Optimal bounds for ancient caloric functions

TL;DR

This paper establishes bounds on the dimension of ancient caloric functions with polynomial growth on manifolds, linking it to harmonic functions and confirming sharp bounds under Yau's conjecture.

Contribution

It provides the first bounds for ancient caloric functions on manifolds with polynomial volume growth, connecting them to harmonic functions and Yau's conjecture.

Findings

Bound on the dimension of ancient caloric functions proportional to polynomial growth degree

Sharp bounds achieved under Yau's 1974 conjecture

Extension of harmonic function results to caloric functions

Abstract

For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions on any space where Yau's 1974 conjecture about polynomial growth harmonic functions holds.

Equations

$$\underset{B_R(p)}{\sup }|u|\le C_u(1+R{)}^d\text{ for all }R.$$

$$\underset{B_R(p)\times [-R^2,0]}{\sup }|u|\le C_u{\left(1+R\right)}^d\text{ for all R}.$$

$$\dim {\mathcal{P}}_{2k}(M)\le \sum _{i=0}^k\dim {\mathcal{H}}_{2i}(M).$$

$$\dim {\mathcal{P}}_{2k}(M)\le (k+1)\dim {\mathcal{H}}_{2k}(M).$$

$$\dim {\mathcal{P}}_d(M)\le C{d}^n.$$

$$c^{-1}{d}^n\le \dim {\mathcal{P}}_d({\mathbb{R}}^n)\le c{d}^n.$$

$$\dim {\mathcal{H}}_d(M)\le C_n{\text{V}}_M{d}^{n-1}+o(d^{n-1}).$$

$$r^2{\int }_{B_{\frac{r}{10}}\times [-\frac{r^2}{100},0]}|\nabla u{|}^2+r^4{\int }_{B_{\frac{r}{10}}\times [-\frac{r^2}{100},0]}{u}_t^2$$

$${\partial }_t\int u^2{\psi }^2$$

$$\le -\int |\nabla u{|}^2{\psi }^2+4\int u^2|\nabla \psi {|}^2.$$

$${\int }_{t=0}{u}^2{\psi }^2-{\int }_{t=-R^2}{u}^2{\psi }^2$$

$${\int }_{-R^2}^0\int |\nabla u{|}^2{\psi }^{2}dt\le {\int }_{t=-R^2}{u}^2{\psi }^2+4{\int }_{-R^2}^0\int u^2|\nabla \psi {|}^{2}dt.$$

$${\int }_{Q_{R/2}}|\nabla u{|}^2\le {\int }_{B_R\times \{t=-R^2\}}{u}^2+\frac{16}{R^2}{\int }_{Q_R}{u}^2.$$

$${\partial }_t\int |\nabla u{|}^2{\psi }^2$$

$$\le -\int u_t^2{\psi }^2+4\int |\nabla u{|}^2|\nabla \psi {|}^2.$$

$${\int }_{t=0}|\nabla u{|}^2{\psi }^2-{\int }_{t=-R^2}|\nabla u{|}^2{\psi }^2\le {\int }_{-R^2}^0\left(-\int u_t^2{\psi }^2+4\int |\nabla u{|}^2|\nabla \psi {|}^2\right)dt.$$

$${\int }_{Q_{R/2}}{u}_t^2\le \frac{16}{R^2}{\int }_{Q_R}|\nabla u{|}^2+{\int }_{B_R\times \{t=-R^2\}}|\nabla u{|}^2.$$

$${\int }_{B_r\times \{t=-r_1^2\}}{u}^2\le \frac{25}{9r^2}{\int }_{-r^2}^0\left({\int }_{B_r}{u}^2\right)dt=\frac{25}{9r^2}{\int }_{Q_r}{u}^2.$$

$${\int }_{Q_{\frac{2r}{5}}}|\nabla u{|}^2\le {\int }_{Q_{\frac{r_1}{2}}}|\nabla u{|}^2\le {\int }_{B_{r_1}\times \{t=-r_1^2\}}{u}^2+\frac{16}{r_1^2}{\int }_{Q_{r_1}}{u}^2\le \frac{20}{r_1^2}{\int }_{Q_r}{u}^2.$$

