Isoperimetry and volume preserving stability in real projective spaces

TL;DR

This paper classifies volume-preserving stable hypersurfaces in real projective spaces, confirming a longstanding conjecture and extending previous results, while also deriving a Willmore-type inequality for antipodal invariant hypersurfaces.

Contribution

It provides a complete classification of stable hypersurfaces in real projective spaces and confirms a conjecture from 1988, extending prior work to higher dimensions.

Findings

Solutions are tubular neighborhoods of projective subspaces.

Confirmed Burago and Zalgaller's conjecture from 1988.

Derived a Willmore-type inequality for antipodal invariant hypersurfaces.

Abstract

We classify the volume preserving stable hypersurfaces in the real projective space $\mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $\mathbb{RP}^k\subset \mathbb{RP}^n$ (starting with points). This confirms a conjecture of Burago and Zalgaller from 1988 and extends to higher dimensions previous result of M. Ritor\'{e} and A. Ros on $\mathbb{RP}^3$. We also derive an Willmore type inequality for antipodal invariant hypersurfaces in $\mathbb{S}^n$.

Equations

$$\mathcal{A}=\{x\in {\mathbb{S}}^{n+1}:\sum _{i=1}^r{x}_i^2>a\}$$

$$\displaystyle\int_{\Sigma}\bigg{(}1+H^{2}\bigg{)}^{\frac{n}{2}}d_{\Sigma}\geq\sigma_{n}$$

$$N(x)=\bigg{(}-\sin(r)\,z,\cos(r)\,w\bigg{)}.$$

$$A=\left[\begin{array}{cc}-\frac{\sin (r)}{\cos (r)}{I}_{n_1}& 0\\ 0& \frac{\cos (r)}{\sin (r)}{I}_{n_2}\end{array}\right].$$

$$L=\Delta +n_1+n_2+n_1\frac{{\sin }^2(r)}{{\cos }^2(r)}+n_2\frac{{\cos }^2(r)}{{\sin }^2(r)}.$$

$$\frac{k_1(k_1+n_1-1)}{{\cos }^2(r)}+\frac{k_2(k_2+n_2-1)}{{\sin }^2(r)},$$

$$\sqrt{\frac{n_2}{n_1+2}}\le \tan (r)\le \sqrt{\frac{n_2+2}{n_1}}.$$

$$A^2+\beta A-Id\equiv 0.$$

$$I(f,f)=-{\int }_{\Sigma }fLf{d}_{\Sigma }:={\int }_{\Sigma }|\nabla f{|}^2-(Ri{c}_M(N,N)+|A{|}^2)f^2{d}_{\Sigma }\ge 0.$$

$${\Delta }_{\Sigma }x+nx=nHN\text{and}{\Delta }_{\Sigma }N+|A{|}^{2}N=nHx.$$

$${\Delta }_{\Sigma }⟨N,a⟩$$

$$L(V)=\sum _{i=1}^{n+2}L(⟨V,e_k⟩)e_k,$$

$$\Delta (uv)=u\Delta v+v\Delta u+2⟨\nabla u,\nabla v⟩$$

$$X_1=2a^{\top },X_2=X_3=-2A(a^{\top }),\text{and}{X}_4=2A^2(a^{\top }),$$

$$\int_{\Sigma}\bigg{(}\langle x,a\rangle\langle N,b\rangle-\langle x,b\rangle\langle N,a\rangle\bigg{)}d_{\Sigma}=0.$$

$${\Phi }_{a,b}=-⟨x,a⟩x+⟨N,a⟩N+⟨x,b⟩N.$$

$$\displaystyle L\Phi_{a,b}=-(|A|^{2}-n)\bigg{(}\langle N,a\rangle N+\langle x,a\rangle x\bigg{)}+nH\bigg{(}\langle x,b\rangle x+\langle N,b\rangle N\bigg{)}+X,$$

$$X=2\bigg{(}A^{2}(a^{\top})-A(b^{\top})\,-\,a^{\top}\bigg{)}.$$

$$R(b)={\int }_{\Sigma }⟨x,b⟩N{d}_{\Sigma }.$$

$$I(V,V)=-{\int }_{\Sigma }⟨V,LV⟩d_{\Sigma }=-nH{\int }_{\Sigma }⟨x,b⟩⟨N,b⟩d_{\Sigma }=0.$$

$$nH(⟨x,b⟩x+⟨N,b⟩N)-2A(b^{\top })=c.$$

$$0={\int }_{\Sigma }⟨x,b⟩⟨N,b⟩d_{\Sigma }=\frac{nH}{2}{\int }_{\Sigma }⟨x,b{⟩}^2{d}_{\Sigma }.$$

$$\displaystyle I(\Phi_{a,b},\Phi_{a,b})=\int_{\Sigma}(|A|^{2}-n)\bigg{(}-\langle x,a\rangle^{2}+\langle N,a\rangle^{2}\bigg{)}d_{\Sigma}\quad\quad\quad\quad\quad\quad\quad\quad$$

$$\displaystyle-2nH\int_{\Sigma}\bigg{(}\langle N,a\rangle\langle N,b\rangle-\langle x,a\rangle\langle x,b\rangle+\frac{1}{2}\langle N,b\rangle\langle x,b\rangle\bigg{)}\,d_{\Sigma}.$$

$$L{\Phi }_{a,b}=-(|A{|}^2-n)(⟨x,a⟩x+⟨N,a⟩N)+nH(⟨x,b⟩x+⟨N,b⟩N)+X,$$

$$\displaystyle-\int_{\Sigma}\langle\Phi_{a,b},L\Phi_{a,b}\rangle d_{\Sigma}=\int_{\Sigma}(|A|^{2}-n)\bigg{(}-\langle x,a\rangle^{2}+\langle N,a\rangle^{2}+\langle N,a\rangle\langle x,b\rangle\bigg{)}$$

$$\displaystyle-\int_{\Sigma}nH\bigg{(}-\langle x,b\rangle\langle x,a\rangle+\langle N,b\rangle\langle N,a\rangle+\langle N,b\rangle\langle x,b\rangle\bigg{)}.$$

$${\int }_{\Sigma }(|A{|}^2-n)⟨N,a⟩⟨x,b⟩d_{\Sigma }={\int }_{\Sigma }⟨N,b⟩\Delta ⟨x,a⟩-⟨x,b⟩\Delta ⟨N,a⟩$$

$$\displaystyle\quad-n\,H\int_{\Sigma}\bigg{(}\langle N,a\rangle\langle N,b\rangle-\langle x,a\rangle\langle x,b\rangle\bigg{)}d_{\Sigma}.$$

$${\int }_{\Sigma }(|A{|}^2-n)⟨N,a⟩⟨x,b⟩d_{\Sigma }$$

$$\displaystyle-n\,H\int_{\Sigma}\bigg{(}-\langle x,a\rangle\langle x,b\rangle$$

$$Q_{\varphi}(a,b)=-\int_{\Sigma}\bigg{\langle}\Phi_{a,\varphi(a)},L\Phi_{b,\varphi(b)}\bigg{\rangle}\,d_{\Sigma}.$$

