesic and write $(v,\tau)\in\mathcal{F}_{k}$.

The limit of any converging sequence of $k$-geodesics is a $k$-geodesic. Hence $\mathcal{F}_{k}^{\prime}=\mathcal{F}_{k}\cap(TX_{reg}\times\mathbb{R})$ is a closed set in $TX_{reg}\times\mathbb{R}$. Therefore, $\mathcal{F}\cap(TX_{reg}\times\mathbb{R})$ is a countable union of closed subsets $\mathcal{F}_{k}^{\prime}$, hence measurable. Moreover, the restriction $\phi\colon\mathcal{F}_{k}^{\prime}\to TX_{reg}$ is continuous and, therefore, $\phi\colon\mathcal{F}\cap(TX_{reg}\times\mathbb{R})\to TX_{reg}$ is a Borel-measurable map.

Since $\mathcal{M}(X\backslash X_{reg})=0$ the statement follows. ∎

3.4. Liouville property

Denote by $\mathcal{G}$ the set of all vectors $v\in TX$ such that $\phi_{t}(v)$ is defined for all $t\in\mathbb{R}$. Note that $\mathcal{G}$ contains the [math]-section; it is invariant under multiplications by any $\lambda>0$ and it is invariant under the geodesic flow $\phi$. Moreover, $\mathcal{G}\cap TX_{reg}$ is invariant under the involution $I(v)=-v$.

Definition 3.2**.**

We say that an Alexandrov space $X$ has the Liouville property if $\mathcal{M}(TX\setminus\mathcal{G})=0$ and for any $t\in\mathbb{R}$ the geodesic flow $\phi_{t}\colon\mathcal{G}\to\mathcal{G}$ preserves the Liouville measure.

The Liouville property can be checked infinitesimally using the following lemma.

Lemma 3.3**.**

An Alexandrov space $X$ does not have the Liouville property if and only if there is a compact subset $K\subset X$, a positive number $\varepsilon$ and a sequence of positive numbers $r_{m}\to 0$ with the following property. For every $m$, there exists a Borel subset $A_{m}\subset T^{r_{m}}K$ such that

[TABLE]

Here $\phi_{1}(A_{m})$ is the set of all $\phi_{1}(v),v\in A_{m}$, for which $\phi_{1}(v)$ is defined.

Proof.

If at least one $r_{m}$ with the above property exists, then $X$ does not have the Liouville property by definition.

Assume that $X$ does not have the Liouville property. Then, by homogeneity of the geodesic flow, $\phi_{1}$ is either undefined on a subset of $TX$ with positive measure or it does not preserve the measure $\mathcal{M}$. In both cases we can find a compact subset $K_{1}\subset X_{reg}$, a Borel subset $A\subset T^{1}K_{1}$ and $\varepsilon>0$ such that

[TABLE]

Since

[TABLE]

we deduce

[TABLE]

Thus, taking either $A_{\frac{1}{2}}:=\frac{1}{2}\cdot A$ or $A_{\frac{1}{2}}:=\phi_{1}(\frac{1}{2}\cdot A)$ we infer

[TABLE]

The set $A_{\frac{1}{2}}$ constructed above is contained in $T^{\frac{1}{2}}K_{\frac{1}{2}}$, where $K_{\frac{1}{2}}=B(K_{1},{\frac{1}{2}})$.

Iterating the above procedure we obtain, for $r_{m}=\frac{1}{2^{m}}$, a subset $A_{r_{m}}\subset T^{r_{m}}K_{m}$ where $K_{m}=B(K_{m-1},{r_{m}})$ and such that (3.2) holds true.

The claim follows since all $K_{m}$ are contained in the set $B(K_{1},1)$, whose closure is compact, by completeness of $X$. ∎

Remark 3.4*.*

The completeness of the space $X$ is used in the proof of Theorem 1.6 only once, namely in the last line of the above proof.

3.5. Relation with the mm-boundary

Let us interpret the deviation measures $\mathcal{V}_{r}$ from (1.2) in suitable geometric terms.

Let $K\subset X$ be measurable and let $r>0$ be arbitrary. Since $\mathcal{H}^{n}(X\setminus X_{reg})=0$, we have

[TABLE]

Denote now by $U^{r}(K)$ the set of all pairs $(x,y)\in X\times X$ with $x\in K$ and $d(x,y)<r$. By Fubini’s theorem the set $U^{r}(K)$ is $\mathcal{H}^{n}\otimes\mathcal{H}^{n}=\mathcal{H}^{2\cdot n}$ measurable and we have

[TABLE]

Taking both equations together, we see that the signed measure $\mathcal{V}_{r}$ expresses the difference between $\mathcal{H}^{2\cdot n}$ and $\mathcal{M}$. More precisely,

[TABLE]

The following statement is a reformulation of Theorem 1.6.

THEOREM 3.5**.**

If an Alexandrov space $X$ has vanishing mm-boundary then it has the Liouville property.

Proof.

Arguing by contradiction, assume that $X$ does not have the Liouville property. Consider the compact subset $K\subset X$, the positive numbers $\varepsilon,r_{m}$ and the Borel subsets $A_{m}\subset T^{r_{m}}K$ provided by Lemma 3.3.

Let $Y$ be the closure of $B(K,1)$. Recall that $D\subset TX$ is the set of all vectors at which the exponential map is defined. For $r>0$, denote by $D^{r}$ the intersection of $D$ with $T^{r}Y$ and consider the “total exponential map” $E\colon D^{r}\to X\times X$ given by

[TABLE]

As above, let $U^{r}=U^{r}(Y)$ be the set of all pairs $(y,x)\in X\times X$ with $y\in Y$ and $d(x,y)<r$. Note that

[TABLE]

Moreover, for any fixed $x\in Y$, the restriction of $E$ to $D_{x}\cap D^{r}$ is a $(1+Cr^{2})$-Lipschitz continuous map from $D_{x}\subset T_{x}X$ onto the set $U^{r}\cap(\{x\}\times X)$ (see Subsection 2.3). Thus, for all sufficiently small $r$, and any Borel subset $S\subset D_{x}\cap D^{r}$, we have

[TABLE]

Using the definition of the Liouville measure $\mathcal{M}$ and Fubini’s formula for the product measure $\mathcal{H}^{2\cdot n}=\mathcal{H}^{n}\otimes\mathcal{H}^{n}$ on $X\times X$ we obtain for any $\mathcal{M}$-measurable subset $S$ of $D^{r}$

[TABLE]

Due to (3.3), the vanishing of the mm-boundary of $X$ implies

[TABLE]

Thus, up to terms of order higher than $r^{n+1}$, the map $E$ does not increase the measure of subsets, but the total mass of the image coincide with the total mass of the target. Therefore, $E$ is measure preserving up to terms of order higher than $r^{n+1}$ on all subsets of $T^{r}Y$. More precisely, combining the last two formulas we obtain for every $\delta>0$ the existence of some $s>0$ with the following property. For all $0<r<s$ and all measurable subsets $S\subset D^{r}$ we have

[TABLE]

For any measurable subset $S\subset D^{r}\cap TK$ we now claim

[TABLE]

In order to prove (3.7), let $S^{+}$ be the subset of all vectors $v\in S$ for which $\phi_{1}(v)$ exists and is contained in $TX_{reg}$. For all $v\in S^{+}$, we have $-\phi_{1}(v)\in D^{r}$ and, due to (3.1),

[TABLE]

The involution $I(v)=-v$ is $\mathcal{M}$-preserving on $TX_{reg}$. And the involution $J\colon X\times X\to X\times X$ given by $J(x,y)=(y,x)$ preserves $\mathcal{H}^{2\cdot n}$. Therefore, from (3.6) we deduce

[TABLE]

On the other hand, by construction,

[TABLE]

Hence, applying (3.6), we see

[TABLE]

and

[TABLE]

Together with (3.8) this finishes the proof of (3.7).