$${\int }_{B_{\frac{2r}{5}}\times \{t=-{\rho }^2\}}|\nabla u{|}^2\le \frac{25}{3r^2}{\int }_{-\frac{4r^2}{25}}^0\left({\int }_{B_{\frac{2r}{5}}}|\nabla u{|}^2\right)dt=\frac{25}{3r^2}{\int }_{Q_{\frac{2r}{5}}}|\nabla u{|}^2\le \frac{c}{r^4}{\int }_{Q_r}{u}^2.$$

$${\int }_{Q_{\rho /2}}{u}_t^2$$

$${\int }_{Q_{\frac{r}{10^k}}}{\left|{\partial }_t^{k}u\right|}^2\le \frac{c_k}{r^{4k}}{\int }_{Q_r}{u}^2\le \frac{c_k}{r^{4k}}{r}^2\text{Vol}(B_r)\underset{Q_r}{\sup }{u}^2\le C{c}_k{r}^{2-4k}(1+r{)}^{2d+d_V}.$$

$$u=p_0+t{p}_1+\cdots +t^{m-1}{p}_{m-1},$$

$$(1,t_i,t_i^2,\dots ,t_i^k)$$

$$a_0+a_{1}t+\cdots +a_k{t}^k,$$

$$e_j=\sum _i{b}_i^j(1,t_i,t_i^2,\dots ,t_i^k).$$

$$p_j(x)=\sum _i{b}_i^{j}u(x,t_i).$$

$$\Delta p_j=(j+1)p_{j+1}.$$

$$\dim {\mathcal{P}}_{2k}(M)\le \dim {\mathcal{K}}_0+\dim {\mathcal{H}}_{2k}(M).$$

$$\dim {\mathcal{K}}_0\le \dim {\mathcal{K}}_1+\dim {\mathcal{H}}_{2k}(M).$$

$$|p_j(x)|\le C_j\left(1+|x{|}^{2(k-j)}\right).$$

Full text

The authors were partially supported by NSF Grants DMS 1812142 and DMS 1707270.

0. Introduction

Given a complete manifold $M$ and a constant $d$, ${\mathcal{H}}_{d}(M)$ is the linear space of harmonic functions of polynomial growth at most $d$. Namely, $u\in{\mathcal{H}}_{d}(M)$ if $\Delta u=0$ and for some $p\in M$ and a constant $C_{u}$ depending on $u$

[TABLE]

In 1974, S.T. Yau conjectured that ${\mathcal{H}}_{d}(M)$ is finite dimensional for each $d$ when ${\text{Ric}}_{M}\geq 0$. The conjecture was settled in [CM2]; see [CM1]–[CM5] for more results.111For Yau’s 1974 conjecture see: page $117$ in [Ya2], problem $48$ in [Ya3], Conjecture $2.5$ in [Sc], [Ka], [Kz], [DF], Conjecture $1$ in [Li1], and problem (1) in [LiTa], amongst others. In fact, [CM2]–[CM4] proved finite dimensionality under much weaker assumptions of:

1. (1)

A volume doubling bound. 2. (2)

A scale-invariant Poincaré inequality or meanvalue inequality.

The natural parabolic generalization is a polynomial growth ancient solution of the heat equation. A solution of the heat equation is often called a caloric function. Ancient solutions are ones that are defined for all negative $t$ - these are the solutions that arise in a blow up analysis. Given $d>0$, $u\in{\mathcal{P}}_{d}(M)$ if $u$ is ancient, $\partial_{t}u=\Delta\,u$ and for some $p\in M$ and a constant $C_{u}$

[TABLE]

On ${\mathbb{R}}^{n}$, ${\mathcal{P}}_{d}$ is the classical space of caloric polynomials that generalize the Hermite polynomials; see [N], [E1], [E2]. More generally, the spaces ${\mathcal{P}}_{d}(M)$ play a fundamental role in geometric flows, see [CM6]–[CM8]. They were studied by Calle in her 2006 thesis, [Ca1], [Ca2], in the context of mean curvature flow.