Full text

1. Introduction

The isoperimetric problem looks for the regions that minimize perimeter among compact sets of a given fixed volume. The solutions are called isoperimetric regions and their boundaries isoperimetric hypersurfaces. The isoperimetric inequality beautifully describes geodesic balls as the only solutions of the problem in the Euclidean space $\mathbb{R}^{n+1}$. The classical proof based on symmetrization methods works also in the hyperbolic space $\mathbb{H}^{n+1}$ and in the round sphere $\mathbb{S}^{n+1}$.

Due to its variational nature, existence and regularity are relevant parts of the problem. The works of Almgren [2] and Schoen and Simon [35] (see also Morgan [22]) give a satisfactory analytical description: when $M^{n+1}$ is closed or homogeneous, isoperimetric hypersurfaces exist and are smooth except for a closed set of Hausdorff dimension $n-7$. The regular part is a volume preserving stable constant mean curvature hypersurface. In [3, 4] Barbosa, do Carmo and Eschenburg introduced the notion of being volume preserving stable in the broader class of immersed constant mean curvature hypersurfaces and proved that only geodesic balls are stable in the simply connected space forms.

In [31] Ritoré and Ros studied volume preserving stable surfaces in non-simply connected $3$-dimensional space forms. Exploiting the Hopf quadratic differential of the surfaces, they proved that volume preserving stable tori are flat. Moreover, using the Hersch-Yau trick for constructing meromorphic maps, they also showed that sphere and tori are the only possible topologies for a volume preserving stable surface in the real projective space $\mathbb{RP}^{3}$. As a result, the solutions of the isoperimetric problem are either geodesic spheres or tubes about closed geodesics. Inspired by this result, one can consider the projective spaces $\mathbb{P}^{n}(\mathbb{K})$, over the field $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$, the most appealing spaces one would like the isoperimetric problem solved. The following conjecture, for $\mathbb{K}=\mathbb{R}$, was posed by Burago and Zalgaller in 1988:

Conjecture 1.1** (Berger [5], Burago and Zalgaller [6], and Ros [34]).**

The isoperimetric regions in the projective spaces $\mathbb{P}^{n+1}(\mathbb{K})$ are tubular neighborhoods of projective subspaces $\mathbb{P}^{k}(\mathbb{K})\subset\mathbb{P}^{n+1}(\mathbb{K})$.

Our main result address the real projective space of any dimension:

Theorem 1.2**.**

If $\Sigma$ is a compact two-sided volume preserving stable hypersurface in $\mathbb{RP}^{n+1}$, then $\Sigma$ is either a geodesic sphere, a quotient of a Clifford hypersurface $\mathbb{S}^{n_{1}}(R_{1})\times\mathbb{S}^{n_{2}}(R_{2})\subset\mathbb{S}^{n+1}$, or a two-fold covering of the projective subspace $\mathbb{RP}^{n}$.

Corollary 1.3**.**

The isoperimetric regions in the real projective space $\mathbb{RP}^{n+1}$ are tubular neighborhoods of projective subspaces $\mathbb{RP}^{k}\subset\mathbb{RP}^{n+1}$.

The corresponding result in $\mathbb{S}^{n+1}$ states that among antipodal invariant regions of a fixed volume, the least perimeter is

[TABLE]

for some $r\in\{1,\ldots,n+1\}$ and $a\in\mathbb{R}_{+}$. For a discrete version of this result see [12].

The following is a brief account of results on the isoperimetric problem. Rotationally symmetric surfaces were treated in [16, 30] (see also the survey [15]). The spaces $\mathbb{H}^{2}\times\mathbb{R}$, $\mathbb{S}^{2}\times\mathbb{R}$, $\mathbb{H}^{2}\times\mathbb{S}^{1}$, and $\mathbb{S}^{2}\times\mathbb{S}^{1}$ were studied in [14, 27, 28] via symmetrization methods and ODE stability analysis. The case $\mathbb{S}^{1}\times\mathbb{R}^{n}$ is also treated in [28]: spheres and cylinders are the only solutions when $n\leq 7$ (see also [31] when $n=2$). The problem is also solved outside an explicit compact subset in the moduli space of flat $T^{2}\times\mathbb{R}$, see [13, 32]. The $3$-dimensional spherical space forms with large fundamental group were studied by this author in [38]. Finally, we mention the works [18, 24] on the description of small isoperimetric regions in general manifolds and [9, 11] on the uniqueness of large isoperimetric regions in asymptotic flat manifolds with non-negative scalar curvature and positive mass.

The proof of Theorem 1.2 is inspired by the work of do Carmo, Ritoré and Ros in [10] on the classification of two-sided index one minimal hypersurfaces in the real projective space $\mathbb{RP}^{n}$. The question of isoperimetry in $\mathbb{RP}^{n}$ is also raised in [10]. As in the volume preserving case, index one minimal hypersurfaces also minimize area for a certain class of variations. Despite this similarity, the stability analysis for constant mean curvature brings several algebraic difficulties. Although there are many topologies for the index one minimal hypersurfaces, the norm of their second fundamental forms can only take two possible values. This striking fact manifested sharply in the proof given in [10]. In contrast, the norm of the second fundamental form of volume preserving stable hypersurfaces ranges over all non-negative numbers. A key observation that unite all hypersurfaces in Theorem 1.2 is that their second fundamental form obey an equation of the form $A^{2}+\beta A-Id=0$ for some constant $\beta$. This fact is reflected on the choice of vector fields used in the second variation argument. Finally, we observed that the constant $\beta$ has a variational interpretation which we exploited to balance the vector fields. To handle possible singular isoperimetric hypersurfaces in higher dimensions we used a cut-off approach due to Morgan and Ritoré [23] which shows that singularities are negligible for the second variation argument. The isoperimetric profile is discussed in Section 4: we verified in lower dimensions that the profile is given by the perimeter of successive tubular neighborhoods of projective subspaces as conjectured by Berger [5].