Coming back to our subsets $A_{m}\subset T^{r_{m}}K$, we have

[TABLE]

Setting $S_{m}=A_{m}\cap D^{r_{m}}$ we estimate the right hand side as the sum of the following three terms:

[TABLE]

Applying (3.6) and (3.7) this sum is bounded above by $4{\cdot}\delta{\cdot}r_{m}^{n+1}$, for all large $m$.

Therefore

[TABLE]

for all large $m$. Since $\delta$ is an arbitrary positive number, this leads to a contradiction. ∎

3.6. Quasi-geodesics flow

Finally, we discuss some relations with quasi-geodesics, referring the reader to [Pet07] for the basic properties of such curves. Recall, that whenever a unit speed minimizing geodesic $\gamma_{v}\colon[0,a]\to X$ start at a point $x$ in the direction $v$ then this is the unique quasi-geodesic defined on the interval $[0,a]$, [PP96], p.8, thus the same statement is also true for (local) geodesics $\gamma_{v}$.

Using this and the fact that a limit of quasi-geodesics is a quasi-geodesic, it is not difficult to conclude that the partial geodesic flow $\phi\colon\mathcal{F}\cap TX_{reg}\to TX_{reg}$ defined above is continuous. The latter statement slightly strengthening Lemma 3.1.

As in Subsection 3.1, we have a canonical measure $\mathcal{M}_{1}$ on the unit tangent bundle $\Sigma X\subset TX$ of $X$, which we also call the Liouville measure. Whenever $X$ has the Liouville property, then the geodesic flow is defined $\mathcal{M}_{1}\otimes\mathcal{H}^{1}$-almost everywhere on $\Sigma X\times\mathbb{R}$ and preserves $\mathcal{M}_{1}$. In this case for $\mathcal{M}_{1}$-almost each unit direction there exists exactly one quasi-geodesic starting in this direction.

Let now $X$ be an Alexandrov space with topological boundary $\partial X$ and let $Z$ be the doubling $X\sqcup_{\partial X}X$, which is an Alexandrov space without boundary, [Per91]. Quasi-geodesics in $X$ are exactly the projections of the quasi-geodesics in $Z$ under the folding $f\colon Z\to X$. From this we deduce that if $Z$ has the Liouville property, then $\mathcal{M}_{1}$-almost each direction $v\in\Sigma X$ is the starting direction of a unique infinite quasi-geodesic in $X$. Moreover, in this case, the corresponding quasi-geodesic flow preserves $\mathcal{M}_{1}$.

Finally, as an application of Theorem 1.6 and Theorem 1.7 we see that the above assumptions are fulfilled whenever the complement $X\setminus\partial X$ has vanishing mm-boundary. Indeed, in this case the mm-boundary of $Z$ must be concentrated on $\partial X\subset Z$, hence it must be trivial by Theorem 1.7,(3).

4. Surfaces with bounded integral curvature in the sense of Alexandrov

4.1. Preparations

We assume that the reader is familiar with the theory of surfaces with bounded integral curvature; see [AZ67] and [Res93].

Let $X$ be a surface with bounded integral curvature; it is a locally geodesic metric space, homeomorphic to a two-dimensional surface. It has Hausdorff dimension $2$ and the Hausdorff measure $\mathcal{H}^{2}$ is a Radon measure on $X$. There is another signed Radon measure on $X$, the so called curvature measure which will be denoted $\Omega$, [Res93, Section 8]. We will not assume that $X$ is complete.

We will derive Theorem 1.10 as a consequence of the following weak local version of a theorem of Mario Bonk and Urs Lang, [BL03], which relate the curvature measure to the volume of balls.

Lemma 4.1**.**

There exists some $\delta_{0}>0$ with the following property.

Let $X$ be a surface with bounded integral curvature and let $\Omega\in\mathrm{M}(X)$ be its curvature measure. Assume $X$ is homeomorphic to a plane and $|\Omega|(X)<\delta_{0}$. Then for any point $x\in X$, and $r>0$ such that $\bar{B}(x,{r})$ is compact we have

[TABLE]

Proof.

Set $\delta=|\Omega|(B(x,{r}))$. By continuity, it is sufficient to prove that $|1-\frac{b_{s}(x)}{\pi{\cdot}s^{2}}|\leq 3{\cdot}\delta$ for any $s<r$. Using approximations of the metric on $X$ by polyhedral metrics [Res93, Theorem 8.4.3, Theorem 8.1.9], we assume from now on that $X$ is polyhedral and homeomorphic to $\mathbb{R}^{2}$.

Claim: There exists a complete polyhedral surface $\hat{X}$ homeomorphic to a plane, which contains a copy of $B(x,s)$ and such that the curvature measure $\hat{\Omega}$ of $\hat{X}$ satisfies $|\hat{\Omega}|(\hat{X})<3\cdot\delta$.

Once the claim is proven, [BL03] provides us a bi-Lipschitz map $f\colon\hat{X}\to\mathbb{R}^{2}$ with the constant $L\leq 1+\frac{3{\cdot}\delta}{2{\cdot}\pi-3{\cdot}\delta}$. Since $\delta$ is small, an application of (2.1) finishes the proof of the lemma.

It remains to prove the Claim, certainly well-known to experts. Take some $r>t>s$ and consider the compact metric ball $\bar{B}(x,t)\subset B(x,r)$. We may assume that the boundary $S_{t}$ of $\bar{B}(x,t)$ does not contain singular points of $X$. By [Res93, Theorem 9.1, Theorem 9.3], the boundary $S_{t}$ is a (piecewise smooth) Jordan curve, once $\delta_{0}<2{\cdot}\pi$, and the negative part $\kappa^{-}$ of the geodesic curvature $\kappa$ of $S_{t}$ satisfies $|\kappa^{-}|(S_{t})\leq\delta$. Since $X$ is homeomorphic to a plane this implies that $\bar{B}(x,t)$ is homeomorphic to a closed disk $\bar{D}^{2}$ in $\mathbb{R}^{2}$.

We find a polygonal Jordan curve $\Gamma$ in $B(x,t)$ approximating $S_{t}$ such that the negative part of the geodesic curvature of $\Gamma$ is smaller than $2{\cdot}\delta$. Consider the closed Jordan domain $Y$ bounded by $\Gamma$, which can be assumed to contain $B(x,s)$. Now we glue to $Y$ along any edge of $\Gamma$ a flat half-strip. The boundary of the arising polyhedral surface consists of pairs of rays $\gamma_{i}^{\pm}$ emanating from the vertices $V_{1},\dots,V_{k}$ of $\Gamma$. The rays $\gamma_{i}^{\pm}$ enclose an angle equal to $2{\cdot}\pi-\alpha_{i}$, where $\pi-\alpha_{i}$ is the angle of $\Gamma$ at $V_{i}$ measured in $Y$. In order to finish the construction of $\hat{X}$ we glue a flat sector of angle $\alpha_{i}$ between $\gamma_{i}^{\pm}$, if $\alpha_{i}>0$ and we glue $\gamma_{i}^{\pm}$ together if $\alpha_{i}\leq 0$. Since $Y$ was a polyhedral disc, the arising space $\hat{X}$ is a complete polyhedral plane. All of the singularities of $\hat{X}$ are contained in $B(x,s)\cup\{V_{1},\dots,V_{k}\}$. Moreover, by construction, the curvature measure $\hat{\Omega}$ of $\hat{X}$ satisfies

[TABLE]

We deduce,

[TABLE]

This finishes the proof of the claim and of Lemma 4.1. ∎

4.2. Local finiteness of mm-curvature

Now we are ready to prove and prove the following generalization of Theorem 1.10.

THEOREM 4.2**.**

Let $X$ be an Alexandrov surface with integral curvature bounds. Then, equipped with the Hausdorff measure $\mathcal{H}^{2}$, the space $X$ has locally finite mm-curvature.

Proof.