A manifold has polynomial volume growth if there are constants $C$ and $d_{V}$ so that ${\text{Vol}}(B_{R}(p))\leq C\,(1+R)^{d_{V}}$ for some $p\in M$, all $R>0$.222A volume doubling space with doubling constant $C_{D}$ has polynomial volume growth of degree ${\log_{2}C_{D}}$. Our main result is the following sharp inequality:

Theorem \the\fnum.

If $M$ has polynomial volume growth and $k$ is a nonnegative integer, then

[TABLE]

The inequality (0.3) is an equality on ${\mathbb{R}}^{n}$ (see Corollary 2.1 below). Since ${\mathcal{H}}_{d_{1}}\subset{\mathcal{H}}_{d_{2}}$ for $d_{1}\leq d_{2}$, Theorem ‣ 0. Introduction implies:

Corollary \the\fnum.

If $M$ has polynomial volume growth, then for all $k\geq 1$

[TABLE]

Combining this with the bound $\dim{\mathcal{H}}_{d}(M)\leq C\,d^{n-1}$ when ${\text{Ric}}_{M^{n}}\geq 0$ from [CM3] gives:

Corollary \the\fnum.

There exists $C=C(n)$ so that if ${\text{Ric}}_{M^{n}}\geq 0$, then for $d\geq 1$

[TABLE]

The exponent $n$ in (0.5) is sharp: There is a constant $c$ depending on $n$ so that for $d\geq 1$

[TABLE]

Recently, Lin and Zhang, [LZ], proved very interesting related results, adapting the methods of [CM2]–[CM4] to get the bound $d^{n+1}$.

Using parabolic gradient estimates of Li-Yau, [LiY], and Souplet-Zhang, [SoZ], one can show that if $d<2$ and ${\text{Ric}}\geq 0$, then ${\mathcal{P}}_{d}(M)={\mathcal{H}}_{d}(M)$ consists only of harmonic functions of polynomial growth. In particular, ${\mathcal{P}}_{d}(M)=\{{\text{Constant functions}}\}$ for $d<1$ and, moreover, $\dim{\mathcal{P}}_{1}(M)\leq n+1$, by Li and Tam, [LiTa], with equality if and only if $M={\mathbb{R}}^{n}$ by [ChCM].

The exponent $n-1$ is also sharp in the bound for $\dim{\mathcal{H}}_{d}$ when ${\text{Ric}}_{M^{n}}\geq 0$. However, as in Weyl’s asymptotic formula, the coefficient of $d^{n-1}$ can be related to the volume, [CM3]:

[TABLE]

• •

${\text{V}}_{M}$ is the “asymptotic volume ratio” $\lim_{r\to\infty}\,{\text{Vol}}(B_{r})/r^{n}$.

• •

$o(d^{n-1})$ is a function of $d$ with $\lim_{d\to\infty}\,o(d^{n-1})/d^{n-1}=0$.

Combining (0.7) with Corollary ‣ 0. Introduction gives $\dim\,{\mathcal{P}}_{d}(M)\leq C_{n}\,{\text{V}}_{M}\,d^{n}+o(d^{n})$ when ${\text{Ric}}_{M^{n}}\geq 0$.

An interesting feature of these dimension estimates is that they follow from “rough” properties of $M$ and are therefore surprisingly stable under perturbation. For instance, [CM4] proves finite dimensionality of ${\mathcal{H}}_{d}$ for manifolds with a volume doubling and a Poincaré inequality, so we also get finite dimensionality for ${\mathcal{P}}_{d}$ on these spaces. Unlike a Ricci curvature bound, these properties are stable under bi–Lipschitz transformations (cf. [MS]). Moreover, these properties make sense also for discrete spaces, vastly extending the theory and methods out of the continuous world. Recently Kleiner, [K], (see also Shalom-Tao, [ST], [T1], [T2]) used, in part, this in his new proof of an important and foundational result in geometric group theory, originally due to Gromov, [G]. We expect that the proof of Theorem ‣ 0. Introduction extends to many discrete spaces, allowing a wide range of applications.