There is a natural comparison between the isoperimetric profile and the Willmore energy of surfaces as showed by Ros [33, 34]. This connection was investigated further by Marques and Neves in their celebrated proof of the Willmore conjecture in [21]. Following these ideas, we add to the literature an Willmore type inequality for antipodal invariant hypersurfaces in $\mathbb{S}^{n}$ which generalizes the main result in [33]:

Theorem 1.4**.**

Let $\Sigma$ be a compact embedded hypersurface separating $\mathbb{S}^{n+1}$ in two connected regions which are both antipodal invariant. Then

[TABLE]

with equality if, and only if, $\Sigma$ is congruent to the Clifford hypersurface C_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}=\mathbb{S}^{\lfloor\frac{n}{2}\rfloor}\bigg{(}\sqrt{\frac{\lfloor\frac{n}{2}\rfloor}{n}}\bigg{)}\times\mathbb{S}^{\lceil\frac{n}{2}\rceil}\bigg{(}\sqrt{\frac{\lceil\frac{n}{2}\rceil}{n}}\bigg{)}.

Remark 1.5**.**

It is a conjecture of Solomon that among non-totally geodesic minimal hypersurfaces in $\mathbb{S}^{n+1}$, the Clifford hypersurface $C_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}$ has the least area, see [17, 20, 25]. Theorem 1.4 confirms this conjecture in the class of minimal hypersurfaces with antipodal symmetry. In particular, $C_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}$ has the least area among the minimal Clifford hypersurfaces C_{p,n-p}=\mathbb{S}^{p}\bigg{(}\sqrt{\frac{p}{n}}\bigg{)}\times\mathbb{S}^{n-p}\bigg{(}\sqrt{\frac{n-p}{n}}\bigg{)} for every $n$. For partial results concerning Solomon’s conjecture, see Ilmanen and White [17].

Some related results: (i) Theorem 1.2 was obtained in [1] under the additional assumption that the length of the second fundamental form is constant, (ii) Ramírez-Luna [29] proved that the width of $\mathbb{RP}^{n}$ is realized by a Clifford hypersurface.

Acknowledgements. I would like to thank the support and hospitality of the Institute for Advanced Study where this work was conducted and Fernando Codá Marques for organizing the special year Variational Methods in Geometry and for his interest in this work.

This material is based upon work supported by the National Science Foundation under grant No. DMS-1638352.

2. Clifford hypersurfaces

The Clifford hypersurface $T_{r}^{(n_{1},n_{2})}$ in $\mathbb{S}^{n+1}$ is defined as the product $\mathbb{S}^{n_{1}}(\cos r)\times\mathbb{S}^{n_{2}}(\sin r)$, where $n_{1}+n_{2}=n$ and $r$ is constant in $(0,\frac{\pi}{2})$. Each point $x$ in $T_{r}^{(n_{1},n_{2})}$ is written as $x=(\cos(r)z,\sin(r)w)$, where $z\in\mathbb{S}^{n_{1}}$ and $w\in\mathbb{S}^{n_{2}}$. In these coordinates, the unit normal vector field $N$ over $T_{r}^{(n_{1},n_{2})}$ is given by

[TABLE]

The second fundamental form of $\mathbb{S}^{n_{1}}(\cos r)\times\mathbb{S}^{n_{2}}(\sin r)$ is given below:

[TABLE]

Here, $I_{n_{1}}$ and $I_{n_{2}}$ denotes the identity maps in $\mathbb{R}^{n_{1}}$ and $\mathbb{R}^{n_{2}}$ respectively. The principal curvatures of $T_{r}^{(n_{1},n_{2})}$ are $-\frac{\sin(r)}{\cos(r)}$ with multiplicity $n_{1}$ and $\frac{\cos(r)}{\sin(r)}$ with multiplicity $n_{2}$. The family $\{T_{r}^{(n_{1},n_{2})}\}_{r\in(0,\frac{\pi}{2})}$ foliates $\mathbb{S}^{n+1}$ by constant mean curvature hypersurfaces starting at the $n_{1}$-dimensional sphere $T_{0}^{(n_{1},n_{2})}=\mathbb{S}^{n_{1}}\times\{0\}$ and ending at the $n_{2}$-dimensional sphere $T_{\frac{\pi}{2}}^{(n_{1},n_{2})}=\{0\}\times\mathbb{S}^{n_{2}}$. Each Clifford hypersurface is antipodal symmetric and, hence, descends naturally to the real projective space $\mathbb{RP}^{n+1}$. The Jacobi operator $L=\Delta+n+|A|^{2}$ on $T_{r}^{(n_{1},n_{2})}$ has the following expression

[TABLE]

We want to study the stability of $T_{r}^{(n_{1},n_{2})}$ when projected in $\mathbb{RP}^{n+1}$. The eigenvalues of the Laplacian $\Delta$ on $\mathbb{S}^{n_{1}}(\cos(r))\times\mathbb{S}^{n_{2}}(\sin(r))$ are

[TABLE]

where $k_{1}$ and $k_{2}$ are non-negative integers. The eigenfunctions are restrictions to $T_{r}^{(n_{1},n_{2})}$ of products of homogeneous harmonic polynomials of degree $k_{1}$ in $\mathbb{R}^{n_{1}+1}$ with homogeneous harmonic polynomials of degree $k_{2}$ in $\mathbb{R}^{n_{2}+1}$. These eigenfunctions are invariant by the antipodal map only when $k_{1}+k_{2}$ is even. The possible values of $k_{1}$ and $k_{2}$ to obtain the smallest positive eigenvalue $\lambda$ are either $(k_{1},k_{2})=(1,1)$ or $(k_{1},k_{2})=(2,0)$ or $(0,2)$. The stability of $T_{r}^{(n_{1},n_{2})}$ is then equivalent to $\lambda-n-|A|^{2}\geq 0$. This holds whenever