Let again $\Omega$ denote the curvature measure of $X$. Let $\delta_{0}>0$ be sufficiently small and satisfy the conclusion of Lemma 4.1. The statement of Theorem 4.2 is local, so we need to prove it only in a small neighborhood of any point. Thus we may (and will) assume that there is a point $x_{0}\in X$ such that $|\Omega|(X\setminus\{x_{0}\})<\delta_{0}$ and that $X$ is homeomorphic to a plane.

Let $A\subset X$ be compact. Choose some $\varepsilon>0$ such that the closure of $B(A,{2{\cdot}\varepsilon})$ in $X$ is compact and such that, for any $0<2{\cdot}r<\varepsilon$ the inequality $\mathcal{H}^{2}(B(x_{0},{3{\cdot}r}))<\tfrac{1}{\varepsilon}\cdot r^{2}$ holds true; see [Res93, Lemma 8.1.1].

Let $r<\varepsilon$ be arbitrary. For any $x\in B(x_{0},{2{\cdot}r})$ we have

[TABLE]

For any $x\notin B(x_{0},{r})$ we have $|\Omega|(B(x,r))<\delta_{0}$. Thus, by Lemma 4.1,

[TABLE]

For the deviation measures $\mathcal{V}_{r}$ from (1.2) we estimate:

[TABLE]

On the other hand,

[TABLE]

where we have used Lemma 2.1 in the last step. For any $x$ contained in the domain of integration of the last integral, we have $\mathcal{H}^{2}(B(x,r))=b_{r}(x)\leq 2{\cdot}\pi{\cdot}r^{2}$, by Lemma 4.1, once $\delta_{0}$ has been chosen to be sufficiently small. We deduce $|\mathcal{V}_{r}|(A\setminus B(x_{0},{2{\cdot}r}))\leq 6{\cdot}\pi{\cdot}\delta_{0}{\cdot}r^{2}$.

Thus, for some constant $C=C(\varepsilon)$ and all $r<\varepsilon$, we obtain

[TABLE]

This finishes the proof of the theorem. ∎

5. Convex hypersurface

In this section we are going to prove Theorem 1.1.

The proof will follow from Theorem 1.6 by comparing the mm-boundary with the mean curvature measure on convex hypersurfaces.

It is possible to deduce the theorem without a reference to Theorem 1.7, from Lemma 5.2 alone, but Theorem 1.7 shortens the proof.

All results in this section are local, but for simplicity, we consider only closed convex hypersurfaces. The hypersurfaces will always be equipped with the induced intrinsic metric.

We assume that the reader is familiar with the basics of the theory of convex functions and convex geometry.

5.1. Mean curvature

Let $X$ be a convex hypersurface in $\mathbb{R}^{n+1}$. Recall that there exists a Radon measure $\mathcal{K}$ on $X$, called the mean curvature measure; see [Sch93], [Fed59].

The measure $\mathcal{K}$ has the following properties. For smooth hypersurfaces $X$, we have $\mathcal{K}=\kappa\cdot\mathcal{H}^{n}$, where $\kappa$ is the usual mean curvature function of $X$. The mean curvature measure is stable under Hausdorff convergence of convex hypersurfaces in $\mathbb{R}^{n+1}$. If the hypersurface is rescaled by $\lambda$, the mean curvature $\mathcal{K}$ is rescaled by $\lambda^{n-1}$.

A point $x$ in the convex hypersurface $X$ is called smooth if there is a unique supporting hyperplane of $X$ at this point. For any smooth point $x\in X$, any sequence $x_{j}\in X$ converging to $x$ and any sequence of positive numbers $t_{j}$ converging to [math], the sequence of convex hypersurfaces $X_{j}$ obtained from $X$ by the dilatation by the factor $\frac{1}{t_{j}}$ centered at the point $x_{j}$ converges to the tangent hyperplane of $X$ at $x$.

The stability of the mean curvature measures $\mathcal{K}$, vanishing of $\mathcal{K}$ on flat hyperplanes and the behavior of $\mathcal{K}$ under rescalings gives us:

Lemma 5.1**.**

Let $X$ be a convex hypersurface in $\mathbb{R}^{n+1}$. Let $A$ be a compact set of smooth points in $X$ and $\delta>0$. Then there exists some $t>0$ such that

[TABLE]

for any $y\in B(A,{t})$ and any $0<r<t$.

Thus, the following lemma applies to all small balls in a neighborhood of any smooth point.

Lemma 5.2**.**

There exist numbers $\delta_{0},C>0$ depending only on $n$ with the following property. Let $X$ be a convex hypersurface in $\mathbb{R}^{n+1}$. Let $x\in X$ be a point and $r>0$ be such that the mean curvature $\mathcal{K}$ satisfies $\mathcal{K}(B(x,{6{\cdot}r}))<\delta\cdot r^{n-1}$ with $\delta<\delta_{0}$. Then

[TABLE]

Proof.

By rescaling, it suffices to prove the existence of $\delta_{0},C>0$ such that the lemma holds for $r=1$. By approximation, it is sufficient to prove the result for smooth convex hypersurfaces.

Fix a sufficiently small $\varepsilon_{0}>0$. The mean curvature vanishes on $B(x,6)$ if an only if $B(x,6)$ is contained in a flat hyperplane. Due to the stability of $\mathcal{K}$ under convergence, if $\delta_{0}$ is small, then the ball $U=B(x,{5})\subset X$ is close to a flat hyperplane in $\mathbb{R}^{n+1}$. Thus, we may assume that the tangent hyperplanes to points in $U$ are $\varepsilon_{0}$-close to the tangent space $W=T_{x}X\subset\mathbb{R}^{n+1}$. Therefore, $U$ is a graph $U=\{(x,f(x))\}$ of a convex function $f\colon V\to\mathbb{R}$ defined on an open subset $V\subset W$. Moreover, $V$ contains the ball of radius $4$ in $W$ around $x$. Denote by $B(x,2)_{W}$ the ball of radius $2$ in $W$ around $x$. Set

[TABLE]

If $\delta_{0}$ is small, then $a<\varepsilon_{0}$. The orthogonal projection $P\colon U\to V$ is $1$-Lipschitz and the restriction of the inverse $P^{-1}$ to $B(x,2)_{W}$ has Lipschitz constant

[TABLE]

Applying (2.1) we only need to prove that $a<C\cdot\delta$ for a constant $C$.

Denote by $|D^{2}f|$ the largest eigenvalue of the Hessian $D^{2}f$. Since $f$ is convex and $\varepsilon_{0}$ is small, the mean curvature $\mathcal{\kappa}(x)$ at the point $(x,f(x))$ of the graph $U$ of $f$ satisfies $\kappa(x)\geq\frac{n}{2}\cdot|D^{2}f|$. Hence, the conclusion follows from the following statement.

Claim: Let $f:B\to\mathbb{R}$ be a smooth convex function on the open ball $B=B(0,4)\subset\mathbb{R}^{n}$. If $f(0)=|\nabla f(0)|=0$ then, for some $C=C(n)>0$,

[TABLE]

By convexity, it is sufficient to find some $C=C(n)>0$ with

[TABLE]

see also [EG15, Theorem 6.7].

First note that $f(z)\geq 0$ for all $z$ since $f(0)=|\nabla f(0)|=0$ and $f$ is convex.

In order to verify (5.2), we can multiply the function $f$ by a constant and assume that $f$ takes its maximum on the closed ball $\bar{B}(0,3)$ at the point $y_{0}$ and $f(y_{0})=1$. Convexity of $f$ implies that $|y_{0}|=3$. Since $f(0)=0$ and $f$ is convex, we must have $f(y)\leq\frac{1}{3}$ for all $y\in B(0,1)$.