1. Ancient solutions of the heat equation

The next lemma gives a reverse Poincaré inequality for the heat equation (cf. [M]).

Lemma \the\fnum.

There is a universal constant $c$ so that if $u_{t}=\Delta\,u$, then

[TABLE]

Proof.

Let $Q_{R}$ denote $B_{R}\times[-R^{2},0]$ and $\psi$ be a cutoff function on $M$. Since $u_{t}=\Delta\,u$, integration by parts and the absorbing inequality $4ab\leq a^{2}+4b^{2}$ give

[TABLE]

Integrating this in time from $-R^{2}$ to [math] gives

[TABLE]

In particular, we get

[TABLE]

Let $|\psi|\leq 1$ be one on $B_{R/2}$, have support in $B_{R}$, and satisfy $|\nabla\psi|\leq 2/R$, so we get

[TABLE]

Next, we argue similarly to get a bound on $u_{t}^{2}$. Namely, differentiating, then integrating by parts and using that $u_{t}=\Delta\,u$ gives

[TABLE]

Integrating (1) in time from $-R^{2}$ to [math] gives

[TABLE]

Letting $\psi$ be as above, we conclude that

[TABLE]

Next, choose some $r_{1}\in[4r/5,r]$ with

[TABLE]

Applying (1.5) with $R=r_{1}$ and using the bound (1.9) at $r_{1}$ gives

[TABLE]

For simplicity, $c$ is a constant independent of everything that can change from line to line. It follows from (1.10) that there must exist some $\rho\in[r/5,2r/5]$ so that

[TABLE]

Now applying (1.8) with $R=\rho$ and using (1.10) and (1.11) gives

[TABLE]

∎

Corollary \the\fnum.

If ${\text{Vol}}(B_{R})\leq C\,(1+R)^{d_{V}}$ and $u\in{\mathcal{P}}_{d}(M)$, then $\partial_{t}^{k}u\equiv 0$ for $4k>2d+d_{V}+2$.

Proof.

Since the metric on $M$ is constant in time, $\partial_{t}-\Delta$ commutes with $\partial_{t}$ and, thus, $(\partial_{t}-\Delta)\partial^{j}_{t}u=0$ for every $j$. Let $Q_{R}$ denote $B_{R}\times[-R^{2},0]$. Applying Lemma 1 to $u$ on $Q_{r}$ for some $r$, then to $u_{t}$ on $Q_{\frac{r}{10}}$, etc., we get a constant $c_{k}$ depending just on $k$ so that

[TABLE]

Since $4k>2d+d_{V}+2$, the right-hand side goes to zero as $r\to\infty$, giving the corollary. ∎

We will prove Corollary ‣ 0. Introduction next, though it will eventually be a corollary of Theorem ‣ 0. Introduction.

Proof of Corollary ‣ 0. Introduction.

Choose an integer $m$ with $4m>2k+d_{V}+2$. Corollary 1 gives that $\partial_{t}^{m}u=0$ for any $u\in{\mathcal{P}}_{2k}(M)$. Thus, any $u\in{\mathcal{P}}_{2k}(M)$ can be written as

[TABLE]

where each $p_{j}$ is a function on $M$. Moreover, using the growth bound $u\in{\mathcal{P}}_{2k}(M)$ for $t$ large and $x$ fixed, we see that $p_{j}\equiv 0$ for any $j>k$. (See theorem $1.2$ in [LZ] for a similar decomposition under more restrictive hypotheses and [KoT] for a splitting result for ancient positive solutions on homogeneous spaces.)