[TABLE]

Remark 2.1**.**

To each Clifford hypersurface $\mathbb{S}^{n_{1}}(\cos(r))\times\mathbb{S}^{n_{2}}(\sin(r))$, there exists a number $\beta\in\mathbb{R}$, which depends on $n_{1}$, $n_{2}$, and $r$, such that the the second fundamental form $A$ satisfies:

[TABLE]

Indeed, if $\beta$ solves the equation (2.1), then $\frac{\sin^{2}(r)}{\cos^{2}(r)}-\beta\frac{\sin(r)}{\cos(r)}-1=0$ and $\frac{\cos^{2}(r)}{\sin^{2}(r)}+\beta\frac{\cos(r)}{\sin(r)}-1=0$. Since these equations are equivalent, the claim follows. Taking the trace in both sides of (2.1) gives $|A|^{2}-n+\beta\,nH=0$.

3. Results

3.1. Volume preserving stable

A two-sided isometric immersion $\phi:\Sigma^{n}\rightarrow M^{n+1}$ has constant mean curvature if, and only if, $\phi$ is a critical point of the area functional for volume preserving variations, see [4, Section 2]. The mean curvature $H$ is defined by $tr(A)=n\,H$, where $A$ is the second fundamental form of $\Sigma$. Recall that $A:T_{x}\Sigma\rightarrow T_{x}\Sigma$ is defined by $A=-\overline{\nabla}N$, where $N$ is a unit normal vector over $\Sigma$. The critical point $\phi$ is called volume preserving stable if the second derivative of the area is non-negative for such variations.

Equivalently, $\phi$ is volume preserving stable if for every $f\in C^{\infty}(\Sigma)$ with compact support such that $\int_{\Sigma}f\,d_{\Sigma}=0$, we have

[TABLE]

$L$ is the Jacobi operator of $\Sigma$ defined as $L=\Delta_{\Sigma}+Ric_{M}(N,N)+|A|^{2}$.

Let $\phi:\Sigma^{n}\rightarrow\mathbb{S}^{n+1}$ be a two-sided constant mean curvature immersion.

Lemma 3.1**.**

The position vector $x$ and the unit normal vector $N$ along $\phi:\Sigma\rightarrow\mathbb{S}^{n+1}$ satisfy the following equations:

[TABLE]

Proof.

Recall first that $\Delta_{\Sigma}x=n\overrightarrow{H_{E}}$ where $\overrightarrow{H_{E}}$ is the mean curvature vector of $\Sigma$ in $\mathbb{R}^{n+2}$. On the other hand, $\overrightarrow{H_{E}}=-x+\overrightarrow{H}$, where $\overrightarrow{H}$ is the mean curvature vector of $\Sigma$ in $\mathbb{S}^{n+1}$. If $a$ is a constant vector in $\mathbb{R}^{n+2}$, then $\Delta_{\Sigma}\langle N,a\rangle=e_{i}\,e_{i}\langle N,a\rangle-\nabla_{e_{i}}e_{i}\,\langle N,a\rangle$, where $\{e_{1},\ldots,e_{n}\}$ is an orthonormal basis of $\Sigma$. A direct computation gives:

[TABLE]

The Codazzi equation implies that $div_{\Sigma}A=0$ since $H$ is constant. Therefore, $\Delta_{\Sigma}\langle N,a\rangle=-|A|^{2}\,\langle N,a\rangle+nH\,\langle x,a\rangle$. ∎

Lemma 3.2**.**

If $a$ is a constant vector in $\mathbb{R}^{n+2}$ and $\phi:\Sigma^{n}\rightarrow\mathbb{S}^{n+1}$ has constant mean curvature $H$, then

1. (1)

$L\langle x,a\rangle x=(|A|^{2}-n)\langle x,a\rangle x+nH\langle N,a\rangle x+nH\langle x,a\rangle N+X_{1}$*; * 2. (2)

$L\langle x,a\rangle N=nH\langle N,a\rangle N+nH\langle x,a\rangle x+X_{2}$*; * 3. (3)

$L\langle N,a\rangle x=nH\langle N,a\rangle N+nH\langle x,a\rangle x+X_{3}$*; * 4. (4)

$L\langle N,a\rangle N=(-|A|^{2}+n)\langle N,a\rangle N+nH\langle N,a\rangle x+nH\langle x,a\rangle N+X_{4}$*. *

where $X_{i}:\Sigma\rightarrow\mathbb{R}^{n+2}$ is a vector field tangent to $\Sigma$ for each $i=1,\ldots,4$.

Proof.

The Jacobi operator evaluated in a vector valued function $V:\Sigma\rightarrow\mathbb{R}^{n+2}$ is understood to be computed on each coordinate:

[TABLE]

where $\{e_{1},\ldots,e_{n+2}\}$ is the standard orthonormal basis in $\mathbb{R}^{n+2}$. Hence, for the vector fields in the lemma, each summand $\langle V,e_{k}\rangle$ is a product of functions $u$ and $v$. The lemma follows from combining the formula

[TABLE]

with the identities given in (3.2). The terms associated to the inner product of gradients corresponds to the tangent vector fields $X_{i}$ in the lemma. They are given below:

[TABLE]

where $a^{\top}$ denotes the projection of $a\in\mathbb{R}^{n+2}$ onto $T_{x}\Sigma$. ∎

Lemma 3.3**.**

If the projection of $\Sigma$ is volume preserving stable in $\mathbb{RP}^{n+1}$, then $f:\Sigma\rightarrow\mathbb{R}$ given by $f(x)=\langle x,a\rangle\langle N,b\rangle-\langle x,b\rangle\langle N,a\rangle$ satisfies $L(f)=0$ and

[TABLE]

Proof.