By convexity and the choice of $y_{0}$, the restriction of $f$ to the supporting hyperplane $H$ of $\bar{B}(0,3)$ at $y_{0}$ is bounded from below by $1$. Consider the ball $S$ of radius $\frac{1}{2}$ in $H$ around $y_{0}$. For any point $z\in S$ consider the restriction

[TABLE]

to the segment of length $6$ starting at $z$ orthogonal to $H$. Then

[TABLE]

Thus for some $t\in(0,3)$ we have $f_{z}^{\prime}(t)\leq-\frac{2}{9}$ and for some $t\in(3,6)$ we have $f_{z}^{\prime}(t)\geq-\frac{1}{9}$. Therefore

[TABLE]

Integrating over $S$ we obtain by Fubini’s theorem a uniform positive lower bound on $\int_{B}|D^{2}f|$. This finishes the proof of (5.2). Hence the claim and Lemma follow. ∎

5.2. The proof

The next theorem is the first part of Theorem 1.8; the second part follows from Theorem 4.2. In combination with Theorem 1.6 it also finishes the proof of Theorem 1.1.

THEOREM 5.3**.**

Let $X$ be a convex hypersurface in $\mathbb{R}^{n+1}$. Then it has vanishing mm-boundary.

Proof.

Since $X$ has locally finite mm-boundary by Theorem 1.7, it suffices to prove that any partial limit measure $\nu$ of a sequence $\frac{1}{r_{j}}\cdot\mathcal{V}_{r_{j}}$ for $r_{j}\to 0$ must be the zero measure.

Fix a partial limit measure $\nu$. Due to Theorem 1.7, $\nu(A)=0$ for any Borel subset $A\subset X$ with $\mathcal{H}^{n-1}(A)<\infty$. Let $Y\subset X$ be the set of smooth points of $X$. The complement $X\setminus Y$ is a countable union of subsets with finite $(n-1)$-dimensional Hausdorff measure (see [Zaj79] and [Sch93, Theorem 1.4]) therefore $\nu(X\setminus Y)=0$. Therefore, it is sufficient to prove $\nu(A)=0$ for any compact subset $A\subset Y$.

Fix a compact subset $A\subset Y$ and let $\delta>0$ be an arbitrary sufficiently small number. Consider a positive $1>t>0$ provided by Lemma 5.1. Let $U$ be the open set $B(A,t)$.

Assume $0<r<t$. Applying Lemma 5.2, for $x\in U$ we get

[TABLE]

we have used Lemma 2.1 in the second and Bishop–Gromov inequality in the last inequality. Hence

[TABLE]

Since $\delta$ can be chosen arbitrary small, we obtain $|\nu|(A)=0$.

This finishes the proof of the claim and, therefore, of Theorem 5.3. ∎

6. An integral inequality for Riemannian metrics

6.1. The smooth case

We start by estimating from above the deviation measure $\mathcal{V}_{r}$ on a smooth Riemannian manifold in terms of the first derivatives of the metric. We do not know how to prove a similar estimate from below, see Problem 8.3. However, for the applications to Alexandrov spaces discussed in the next section, the estimate from below is a consequence of the theorem of Bishop–Gromov.

For a smooth Riemannian metric $g$ defined on an open subset $U\subset\mathbb{R}^{n}$ we denote by $|g^{\prime}|\colon U\to[0,\infty)$ the sum $\Sigma_{i,j,k}|\frac{\partial}{\partial\,x_{k}}g_{ij}|$.

PROPOSITION 6.1**.**

There exists a constant $C=C(n)>1$ with the following property. Let $U\subset\mathbb{R}^{n}$ be an open subset with a smooth Riemannian metric $g$ which is $(1+\frac{1}{C})$-bi-Lipschitz to the background Euclidean metric. Let $A\subset U$ be a Borel subset. Let $r>0$ be such that $B(A,{2{\cdot}r})$ is relatively compact in $U$. Then

[TABLE]

Proof.

We will denote by $C_{i}$ various (explicit) constants which depend only on $n$.

We will use the following notations. By $|\cdot|$ and $\mathcal{L}^{n}$ we denote respectively the norm and the Lebesgue measure on the background $\mathbb{R}^{n}$. For $x\in U$ we denote by $g_{x}$ the Riemannian tensor at the point $x$ and by $|\cdot|_{x}$ the corresponding norm. The Hausdorff measure of the Riemannian metric $g$ has the form $u\cdot\mathcal{L}^{n}$, with $u=\sqrt{det(g_{ij})}$.

For $x\in\mathbb{R}^{n}$, we consider the function $K\colon U\to[0,\infty)$ given by

[TABLE]

By smoothness of the determinant and the square root, we find a constant $C_{1}$ such that for all $x\in U$ we have

[TABLE]

We fix $A\subset X$ and $r>0$ as in the formulation of the proposition. For $x\in U$ denote by $B_{x}$ the metric ball $B(x,r)$ in $U$. By $B^{x}$ we denote the metric ball of radius $r$ in the Euclidean norm $|\cdot|_{x}$. In this Euclidean metric the ball $B^{x}$ has measure

[TABLE]

Thus, in order to estimate the deviation measure $\mathcal{V}_{r}$, we only need to control the summands on the right of the following inequality:

[TABLE]

We may assume that the bi-Lipschitz constant $1+\frac{1}{C}$ is close to 1, so that $\frac{1}{2}<u<2$. Moreover, we may assume $B_{x}$ and $B^{x}$ are contained in the ball of radius $\frac{4}{3}r$ around $x$ with respect to the background Euclidean metric.

In order to bound the first summand, for $x\in A$ and $|v|_{x}=1$, we set $l^{x}_{v}$ to be the length of the segment $[x,x+v]$ in the Riemannian metric $g$. Then we compute

[TABLE]

Observe now that the intersection of $B^{x}\setminus B_{x}$ with the ray starting in $x$ in the direction of $v$ has $\mathcal{H}^{1}$-measure ( with respect to the norm $|\cdot|_{x}$) at most $2\cdot(l_{v}^{x}-r)$, once the bi-Lipschitz constant $(1+\frac{1}{C})$ is close to $1$. Integrating in polar coordinates over the ball $B^{x}\subset(\mathbb{R}^{n},|\cdot|_{x})$ we infer:

[TABLE]

where $S^{n-1}_{x}$ is the unit sphere in $(\mathbb{R}^{n},|\cdot|_{x})$.

To get a similar estimate of the other summand in (6.2), we only need to recall the following inequality from [EG15, Lemma 4.1], valid for any $\mathcal{C}^{1}$ function $u$ on a Euclidean ball

[TABLE]

Taking both estimates together with (6.1), embedding $B_{x}$ and $B^{x}$ in slightly larger Euclidean balls and using that $\frac{1}{2}<u<2$, we conclude:

[TABLE]

We divide both sides by $\omega_{n}\cdot r^{n}$ and integrate over $A$. Using that the bi-Lipschitz constant is close to $1$, we see:

[TABLE]

where we have used Lemma 2.1 in the second inequality. This finishes the proof of Proposition 6.1. ∎

6.2. Functions of bounded variations

Let $U$ be an open subset of $\mathbb{R}^{n}$. A function $f\in L^{1}(U)$ is of class BV (bounded variation) if its first partial derivatives, $\frac{\partial f}{\partial x^{i}}$ (here and below always in the sense of distributions) are signed Radon measures with finite mass $|\frac{\partial f}{\partial x^{i}}|(U)$. We denote by $[Df]$ the Radon measure $\sum_{i}|\frac{\partial f}{\partial x^{i}}|$ on $U$. If $f\colon U\to\mathbb{R}$ is a BV function, which is continuous on a subset $R\subset U$ with $\mathcal{H}^{n-1}(U\setminus R)=0$ then the Radon measure $[Df]$ vanishes on all Borel subsets $A\subset U$ with $\mathcal{H}^{n-1}(A)<\infty$, [GL80].

Let $f\colon U\to\mathbb{R}$ be of class BV. Then for $\mathcal{H}^{n}$-almost every point $x\in U$ there exists an affine function $\hat{f}_{x}\colon\mathbb{R}^{n}\to\mathbb{R}$, such that for the BV function $h_{x}=f-\hat{f}_{x}$ we have

[TABLE]

see [EG15, Theorem 6.1 (2),(3)] for the second and the Hölder inequality and [EG15, Theorem 6.1 (1)] for the first inequality.