We will show next that the functions $p_{j}$ grow at most polynomially of degree $d$. Fix distinct values $-1<t_{1}<t_{2}<\dots<t_{k}<t_{k+1}=0$. We claim that the $k+1$-vectors

[TABLE]

are linearly independent in ${\mathbb{R}}^{k+1}$ for $i=1,\dots,k+1$. If this was not the case, then there would be some (non-trivial) $(a_{0},\dots,a_{k})\in{\mathbb{R}}^{k+1}$ that is orthogonal to all of them. But this means that there would be $k+1$ distinct roots to the degree $k$ polynomial

[TABLE]

which is impossible, and the claim follows. Let $e_{j}\in{\mathbb{R}}^{k+1}$ be the standard unit vectors. Since the $(1,t_{i},t_{i}^{2},\dots,t_{i}^{k})$’s span ${\mathbb{R}}^{k+1}$, we can choose coefficients $b^{j}_{i}$ so that for each $j$

[TABLE]

It follows from (1.14) and (1.17) that

[TABLE]

Since $u\in{\mathcal{P}}_{2k}(M)$, (1) implies that each $p_{j}$ is a linear combination of functions that grow polynomially of degree at most $2k$ and, thus, $p_{j}$ grows polynomially of degree at most $2k$.

Since $u$ satisfies the heat equation, it follows that $\Delta p_{k}=0$ and

[TABLE]

Thus, we get a linear map $\Psi_{0}:{\mathcal{P}}_{2k}(M)\to{\mathcal{H}}_{2k}(M)$ given by $\Psi_{0}(u)=p_{k}$. Let ${\mathcal{K}}_{0}={\text{Ker}}(\Psi_{0})$. It follows from this that

[TABLE]

If $u\in{\mathcal{K}}_{0}$, then $p_{k}=0$ and $\Delta p_{k-1}=0$, so we get a linear map $\Psi_{1}:{\mathcal{K}}_{0}\to{\mathcal{H}}_{2k}(M)$ given by $\Psi_{1}(u)=p_{k-1}$. Let ${\mathcal{K}}_{1}$ be the kernel of $\Psi_{1}$ on ${\mathcal{K}}_{0}$. It follows as above that

[TABLE]

Repeating this $k+1$ times gives the theorem. ∎

Lemma \the\fnum.

If $u\in{\mathcal{P}}_{2k}(M)$ can be written as $u=p_{0}(x)+t\,p_{1}(x)+\dots+t^{k}\,p_{k}(x)$, then

[TABLE]

Proof.

By assumption, there is a constant $C$ so that

[TABLE]

Following the proof of Corollary ‣ 0. Introduction, fix $-1<t_{1}<t_{2}<\dots<t_{k}<t_{k+1}=-\frac{1}{2}$ and coefficients $b^{j}_{i}$ so that (1.17) holds for each $j$. Observe that (1.17) gives for each $j$

[TABLE]

Thus, given $R>2$ and $x\in B_{R}$, we get that

[TABLE]

From this, we conclude that $\sup_{B_{R}}\,|p_{j}|\leq 3\,\tilde{C}\,R^{2k-2j}$. ∎

Proof.

(of Theorem ‣ 0. Introduction). Following the proof Corollary ‣ 0. Introduction, each $u\in{\mathcal{P}}_{2k}(M)$, can be expanded as $u=p_{0}(x)+t\,p_{1}(x)+\dots+t^{k}\,p_{k}(x)$. By Lemma 1, the linear map $\Psi_{0}:{\mathcal{P}}_{2k}(M)\to{\mathcal{H}}_{2k}(M)$ given by $\Psi_{0}(u)=p_{k}$ actually maps into ${\mathcal{H}}_{0}(M)$ and, thus,

[TABLE]

Similary, Lemma 1 implies that the map $\Psi_{1}$ maps the kernel of $\Psi_{0}$ to ${\mathcal{H}}_{2}(M)$. Applying this repeatedly gives the theorem. ∎

2. Caloric polynomials

It is a classical fact that ${\mathcal{P}}_{d}({\mathbb{R}}^{n})$ consists of caloric polynomials, i.e., polynomials in $x,t$ that satisfy the heat equation ([E1], [E2], [N]). We compute the dimensions of these spaces.