It follows from Lemma 3.2 that $L(f)=0$. One can easily check that $I(f+t,f+t)<0$ for every constant $t\neq 0$. Hence, $\int_{\Sigma}fd_{\Sigma}=0$. ∎

Given two vectors $a,b\in\mathbb{R}^{n+2}$, we consider the vector valued function $\Phi_{a,b}:\Sigma\rightarrow\mathbb{R}^{n+2}$ defined by:

[TABLE]

Lemma 3.4**.**

The Jacobi operator evaluated on $\Phi_{a,b}$ is given by:

[TABLE]

where $X$ is the tangent vector field over $\Sigma$ given by:

[TABLE]

Next lemma generalizes an assertion for minimal hypersurfaces in [10].

Lemma 3.5**.**

Let $R$ be the linear map $R:\mathbb{R}^{n+2}\rightarrow\mathbb{R}^{n+2}$ defined by

[TABLE]

If the projection of $\Sigma$ is volume preserving stable in $\mathbb{RP}^{n+1}$ and it is not totally geodesic, then $R$ is an isomorphism.

Proof.

If $R$ is not an isomorphism, then there exists a non-zero vector $b\in\mathbb{R}^{n+2}$ such that $R(b)=0$. This suggest using the vector field $V=\langle x,b\rangle N$ in the stability inequality. It follows from Lemma 3.2 that

[TABLE]

Since $\Sigma$ is stable, $LV=c$, for some vector $c\in\mathbb{R}^{n+2}$. It follows from Lemma 3.2 that

[TABLE]

In particular, $\langle x,c-nHb\rangle=0$ which implies $c=nHb$ since $\Sigma$ is not totally geodesic. Moreover, since $\Sigma$ is invariant by the antipodal map and $-2A(b^{\top})=c^{\top}$, we obtain that $2\langle N,b\rangle=nH\langle x,b\rangle$. If $H\neq 0$, then

[TABLE]

This is a contradiction since $\Sigma$ is not totally geodesic. On the other hand, if $H=0$, then $\langle N,b\rangle=0$. Hence, $\nabla^{2}(\langle x,b\rangle)=-\langle x,b\rangle Id$. By Obata’s Theorem in [26], $\Sigma$ is congruent to a round sphere. The Gauss equation then implies that $\Sigma$ is totally geodesic, contradiction. ∎

4. Proof of Theorem 1.2

Proof.

Let $\widetilde{\varphi}:\widetilde{\Sigma}\rightarrow\mathbb{RP}^{n+1}$ denote the stable immersion of $\widetilde{\Sigma}$ in $\mathbb{RP}^{n+1}$. Using locally constant functions we conclude from the stability assumption that $\widetilde{\Sigma}$ is connected. If there exists a lift $\varphi:\widetilde{\Sigma}\rightarrow\mathbb{S}^{n+1}$, then it follows from [4] that $\widetilde{\Sigma}$ is totally umbilical and hence, a geodesic sphere. If such lift does not exist, then there exist a orientable double covering $\Sigma\rightarrow\widetilde{\Sigma}$ and an isometric immersion $\varphi:\Sigma\rightarrow\mathbb{S}^{n+1}$ such that $\varphi\circ s=-\varphi$, where $s$ is the involution associated to the covering $\Sigma\rightarrow\widetilde{\Sigma}$. The two-sided assumption implies that $\Sigma$ is orientable and $N\circ s=-N$. The volume preserving stability of $\widetilde{\Sigma}$ implies that $I(u,u)\geq 0$ for every function $u:\Sigma\rightarrow\mathbb{R}$ which is even, i.e., $u\circ s=u$, and satisfies $\int_{\Sigma}u\,d_{\Sigma}=0$. Note that the vector field $\Phi_{a,b}$ defined in (3.3) satisfies $\Phi_{a,b}\circ s=\Phi_{a,b}$.

Lemma 4.1**.**

[TABLE]

Proof.

Recall from Lemma 3.4 that

[TABLE]

where $X$ is a tangent vector over $\Sigma$. A straightforward computation gives

[TABLE]

We have from the identities (3.2) that

[TABLE]

By Stoke’s Theorem, the first integral in the right hand side above is zero. Hence,

[TABLE]

The lemma follows by substituting formula (4.1) in the expression $-\int_{\Sigma}\langle\Phi_{a,b},L\Phi_{a,b}\rangle\,d_{\Sigma}$ above. ∎

We address first the case where the mean curvature $H$ is positive.

For every linear map $\varphi:\mathbb{R}^{n+2}\rightarrow\mathbb{R}^{n+2}$ we consider the symmetric quadratic form $Q_{\varphi}:\mathbb{R}^{n+2}\times\mathbb{R}^{n+2}\rightarrow\mathbb{R}$ given by

[TABLE]

Let $\mathcal{M}$ be the set $\mathcal{M}=SO(n+2)\times\mathbb{S}^{n+1}$ and let $F:\mathcal{M}\times\mathbb{R}\rightarrow\mathbb{R}$ be the function defined by $F(\varphi,a,\beta)=Q_{\beta\cdot\varphi}(a,a)$.

Lemma 4.2**.**

If $(\varphi_{0},a_{0},\beta_{0})\in\mathcal{M}\times\mathbb{R}$ is a critical point of $F$ and the mean curvature $H$ is positive, then

[TABLE]

Proof.

The tangent space $T_{\varphi_{0}}SO(n+2)$ is given by the linear space $\{\varphi_{0}\cdot K\in\mathcal{L}(\mathbb{R}^{n+2}):\,K^{\top}=-K\}$. Taking the derivative of $F(\varphi,a,\beta)$ with respect to $\varphi$ and recalling the expression of $Q_{\varphi}(a,a)$ in Lemma 4.1 we have

[TABLE]

Since $\{\varphi_{0}\cdot K(a_{0}):\,K^{\top}=-K\}=\langle\varphi_{0}(a_{0})\rangle^{\perp}$, we conclude that the vector $\int_{\Sigma}\Phi_{a_{0},\beta_{0}\varphi_{0}(a_{0})}\,d_{\Sigma}$ is parallel to $\varphi_{0}(a_{0})$ unless $\beta_{0}=0$. On the other hand, we also have

[TABLE]

Hence, if $\beta_{0}\neq 0$, then $\int_{\Sigma}\Phi_{a_{0},\beta_{0}\varphi_{0}(a_{0})}\,d_{\Sigma}=0$. If $\beta_{0}=0$, then (4.2) is also true when $\varphi_{0}$ is replaced by any orthogonal map $\varphi$. This implies $\int_{\Sigma}\Phi_{a_{0},0}\,d_{\Sigma}=0$ as well. This completes the proof of the lemma. ∎