6.3. Almost Riemannian metric spaces

The following definition provides a suitable description of a large part of any Alexandrov space, see Section 7.

Let $C=C(n)$ be the constant determined in Proposition 6.1. We will call a locally geodesic metric space $X$ an almost Riemannian metric space if it has the following properties (see [AB15] for a careful discussion of such $DC_{0}$-Riemannian manifolds in the language of [AB15] and [Per95]):

1. (1)

There is a Borel subset $R\subset X$, called the subset of regular points with $\mathcal{H}^{n-1}(X\setminus R)=0$. 2. (2)

Any minimizing geodesic $\gamma$ in $X$ can be approximated by curves $\gamma_{i}$ in $R$, such that the lengths of $\gamma_{i}$ converge to the length of $\gamma$. 3. (3)

For any $x\in X$, there is a neighborhood $U$ of $x$, called a regular chart, and a bi-Lipschitz map $\phi\colon U\to O$ onto an open subset $O\subset\mathbb{R}^{n}$, with the bi-Lipschitz constant less than $(1+\frac{1}{C})$. 4. (4)

There is a continuous Riemannian tensor $g_{ij}$ on $\phi(U\cap R)$ such that $g_{ij}$ is a function of bounded variation on $O$ for each $1\leq i,j\leq n$. 5. (5)

The length of any curve $\gamma\subset R$ can be computed as the length of $\phi(\gamma)$ via this Riemannian tensor $g$.

For any regular chart $U$ as above, we set $\mathcal{N}_{0}(U)$ to be the Radon measure $[g^{\prime}]$ on $U$ given as the sum of the Radon measures $[Dg_{ij}]$ over the coordinates $g_{ij}$ of the metric tensor $g$. For an almost Riemannian metric space $X$, we define an outer measure $\mathcal{N}$ on $X$ in the following way. For a subset $A\subset X$, we consider all coverings $A\subset\cup U_{i}$ by countably many regular charts $U_{i}$ and let $\mathcal{N}(A)$ to be the infimum of the sums $\sum_{i}\mathcal{N}_{0}(U_{i})$ over all such coverings. This is indeed an outer measure, which takes finite values on compact subsets. Since $\mathcal{N}$ satisfies the Caratheodory criterion, [EG15, Theorem 1.9], it is indeed a Radon measure. We will call $\mathcal{N}$ the minimal metric derivative measure on the almost Riemannian metric space $X$.

Lemma 6.2**.**

Let $X^{n}$ be a almost Riemannian metric space and let $\mathcal{N}$ be its minimal metric derivative measure. Then $\mathcal{N}(A)=0$ for any Borel subset $A\subset X$ with $\mathcal{H}^{n-1}(A)<\infty$. There exists a Borel subset $C\subset X$ of full $\mathcal{H}^{n}$-measure in $X$ with $\mathcal{N}(C)=0$, thus $\mathcal{N}$ is absolutely singular with respect to $\mathcal{H}^{n}$.

Proof.

Clearly, both claims are local. Hence we need to verify them only in a regular chart $U$, which we identify with its image $\phi(U)\subset\mathbb{R}^{n}$. The first statement follows directly from the continuity of the metric tensor $g$ on the subset $U\cap R$ and the result of [GL80] cited above.

In order to verify the second claim we only need to show the following statement; see also [EG15, Section 1.6]. For almost all $x\in U$ there is another regular chart $x\in V$, such that the derivative measure $[h^{\prime}]$ of the Riemannian tensor $h$ in this chart $V$, has $n$-dimensional density [math] at $x$, thus

[TABLE]

Here and below, the ball $B(x,r)$ over which we integrate can be equally considered with respect to the Euclidean or to the original metric on $U$, since both are bi-Lipschitz equivalent. In order to prove (6.4), we follow [Per95, Section 4.2] and consider the Riemannian tensor $g$ of the original chart $U$. Applying  (6.3) to the coordinates of $g$, we find for $\mathcal{H}^{n}$-almost all $x\in U$ a smooth symmetric $2$-tensor $\hat{g}=\hat{g}_{x}$ on $U$ such that for $u=g-\hat{g}$ we have:

[TABLE]

The first statement implies that $\hat{g}$ is indeed a Riemannian metric in a neighborhood $U_{0}$ of $x$.

Fix such a point $x$, neighborhood $U_{0}$ and $\hat{g}$. Consider a small neighborhood $W$ of [math] in $\mathbb{R}^{n}$ and let $\xi\colon W\to U$ be the exponential map with respect to the metric $\hat{g}$. Then $\xi(0)=x$, $D\xi(0)=Id$ and the pull-back Riemannian metric $\hat{h}=\xi^{\ast}(\hat{g})$ has zero derivative at [math]. Since $D\xi$ is the identity, the bi-Lipschitz constant of the restriction $F=\xi^{-1}\circ\phi$ to a sufficiently small neighborhood $V$ of the point $x$ is still less than $(1+\frac{1}{C})$. Hence, $F\colon V\to\mathbb{R}^{n}$ is a regular chart.

The Riemannian tensor $h$ in this chart equals $\hat{h}+\xi^{\ast}(g-\hat{g})$. Now, $D\hat{h}(0)=0$, thus (6.4) holds for $\hat{h}$ instead of $h$. For the other summand $\xi^{\ast}(u)$, the density estimate (6.4) follows from (6.5) and the fact that $\xi$ is a $C^{2}$-diffeomorphism if $W$ is sufficiently small. This finishes the proof of Lemma 6.2. ∎

6.4. The upper bound on the deviation measures

Continuing to denote by $C=C(n)$ the constant from Proposition 6.1 we show:

Corollary 6.3**.**

Let $U$ regular chart of an almost Riemannian metric space $X$. Identifying $U$ with its image $O=\phi(U)$, let $g$ be the metric tensor and the measure $\mathcal{N}_{0}=[Dg]$ the derivative of the metric tensor. For any Borel subset $A\subset U$ and any $r$ such that $B(A,{3{\cdot}r})$ is relatively compact in $U$ we have

[TABLE]

Proof.

Consider a relatively compact open subset $V\subset U$, which contains $B(A,{2{\cdot}r})$. Apply (coordinatewise) the standard mollifying construction to the Riemannian tensor $g$. For all small positive $\varepsilon$, we thus obtain smooth metrics $g_{\varepsilon}$ on $V$ with the following properties. The total derivatives $|g_{\varepsilon}^{\prime}|$, considered as measures, satisfy $|g_{\varepsilon}^{\prime}|\leq\mathcal{N}_{0}$ on $V$, [Zie89, Theorem 5.3.1]. Since $g$ is pointwise $\frac{1}{C}$-close to the background Euclidean inner product, the same is true for $g_{\varepsilon}$. For all sufficiently small $\varepsilon$ the $2{\cdot}r$-tubular neighborhood around $A$ with respect to $g_{\varepsilon}$ is contained in the $3{\cdot}r$-tubular neighborhood around $A$ with respect to the original distance in $X$. Moreover, $g_{\varepsilon}$ converges to $g$ pointwise at all points of $R$, [Zie89, Theorem 1.6.1].

Denote by $d_{\varepsilon}$ the distance function induced by $g_{\varepsilon}$. From the last statement and the properties (2),(5) in the definition of an almost Riemannian metric space we deduce that

[TABLE]

Finally, the Hausdorff measures of the Riemannian metrics $g_{\varepsilon}$ converge on $V$ to the Hausdorff measure of $V$ with respect to the original metric.

Now the result follows directly from Proposition 6.1 applied to the metrics $g_{\varepsilon}$, by letting $\varepsilon$ go to [math]. ∎

As a consequence of Corollary 6.3, the minimal metric derivative measure bounds from above the deviations measure $\mathcal{V}_{r}$ on any almost Riemannian metric space:

Lemma 6.4**.**

Let $X$ be a almost Riemannian metric space with the metric derivative measure $\mathcal{N}$. Then for any compact subset $A\subset X$, there exists some $r_{0}>0$ such that for all $r<r_{0}$ we have

[TABLE]

Proof.