Given a polynomial in $x$ and $t$, define its parabolic degree by considering $t$ to have degree two. Thus, $x_{1}^{m_{1}}x_{2}^{m_{2}}t^{m_{0}}$ has parabolic degree $m_{1}+m_{2}+2m_{0}$. A polynomial in $x,t$ is homogeneous if each monomial has the same parabolic degree. Let $A_{p}^{n}$ denote the homogeneous degree $p$ polynomials on ${\mathbb{R}}^{n}$. The parabolic homogeneous degree $p$ polynomials ${\mathcal{A}}^{n}_{p}$ are

[TABLE]

Lemma \the\fnum.

For each positive integer $p$, we have $\dim\left({\mathcal{P}}_{p}({\mathbb{R}}^{n})\cap{\mathcal{A}}^{n}_{p}\right)=\dim A^{n}_{p}$ and

[TABLE]

Proof.

Observe that $\partial_{t}$ and $\Delta$ map ${\mathcal{A}}^{n}_{p}$ to ${\mathcal{A}}^{n}_{p-2}$. Moreover, given any $u\in{\mathcal{A}}^{n}_{p-2}$, we have

[TABLE]

Therefore, the map $(\partial_{t}-\Delta):{\mathcal{A}}^{n}_{p}\to{\mathcal{A}}^{n}_{p-2}$ is onto. Since the kernel of this map is ${\mathcal{P}}_{p}({\mathbb{R}}^{n})\cap{\mathcal{A}}^{n}_{p}$, we conclude that

[TABLE]

This gives both claims. ∎

Lemma \the\fnum.

If $p\geq n$, then

[TABLE]

Proof.

To get the upper bound, we use that $p\geq n$ to get

[TABLE]

The lower bound follows similarly since $\frac{(p+n-1)!}{p!\,(n-1)!}\geq\frac{p^{n-1}}{(n-1)!}$. ∎

The dimension bounds for ${\mathcal{P}}_{d}({\mathbb{R}}^{n})$ in (0.6) follow by combining Lemmas 2 and 2.

2.1. Harmonic polynomials

For each $j$, the Laplacian gives a linear map $\Delta:A^{n}_{j}\to A^{n}_{j-2}$. The kernel $H^{n}_{j}\subset A^{n}_{j}$ of this map is the linear space of homogeneous harmonic polynomials of degree $j$ on ${\mathbb{R}}^{n}$. The next lemma shows that this map is onto:

Lemma \the\fnum.

For each $d$, the map $\Delta:A^{n}_{d+2}\to A^{n}_{d}$ is onto.

Proof.

Take an arbitrary $u\in A^{n}_{d}$. For each nonnegative $\ell\leq d/2$, define $u_{\ell}$ and $v_{\ell}$ by

[TABLE]

Note that $u_{0}=u$. We will use repeatedly that if $v\in A^{n}_{k}$, then homogeneity gives

[TABLE]

Using this and $\Delta\,|x|^{2}=2n$, we get for each $\ell$ that

[TABLE]

Thus, if we define positive constants $c_{\ell}=(\ell+1)\,\left(2n+4d-4\ell\right)$, then we have that

[TABLE]

Let $k$ be the greatest integer less than or equal to $\frac{d}{2}$. Note that $u_{k+1}=v_{k+1}\equiv 0$. It follows from this and (2.11) that

[TABLE]

is a nonzero multiple of $u_{0}=u$, giving the lemma. ∎

Corollary \the\fnum.

For each positive integer $k$, we have $\dim H^{n}_{k}=\dim A^{n}_{k}-\dim A^{n}_{k-2}$ and

[TABLE]

Proof.

Note that $\Delta:A^{n}_{j}\to A^{n}_{j-2}$ gives a linear map with kernel equal to $H^{n}_{j}$. The map is onto by Lemma 2.1, giving the first claim. Summing the first claim gives (2.13). ∎

Corollary \the\fnum.

For each $k$, (0.3) is an equality on ${\mathbb{R}}^{n}$.

Proof.

Corollary 2.1 and Lemma 2 give

[TABLE]

∎