Our goal is to prove existence of a critical point $(\varphi_{0},a_{0},\beta_{0})$ for $F$ using the standard mountain pass variational theory. For this we consider the max-min number defined below:

[TABLE]

where $[\mathcal{M}]$ is the class of maps $\phi:\mathcal{M}\rightarrow\mathcal{M}\times\mathbb{R}$ that are homotopic to $\phi_{0}(a,\varphi)=(a,\varphi,0)$. First, let us show that $m_{1}\leq 0$. Indeed, each connected component of

[TABLE]

is unbounded in the directions $\beta\rightarrow\pm\infty$. Hence, every $\phi\in[\mathcal{M}]$ must satisfy $\phi(\mathcal{M})\cap Z\neq\emptyset$. The claim now follow since the first eigenvalue of $Q_{\beta\cdot Id}$ is non-positive for every $\beta$. Indeed, if $\{e_{1},\ldots,e_{n+2}\}$ is an orthonormal basis of $\mathbb{R}^{n+2}$, then

[TABLE]

In particular, $m_{1}$ is well defined. Note that if $m_{1}$ is a critical value of $F$, then it will follow from Lemma 4.2 and the volume preserving stability of $\Sigma$ that $m_{1}\geq 0$, and hence $m_{1}=0$. Next we study the behavior of $F$ at the ends of $\mathcal{M}\times\mathbb{R}$:

Lemma 4.3**.**

There exists constants $C_{1}>0$ and $C_{2}>0$ such that $|\nabla F|(\varphi,a,\beta)\geq C_{1}$ for every $|\beta|\geq C_{2}$.

Proof.

Assume $(\varphi_{l},a_{l},\beta_{l})$ is a sequence of points in $\mathcal{M}\times\mathbb{R}$ satisfying $\lim_{l\rightarrow\infty}|\beta_{l}|=\infty$ and such that $|\nabla F|(\varphi_{l},a_{l},\beta_{l})=\varepsilon_{l}\rightarrow 0$. We have by Lemma 4.2 that

[TABLE]

Note by compactness of $\mathcal{M}$ that $(\varphi_{l},a_{l})$ has a convergent subsequence. Since $|\beta_{l}|\rightarrow\infty$, we conclude that

[TABLE]

This is a contradiction since $\varphi_{l}(a_{l})\in\mathbb{S}^{n+1}$ and the linear map $R$ is an isomorphism by Lemma 3.5. ∎

By Lemma 4.3, the functional $F$ satisfies the Palais-Smale condition (P.-S.): namely, any sequence satisfying $|F(u_{i})|\leq c$ and $|\nabla F(u_{i})|\rightarrow 0$ has a convergent subsequence. The following finite dimensional min-max principle is proved in Struwe [37]:

Min-max principle. Suppose $M$ is a complete Riemannian manifold and $F\in C^{\infty}(M)$ satisfies (P.-S.). Also suppose that $\mathcal{E}$ is a collection of sets which is invariant with respect to any smooth semi-flow $\Psi:M\times[0,\infty)\rightarrow M$ such that $\Psi(\cdot,0)=id$, $\Psi(\cdot,t)$ a diffeomorphism of $M$ for any $t\geq 0$, and $F(\Psi(u,t))$ is non-decreasing in $t$ for any $u\in M$. If

[TABLE]

is finite, then $\mu$ is a critical value of $F$.

Let us now address the case $H=0$ proved in [10]. The proof is included for completeness. It follows from Lemma 3.5 the existence of an unique linear map $\varphi_{1}:\mathbb{R}^{n+2}\rightarrow\mathbb{R}^{n+2}$ such that

[TABLE]

In other words, the vector field $\Phi_{a,\varphi_{1}(a)}$ defined in (3.3) is balanced. By Lemma 4.1, we have

[TABLE]

for every $a\in\mathbb{R}^{n+2}$. On the other hand, if $\{e_{1},\ldots,e_{n}\}$ is an orthonormal basis of $\mathbb{R}^{n+2}$ with $e_{1}=\frac{a}{|a|}$, then

[TABLE]

Therefore, $I(\Phi_{a,\varphi_{1}(a)},\Phi_{a,\varphi_{1}(a)})=0$ for every vector $a\in\mathbb{R}^{n+2}$.

Despite the conclusion obtained for $H>0$ be weaker than the one obtained for $H=0$, it is enough to complete the proof of the theorem:

Assertion. If there are vectors $a_{0}\in\mathbb{R}^{n+2}$ and $b_{0}\in\mathbb{R}^{n+2}$ such that $I(\Phi_{a_{0},b_{0}},\Phi_{a_{0},b_{0}})=0$ and

[TABLE]

then $\Sigma$ is congruent to either a geodesic sphere or a Clifford hypersurface $\mathbb{S}^{n_{1}}(R_{1})\times\mathbb{S}^{n_{2}}(R_{2})\subset\mathbb{S}^{n+1}$.

Proof. It follows from the variational characterization of volume preserving stable hypersurfaces that $L\Phi_{a_{0},b_{0}}=c$ for some vector $c\in\mathbb{R}^{n+2}$. By Lemma 3.4, this is equivalent to

[TABLE]

If we use the notation $r(x)=(|A|^{2}-n)(x)$, then

[TABLE]

for every $x\in\Sigma$. In particular,

[TABLE]

Let $x_{0}\in\Sigma$ be the point of maximum for the function $r(x)$. It follows from equations (4.3) and (4.4) that $c=nH\,b_{0}-r(x_{0})\,a_{0}$. Similarly, if $x_{1}\in\Sigma$ is the point of minimum for $r(x)$, then equations (4.3) and (4.4) implies that $c=nH\,b_{0}-r(x_{1})\,a_{0}$. Hence,

[TABLE]

and $r(x)=(|A|^{2}-n)(x)$ is a constant function. Consequently, we can find a constant $\beta_{0}$ for which $|A|^{2}-n-\beta_{0}nH=0$. Therefore,