Cover $A$ by finitely many regular charts $U_{i}$ such that $\sum\mathcal{N}_{0}(U_{i})$ is sufficiently close to $\mathcal{N}(A)$. Since the covering dimension of $X$ is $n$, we find a finite covering $V_{j}$ of $A$, which refines the first covering but has intersection multiplicity less than $(n+2)$. Considering each $V_{j}$ as a subchart of the corresponding chart $U_{i}$ we see that

[TABLE]

Consider $r_{0}>0$ such that for any $x\in A$ the ball $B(x,{4{\cdot}r_{0}})$ is contained in one of the sets $V_{j}$. Denote by $A_{j}$ the set of all such $x$. Then $\mathcal{V}_{r}(A)\leq\sum\mathcal{V}_{r}(A_{j})$ and, due to Corollary 6.3, $\mathcal{V}_{r}(A_{j})\leq 2\cdot C\cdot N_{0}(V_{j})$. Combining these inequalities finishes the proof. ∎

7. Alexandrov spaces

7.1. Strained points

Strainers and strainer maps are basic tools for Alexandrov spaces; see also [BGP92], [OS94], [KMS01] and will play an important role in the proof of Theorem 1.7.

Let us list main properties of the subsets of strained points. We fix a natural number $n$. Then for all sufficiently large $A$ and any $0<r,\delta\leq\frac{1}{A^{2}}$ the following properties hold true for all $n$-dimensional Alexandrov spaces $X$ of curvature $\geq-1$:

1. (1)

The set $X_{r,\delta}$ of points in $X$ which have an $Ar$-long $(n,\delta)$-strainer is open in $X$. For $s<r$, we have $X_{r,\delta}\subset X_{s,\delta}$, [BGP92, 9.7]. 2. (2)

Assume a sequence $(X_{i}^{n},x_{i})$ of Alexandrov spaces of curvature $\geq-1$ converges to an $n$-dimensional Alexandrov space $(X,x)$ in the pointed Gromov–Hausdorff topology. If $x\in X_{r,\delta}$ then, for all large $i$, the point $x_{i}$ has an $Ar$-long $(n,\delta)$-strainer in $X_{i}$. 3. (3)

Rescaling $X$ with a constant $\lambda\geq 1$ sends the subset $X_{r,\delta}$ to a subset of $(\lambda X)_{\lambda r,\delta}$ of the rescaled Alexandrov space $\lambda X$. 4. (4)

The union $X_{\delta}:=\cup_{r>0}X_{r,\delta}$ contains the set $X_{reg}$ of all regular points of $X$. The Hausdorff dimension of the set $X\setminus(X_{\delta}\cup\partial X)$ is at most $n-2$ [BGP92, 10.6, 10.6.1, 12.8]. 5. (5)

For any point $x\in X_{r,\delta}$ there are natural distance coordinates $\phi\colon B(x,{3{\cdot}r})\to\mathbb{R}^{n}$ which are $(1+\varepsilon)$-bi-Lipschitz onto an open subset $O\subset\mathbb{R}^{n}$. Here, $\varepsilon\to 0$ as $A\to\infty$, [BGP92, 9.4]. 6. (6)

The chart $\phi$ can be smoothed to satisfy the following property, [OS94, Theorem B]. There exists a continuous Riemannian metric $g$ on $\phi(X_{reg}\cap B(x,{3{\cdot}r}))\subset O$ such that for any curve $\gamma\subset X_{reg}\cap B(x,{3{\cdot}r})$ its length coincides with the length of $\phi(\gamma)$ with respect to the Riemannian metric $g$. 7. (7)

The metric tensor $g$ on a chart $O$ defined above is of bounded variation on $O$, [Per95, 4.2] (see also [AB15]).

The last three statements in the above list together with the density and convexity of the set $X_{reg}$ of regular points imply the following.

Corollary 7.1**.**

In the above notations, the subset $X_{\delta}\subset X$ is an almost Riemannian metric space, once $A$ is sufficiently large.

In fact, the arguments in [Per95, 4.2], provide a slightly more precise version of (7) in the above list:

Lemma 7.2**.**

In the notations above, the constant $A$ can be chosen sufficiently large, so that the following holds true. The derivative measure $[g^{\prime}]$ of the Riemannian tensor $g$ in the canonical distance chart $O$ satisfies $[g^{\prime}](\hat{O})\leq A\cdot r^{n-1}$, where $\hat{O}$ is the image $\phi(B(x,{2{\cdot}r}))\subset O=\phi(B(x,{3{\cdot}r}))$.

Proof.

We only sketch the proof, referring to [Per95] for details. First we fix $r=\frac{1}{A^{2}}$.

The fact that $g$ has bounded variation in the chart $O$ follows in [Per95, Section 4.2], by writing the coordinates of $g$ as a universal smooth map $\Phi(f_{1},\dots,f_{\alpha})$ of a finite number of distance functions $f_{j}$ on $X$ and their partial derivatives, both expressed in the chart $\phi$. It is shown in [Per95, Section 3], that any such distance function $f_{j}$ is expressed in the chart $O$ as a difference of two $L$-Lipschitz and $\lambda$-concave functions, where $L,\lambda$ depends only on the semi-concavity of the corresponding distance functions in $X$. Since we have fixed $r>0$, these numbers $\lambda,L$ can be chosen independently of $X$. Thus, $f_{j}$ can be written in the chart $O$ as the difference of two convex functions with universal Lipschitz constants $L^{\prime}$. Therefore, for any unit vector $v\in\mathbb{R}^{n}$, we have a uniform bound on the total mass of the Radon measure $[\frac{\partial^{2}f_{j}}{\partial^{2}v}](\hat{O})$. This implies that all partial second derivatives of $f$ have uniformly bounded mass on $\hat{O}$; see also [EG15, Theorem 6.8].

From this we deduce a uniform bound $A^{\prime}$ on the total mass $[g^{\prime}](\hat{O})$, for the fixed value of $r_{0}=\frac{1}{A^{2}}$.

For any $r<r_{0}$ we rescale the space by $\frac{r_{0}}{r}$. The total mass of the Riemannian tensor $g$ is then rescaled by $(\frac{r_{0}}{r})^{n-1}$. Thus,

[TABLE]

We finish the proof by replacing $A$ by $\max(A,A^{\prime}r_{0}^{1-n})$. ∎

Now we use Corollary 6.3 to conclude:

PROPOSITION 7.3**.**

Let $C=C(n),A=A(n)$ be the constants from Proposition 6.1 and Lemma 7.2. For any point $x\in X_{r,\delta}$, any $s<r$ and any Borel subset $K\subset B(x,r)$ the deviation measure $\mathcal{V}_{s}$ satisfies $\mathcal{V}_{s}(K)\leq 2\cdot C\cdot A\cdot r^{n-1}.$

7.2. Decomposition in good balls

Let the constant $A$ be as above. A ball $B(x,r)$ in $X^{n}$ will be called good if $x\in X_{r,\delta}$. A ball $B(x,r)$ in $X$ will be called bad if it is not good.

In this subsection we give a controlled covering result; see also Problem 8.10.

PROPOSITION 7.4**.**

Let $X^{n}$ be an $n$-dimensional Alexandrov space without boundary. For every compact $W\subset X$, every $\alpha>n-2$ there exists a positive number $q=q(W,\alpha)>0$ with the following property. For every $x\in W$ and every $s<1$ there exists a countable collection of good balls $B_{m}=B(x_{m},{r_{m}})\subset X$ such that

1. (1)

$r_{m}<s$* for all $m$.* 2. (2)

\mathcal{H}^{n}\big{(}B(x,s)\setminus(\cup_{m}B_{m})\big{)}=0. 3. (3)

$\sum_{m}r_{m}^{\alpha}<q\cdot s^{\alpha}$.

The proof will be obtained by a recursive application of the following lemma.