[TABLE]

for every $a\in\mathbb{R}^{n+2}$. The second equality follows from the integration by parts in (4.1). If $H=0$, then $|A|^{2}\equiv n$ (Lemma 3.3 and Lemma 3.5). By the result of Chern, do Carmo, and Kobayashi [8] (see also Lawson [19]), $\Sigma$ is congruent to the Clifford hypersurface $\mathbb{S}^{p}(\sqrt{\frac{p}{n}})\times\mathbb{S}^{n-p}(\sqrt{\frac{n-p}{n}})$ for some $p$. From now on, $H>0$. Since the quadratic in the right hand side above is symmetric (Lemma 3.3), we obtain that

[TABLE]

for every $a\in\mathbb{R}^{n+2}$. In other words, the vector fields $\Phi_{a,\beta_{0}a}$ are balanced. On the other hand, we have by Lemma 4.1 that

[TABLE]

Hence, $I(\Phi_{a,\beta_{0}a},\Phi_{a,\beta_{0}a})=0$ for every $a\in\mathbb{R}^{n+2}$. It follows from the volume preserving stability of $\Sigma$ that $L(\Phi_{a,\beta_{0}\cdot a})=c$. Since $\beta_{0}$ satisfies $|A|^{2}-n-\beta_{0}nH=0$, we conclude simultaneously that both $c$ and the vector field $X$ from Lemma 3.4 are zero:

[TABLE]

for every $a\in\mathbb{R}^{n+2}$. Hence, $A^{2}-\beta_{0}\,A-Id=0$ as a $2$-tensor. As a consequence, the principal curvatures of $\Sigma$ are constant and equal to:

[TABLE]

In other words, $\Sigma$ is an isoparametric hypersurface with at most two principal curvatures. If the principal curvatures are equal, then $\Sigma$ is totally umbilical and, hence, a geodesic sphere. If the principal curvatures are $\mu_{+}$ with multiplicity $n_{1}$ and $\mu_{-}$ with multiplicity $n_{2}$, then $\Sigma$ is congruent to the Clifford hypersurface $\mathbb{S}^{n_{1}}(\cos(r))\times\mathbb{S}^{n_{2}}(\sin(r))$ for some $r\in(0,\frac{\pi}{2})$, see Theorem 3.29 in [7]. ∎

4.1. The isoperimetric problem

In this section we use Theorem 1.2 to discuss the isoperimetric problem in $\mathbb{RP}^{n}$.

Let $M$ be a Riemannian manifold. We denote the $(n+1)$-dimensional Hausdorff measure of a region $\Omega\subset M$ by $|\Omega|$. The $n$-dimensional Hausdorff measure of $\partial\Omega$ is denoted by $|\partial\Omega|$. The class of regions considered here are those of finite perimeter, see [6]. A region $\Omega\subset M$ is called an isoperimetric region if

[TABLE]

In this case, the hypersurface $\Sigma=\partial\Omega$ is called an isoperimetric hypersurface. For a recent reference on the regularity of isoperimetric hypersurfaces see [22]:

Theorem 4.4**.**

If $(M^{n+1},g)$ is a closed or homogeneous Riemannian manifold, then for any $0<t<\text{vol}(M)$ there exists an isoperimetric region $\Omega$ satisfying $|\Omega|=t$. Moreover, $\Sigma=\partial\Omega$ is smooth up to a closed set of Hausdorff dimension $n-7$. The regular part is a constant mean curvature volume preserving stable hypersurface.

Lemma 4.5** (Morgan-Ritoré [23]).**

Let $\Sigma^{k}$ be a smooth, embedded submanifold of bounded mean curvature in $\mathbb{R}^{n+1}$ with compact singular set $\Sigma_{0}=\overline{\Sigma}-\Sigma$, satisfying $\mathcal{H}^{k-2}(\Sigma_{0})=0$. Then, given $\varepsilon>0$, there is a smooth function $\varphi_{\varepsilon}:\overline{\Sigma}\rightarrow[0,1]$ supported in $\Sigma$ such that

1. (1)

$\mathcal{H}^{k}(\{\varphi_{\varepsilon}\neq 1\})<\varepsilon$; 2. (2)

$\int_{\Sigma}|\nabla\varphi_{\varepsilon}|^{2}\,d_{\Sigma}<\varepsilon$; 3. (3)

$\int_{\Sigma}|\Delta\varphi_{\varepsilon}|\,d_{\Sigma}<\varepsilon$.

As pointed out in [23], Lemma 4.5 extends the stability inequality to the set of smooth bounded functions $f:\Sigma\rightarrow\mathbb{R}$ with mean zero on $\Sigma$ and gradient in $L^{2}(\Sigma)$:

[TABLE]

Indeed, given such $f$, we construct the function $f_{\varepsilon}=(\varphi_{\varepsilon}f)^{+}-t_{\varepsilon}(\varphi_{\varepsilon}f)^{-}$ where $t_{\varepsilon}$ is a constant so that $f_{\varepsilon}$ has mean zero in $\Sigma$. Since support of $f_{\varepsilon}$ is in the regular part of $\Sigma$, (4.6) holds for $f_{\varepsilon}$. The claim now follows by taking $\varepsilon\rightarrow 0$ and applying Lemma 4.5.

The vector valued function $\Phi_{a,b}$ used in the proof of Theorem 1.2 is smooth and bounded on $\Sigma$; let us show that its gradient is in $L^{2}(\Sigma)$. By the Cauchy-Schwarz inequality, $|\nabla\langle\Phi_{a,b},e_{j}\rangle|$ is bounded from above by linear combinations of the principal curvatures $k_{i}$ of $\Sigma$. Therefore, it suffices showing that $|A|\in L^{2}(\Sigma)$. As in [23], let $u:\mathbb{RP}^{n+1}\rightarrow[0,1]$ be such that $u\equiv 1$ in a neighborhood $V$ of the singular set $\Sigma_{0}$, $\overline{\nabla}u$ bounded, and $\int_{\Sigma}u\,d_{\Sigma}=0$. Since the stability inequality (4.6) holds for $u$, we obtain that

[TABLE]

In particular, the index form (4.6) is well defined on each coordinate of $\Phi_{a,b}$. Moreover, the integration by parts (4.1) follows directly from Lemma 4.5.