Lemma 7.5**.**

There is an integer $N=N(W,\alpha)$ with the following property. For any $p\in W$ and $\rho<1$ the ball $B(p,{\rho})$ can be covered by at most $N$ balls $B_{i}=B(x_{i},{r_{i}})$ such that $r_{i}<\rho$, for all $i$, and

[TABLE]

where $i\in\mathop{\rm BAD}\nolimits$ means that $B_{i}$ is a bad ball.

Proof. Assume the contrary. Thus we can find a sequence of balls $K_{l}=B(p_{l},\rho_{l})$ such that $p_{l}\in W$, $\rho_{l}<1$ and one needs at least $l$ balls to cover $K_{l}$, so that the conditions in the lemma are fulfilled.

Taking a subsequence we may assume that the following limit exists in the pointed Gromov–Hausdorff metric.

[TABLE]

Since the points $p_{m}$ range over a compact subset of $X$ and $\rho_{m}<1$, the sequence is non-collapsing, i.e. $Y$ is an $n$-dimensional Alexandrov space. By Perelman’s stability theorem, $\partial Y$ is empty. Therefore, $S:=(Y\setminus Y_{\delta})\cap\bar{B}(p,2)$ is a compact set of Hausdorff dimension $\leq n-2$.

By the definition of Hausdroff dimension, we can cover $S$ by a finite number of balls $B_{i}=B(x_{i},{r_{i}})$ such that

[TABLE]

Any point in the remaining compact set $K\backslash(\cup_{i}B_{i})$ is contained in $Y_{\delta}$. Therefore a small ball centered at any point of this set is good. By compactness, we can cover $K\backslash(\cup_{i}B_{i})$ by a finite number of good balls. Let $N$ be the total number of balls in the obtained covering of $K$.

Lifting the constructed covering to $K_{l}$, for all large $l$, we cover the ball $K_{l}$ by at most $N$ balls satisfying the conditions of the lemma. This contradiction to our assumption finishes the proof of the lemma. ∎

Proof of Proposition 7.4.

Cover $B(x,{s})$ by $N$ balls as in Lemma 7.5 and call this covering $\mathcal{F}_{1}$. Now cover every bad ball from the covering $\mathcal{F}_{1}$ by at most $N$ balls provided by Lemma 7.5. Together with the good balls from $\mathcal{F}_{1}$ the new balls define a covering $\mathcal{F}_{2}$ of $B(x,s)$. Proceeding in this way define for each natural number $k$ a covering $\mathcal{F}_{k}$ of $B(x,s)$.

Denote by $g_{l}^{+}$ and $g_{l}^{-}$ the sum of $r_{i}^{\alpha}$ over good, respectively bad balls $B(x_{i},{r_{i}})$ in the covering $\mathcal{F}_{l}$. Then, by construction, $g_{l+1}^{-}<\frac{1}{2}g_{l}^{-}$ and $g_{l+1}^{+}\leq g_{l}^{+}+N\cdot g_{l}^{-}$. Therefore, $g_{l}^{-}\leq 2^{-l}\cdot g_{1}^{-}$ and $g_{l}^{+}$ is uniformly bounded form above. The volume of the union of bad balls in $\mathcal{F}_{l}$ is at most $g_{l}^{-}$ and converges to [math] as $l$ goes to $\infty$.

Let $\mathcal{F}$ be the set of all good balls $B_{j}=B(x_{j},{r_{j}})$ from all the coverings $\mathcal{F}_{l}$. Then $\mathcal{H}^{n}(B(x,s)\setminus(\cup_{\mathcal{F}}B_{j})\leq\lim_{l\to\infty}g_{l}^{-}=0$. On the other hand, by construction,

[TABLE]

Setting $q=3\cdot N$ finishes the proof. ∎

7.3. Final step

Now we can provide the

Proof of Theorem 1.7.

Let $X$ be a fixed $n$-dimensional Alexandrov space. By the inequality of Bishop–Gromov, the deviations measures $\mathcal{V}_{r}$ are uniformly bounded from below by a quadratic term in $r$. Thus in order to control the mm-boundary we only need to bound $\mathcal{V}_{r}$ from above on balls in $X$.

Let the constants $A,C$ be chosen as above, so that Proposition 7.3 can be applied.

Let us first assume that $\partial X$ is empty. Let $W\subset X$ be an arbitrary compact subset. Fix $\alpha=n-\tfrac{3}{2}$ and choose the constant $q$ as in Proposition 7.4. For any $x\in W$ and $s<1$ consider the good balls $B_{i}=B(x_{i},{r_{i}})$ provided by Proposition 7.4 and set $K^{\prime}=\cup_{i}B_{i}$. Let $r<\frac{1}{A^{2}}$ be sufficiently small. Since $\mathcal{H}^{n}(K\setminus K^{\prime})=0$ we have $\mathcal{V}_{r}(K)=\mathcal{V}_{r}(K^{\prime})$.

For all $m$ with $r<r_{m}$, we apply Proposition 7.3 and infer

[TABLE]

On the other hand, for $r_{m}<r$, we have

[TABLE]

Summing up and using $r_{m}^{n-1}<r_{m}^{\alpha}$ we obtain

[TABLE]

This proves that $X$ has locally finite mm-boundary. As already mentioned and used above, any signed Radon measure $\nu$ obtained as a limit of a sequence $\mathcal{V}_{r_{j}}/r_{j}$ for some $r_{j}\to 0$ must be non-negative, hence a Radon measure. We fix such $\nu$.

Inequality (7.1) implies that $\nu$ has finite $\alpha$-dimensional density at every point of $X$, in particular, $\nu$ vanishes on subsets of Hausdorff dimension $\leq n-2$. Thus $\nu(X\setminus X_{\delta})=0$.

Recall that $X_{\delta}$ is an almost Riemannian space. Denote, by $\mathcal{N}$ its minimal metric derivative measure. We extend it to a measure on all of $X$ (still denoted by $\mathcal{N}$) by setting it to be [math] on $X\setminus X_{\delta}$. By Lemma 6.4 the Radon measure $\nu$ is absolutely continuous with respect to $\mathcal{N}$ on compact subsets of $X_{\delta}$. Now (1) and (3) of Theorem 1.7 follow from Lemma 6.2. This finishes the proof in the case $\partial X=\varnothing$.

Assume now that $\partial X\neq\varnothing$ and consider the doubling $Y=X\sqcup_{\partial X}X$ of $X$. Consider $X$ as a convex subset of $Y$ and let $K\subset X$ be compact. We find a constant $L>0$ such that for all sufficiently small $r>0$, we have $\mathcal{H}^{n}(K\cap B(r,{2{\cdot})}(\partial X))\leq L\cdot r$. (This follows, for example, by the coarea formula using the Lipschitz properties of the gradient flow of the distance function $d(\cdot,\partial X)$ which is semiconcave.) On the other hand, for $x\in K\setminus B(\partial X,r)$, the volumes of the $r$-ball in $X$ and in $Y$ coincide. Using that $Y$ has locally finite mm-boundary, we deduce that $\mathcal{V}_{r}(K)$ (computed in the space $X$) is bounded from above $L\cdot r+\tilde{\mathcal{V}}_{r}(K)$, where $\tilde{\mathcal{V}}_{r}(K)$ is the deviation measure of $K$ considered as a subset of $Y$. This implies that $\mathcal{V}_{r}/r$ is uniformly bounded for $r\to 0$. Thus $X$ has locally finite mm-boundary as well.

Any limit of a sequence $\mathcal{V}_{r_{j}}/r_{j}$ for some $r_{j}\to 0$ must be again non-negative, hence a Radon measure. Outside of $\partial X$ $\nu$ coincides with the restriction of the corresponding measure defined on $Y$. From the corresponding statement about $Y$ we deduce that $\nu$ is absolutely singular with respect to $\mathcal{H}^{n}$. Moreover, $\nu$ vanishes on subsets $S\subset X\setminus\partial X$ with finite $\mathcal{H}^{n-1}(S)$.

It remains to prove (2), i.e. to show that the restriction of $\nu$ onto $\partial X$ is at least $c\cdot\mathcal{H}^{n-1}$ for universal constant $c=c(n)$. This statement is local on $\partial X$ and needs to be verified only in small neighborhoods of points $x$ whose tangent $T_{x}X$ are isometric to flat halfspaces.

We fix such a point $x\in\partial X$. We further fix a sufficiently small $\varepsilon>0$ and find a small neighborhood $U$ of $x$ in $X$ which is $(1+\varepsilon)$-bi-Lipschitz to a half-ball in the Euclidean space. Choose an arbitrary $s>0$ such that $B(x,{2{\cdot}s})\subset U$. Let $K=\bar{B}(x,s)\cap\partial X$ be the closed ball of radius $s$ in $\partial X$ with respect to the ambient metric. Due to [EG15, Section 1.6], it is sufficient to prove that $\nu(K)\geq c_{0}\cdot s^{n-1}$ for a universal constant $c_{0}$ depending only on the dimension.

In order to prove this inequality, we consider any open neighborhood $V$ of $K$ in $X$. For all small $r>0$, the neighborhood $V$ contains $B(K,{2{\cdot}r})$. Once $\varepsilon$ has been chosen sufficiently small, the ball $B(z,r)$ in $X$ has volume at most $(1-k_{1})\cdot\omega_{n}\cdot r^{n}$, for any point $z\in B(K,{\frac{1}{10}{\cdot}r})$. Here $k_{1}=k_{1}(n)>0$ is a universal constant. Moreover, the set $B(K,{\frac{1}{10}{\cdot}r})$ has volume at least $\frac{1}{20}\cdot r\cdot\omega_{n-1}\cdot s^{n-1}$. Integrating over $V$ (and using the inequality of Bishop–Gromov on the complement of $B(K,{\frac{1}{10}{\cdot}r})$) we deduce:

[TABLE]

for some $k_{3}$ depending only on the volume of $V$ and independent of $r$. Dividing by $r$ and letting it go to [math] we obtain $\nu(V)\geq k_{4}\cdot s^{n-1}$, for a universal constant $k_{4}>0$. Since the neighborhood $V$ of $K$ was arbitrary, we infer the same inequality for $K$ instead of $V$, finishing the proof. ∎

8. Questions and Comments

8.1. Manifolds

The notions of mm-boundary and mm-curvature are very easy to define but difficult to control. For instance, the examples mentioned in the introduction require some amount of computations and estimates. On the other hand, interesting examples seem to be difficult to construct as well. The first question in this direction is:

PROBLEM 8.1**.**

Construct a closed manifold with a continuous Riemannian metric that does not have finite mm-boundary.

The following problem is motivated by our approach to Theorem 1.7 in Sections 6, 7.

PROBLEM 8.2**.**

Let $X$ be an almost Riemannian space. Can the minimal metric derivative measure be non-zero?

In the language of $DC$-calculus as discussed in [AB15], this question can be reformulated as follows. Given a compact subset $K$ on any $DC_{0}$-Riemannian manifold and any $\varepsilon>0$, can one cover $K$ by charts such that the total mass of the derivative of the metric tensor in these coordinates is bounded by $\varepsilon$? Note that the minimal metric derivative measure must vanish if the metric can be locally defined by a Riemannian tensor of class $W^{1,1}$, since the metric derivative measure is absolutely singular with respect to the Hausdorff measure by Lemma 6.2.

The following question is motivated by Lemma 6.4 and potential applications to geodesic flows of spaces with curvature bounded from above; see also Problem 8.12.

PROBLEM 8.3**.**

Let $X$ be an almost Riemannian space. Can one use the minimal metric derivative measure in order to control the deviation measures $\mathcal{V}_{r}$ from below?

8.2. Surfaces and hypersurfaces

The answer to the following question is not trivial in view of Example 1.14.

PROBLEM 8.4**.**

Can one express the mm-curvature of an Alexandrov surface in terms of its curvature measure?

In view of Theorem 1.10 it is reasonable to expect an affirmative answer to the following question

PROBLEM 8.5**.**

Do convex hypersurfaces of $\mathbb{R}^{n}$ have locally finite mm-curvature?

A natural approach to this question is related to the following conjectural generalization of Bonk–Lang theorem [BL03]:

PROBLEM 8.6**.**

Let $X$ be a convex hypersurface sufficiently close to a flat hyperplane. Can we bound the optimal bi-Lipschitz constant for maps into the Euclidean space in terms of the total scalar curvature?

Some natural generalizations of our Theorem 5.3 are possible. Probably, slightly refined arguments can be used to prove that any DC-submanifold of a Euclidean space has vanishing mm-boundary. Using the embedding theorem of Nash, this would also provide an easy generalization of Theorem 5.3 and Theorem 1.1 to convex hypersurfaces of smooth Riemannian manifolds.

8.3. Alexandrov geometry and beyond

As the next generalization of Theorem 1.1, one should study the case of smoothable Alexandrov spaces.

PROBLEM 8.7**.**

Does the mm-boundary vanish in smoothable Alexandrov spaces? Are there relations to scalar curvature measures defined in [LP17]?

Due to the observation after Problem 8.2, the vanishing of mm-boundary would follow from the existence of slightly smoother coordinates than the ones provided by Perelman’s DC-structure.

PROBLEM 8.8**.**

Let $X$ be an Alexandrov space. Can one introduce coordinates on a neighborhood of the set of regular points, such that the metric is locally given by a Riemannian tensor of class $W^{1,1}$?

In the two-dimensional case, the answer to this question is “yes” by the work of Reshetnyak [Res93], see also [AB16].

Due to Theorem 1.6, an affirmative answer to the following question should be expected. A partial answer to it has been announced by Jerome Bertrand.

PROBLEM 8.9**.**

Are there further connections between the size of the mm-boundary of an Alexandrov space $X$, the existence of the geodesic flow and the “average size” of the cut loci of points in $X$?

Should one have a chance to go beyond mm-boundary and towards mm-curvature, one would definitely need to improve the decomposition statement Proposition 7.4, which provides a geometric control of the size of the set of singular points of an Alexandrov space.

PROBLEM 8.10**.**

Can one replace $\alpha>n-2$ by $\alpha=n-2$ in the statement of Proposition 7.4?

An affirmative answer has been announced by Aaron Naber.

It is interesting to understand if our results provide a quantitative version of bi-Lipschitz closeness of small balls to Euclidean balls. It is known [BGP92] that there exists $\kappa(n,\delta)\to 0$ as $\delta\to 0$ such that if $X=X^{n}$ is an Alexandrov space of curvature $\geq-1$, $x\in X$ such that $\omega_{n}{\cdot}r^{n}-\mathcal{H}^{n}(B(x,r))\leq\delta\cdot r^{n}$ then $B(x,{\frac{r}{4}})$ is $(1+\kappa(n,\delta))$-bi-Lipschitz to a Euclidean ball.

PROBLEM 8.11**.**

*Can $\kappa(n,\delta)$ above be chosen of the form $C(n)\cdot\delta$? *

It is natural to look at what happens for curvature bounded above:

PROBLEM 8.12**.**

Can one estimate and use the mm-boundary in geodesically complete spaces with upper curvature bounds to study the geodesic flow?

Finally, it seems reasonable to expect some generalizations to spaces with Ricci curvature bounds, for instance:

PROBLEM 8.13**.**

Can one control the mm-boundary of noncollapsed limits of Riemannian manifolds with Ricci curvature bounded below? Can one expect something like a geodesic flow in this setting?

From the work of Jeff Cheeger and Aaron Naber [CN15] it should follow that on any non-collapsed limit of manifolds with both-sided Ricci curvature bounds, the mm-curvature is locally finite and mm-boundary is zero. Vanishing of the mm-boundary should then imply that the geodesic flow is defined almost everywhere and preserves the Liouville measure by the same argument as in the proof of Theorem 1.6.