These remarks guarantee that the proof of Theorem 1.2 extends to the singular setting except for a minor difference at the rigidity analysis for $L\Phi_{a_{0},b_{0}}=c$. Indeed, we are not allowed to pick the point of maximum for the function $r(x)=(|A|^{2}-n)(x)$ since it is necessarily infinite in the singular case. Nonetheless, the argument up to that point implies that $c=nH\,b_{0}-r(x_{1})a_{0}$, where $x_{1}$ is the point of minimum. From (4.3) we obtain

[TABLE]

for every $x\in\Sigma$. Therefore, $r(x)=(|A|^{2}-n)(x)$ is constant and $\Sigma$ is a regular hypersurface.

The isoperimetric profile of $\mathbb{RP}^{n}$ is the function $I_{\mathbb{RP}^{n}}:[0,vol(\mathbb{RP}^{n})]\rightarrow\mathbb{R}$ defined by $I_{\mathbb{RP}^{n}}(v)=\inf\{|\partial\Omega|:\Omega\subset\mathbb{RP}^{n}\,\,and\,\,|\Omega|=v\}$. Conjecture 1.1 in the form below is due to Berger [5]:

Conjecture 4.6** (Berger [5]).**

The isoperimetric profile of $\mathbb{P}^{n+1}(\mathbb{K})$ is given by the perimeter of successive tubular neighborhoods of projective subspaces $\mathbb{P}^{k}(\mathbb{K})\subset\mathbb{P}^{n+1}(\mathbb{K})$.

Corollary 1.3 confirms the conjecture in the real projective space $\mathbb{RP}^{n}$ except for the word successive. Numeric computations verified this simple description of the profile in the dimensions $n\leq 10$. Figure 1 highlights the word successive.

5. Proof of Theorem 1.4

Arithmetic Mean - Geometric Mean inequality: If $a_{1},\ldots,a_{n}$ are positive numbers, then

[TABLE]

The equality holds if, and only if, all the $a_{i}$’s are equal.

Proof of Theorem 1.4.

Consider the map $F_{t}:\Sigma\rightarrow\mathbb{S}^{n+1}$ given by $F(x,t)=\cos(t)x+\sin(t)N(x)$. Given $p\in\Sigma$, take an orthonormal basis of $T_{p}\Sigma$ by principal directions. If $\{\lambda_{1},\ldots,\lambda_{n}\}$ are the principal curvatures, then the Jacobian of the map $F_{t}$ is

[TABLE]

Assume for the moment that for each $i$ we have $\cos(t)-\lambda_{i}\sin(t)>0$. By the Arithmetic Mean - Geometric Mean Inequality (5.1), we have:

[TABLE]

The last inequality follows from the Cauchy-Schwarz inequality applied to the vectors $\overrightarrow{u}=(\cos(t),\sin(t))$ and $\overrightarrow{v}=(1,-H)$.

Let $U$ and $V$ be such that $\mathbb{S}^{n+1}\backslash\Sigma=U\cup V$. For any $t\in[-\pi,\pi]$, we consider the hypersurfaces $\Sigma_{|t|}=\{p\in U:dist(p,\Sigma)=|t|\}$ and $\Sigma_{-|t|}=\{p\in V:dist(p,\Sigma)=|t|\}$. The distance between $\Sigma$ and a point in $\Sigma_{t}$ is attained by a minimizing geodesic which meets the hypersurface $\Sigma$ orthogonally. This distance is not bigger than the cut function, $c:\Sigma\rightarrow\{0\leq t<\pi\}$. Recall that $c(p)$ is the last positive time, such that the normal geodesic $t\mapsto\cos(t)p+\sin(t)N(p)$ attains the distance to $\Sigma$. Thus, we have that $\Sigma_{t}=\{\cos(t)\,p+\sin(t)\,N(p):p\in\Sigma,\,c(p)\geq t\}$. We also have that $c(p)$ is less than or equal to the first focal value along the normal geodesic at $p$. Hence, for any $0\leq t\leq c(p)$, the Jacobian at $p$, $Jac(F_{t})(p)$, of the map $F_{t}(p)=\cos(t)p+\sin(t)N(p)$ is nonnegative. Therefore,

[TABLE]

It follows from (5.2) and (5.3) that

[TABLE]

By assumption $U$ and $V$ are antipodal invariant. Hence, $U_{t}$ and $V_{t}$ defined by $\mathbb{S}^{n+1}\backslash\Sigma_{t}=U_{t}\cup V_{t}$ are also antipodal invariant. Moreover, the projection of $\{\Sigma_{t}\}_{t\in[-\pi,\pi]}$ in $\mathbb{RP}^{n+1}$ provides an admissible class of sweep out to run the Almgren-Pitts Min-Max Theory. Hence, there exists $t_{0}\in(-\pi,\pi)$ such that

[TABLE]

Combining (5.4), (5.5), and the claim below, we obtain the desired inequality (1.1).

Claim. W(\mathbb{RP}^{n+1})=\bigg{|}\mathbb{S}^{\lfloor\frac{n}{2}\rfloor}\bigg{(}\sqrt{\frac{\lfloor\frac{n}{2}\rfloor}{n}}\bigg{)}\times\mathbb{S}^{\lceil\frac{n}{2}\rceil}\bigg{(}\sqrt{\frac{\lceil\frac{n}{2}\rceil}{n}}\bigg{)}\bigg{|}.

If we have equality in (1.1), then (5.4) and (5.2) become equalities. The equality in (5.2) occurs if, and only if, $\Sigma$ is totally umbilical or both $t_{0}=0$ and $H=0$. By assumption, $\Sigma$ is not totally umbilical and, hence, $\Sigma$ is minimal and $t_{0}=0$. Since we also have equality in (5.5), $\Sigma$ is congruent to \mathbb{S}^{\lfloor\frac{n}{2}\rfloor}\bigg{(}\sqrt{\frac{\lfloor\frac{n}{2}\rfloor}{n}}\bigg{)}\times\mathbb{S}^{\lceil\frac{n}{2}\rceil}\bigg{(}\sqrt{\frac{\lceil\frac{n}{2}\rceil}{n}}\bigg{)}.

Proof of claim: by Zhou [39], the width $W(\mathbb{RP}^{n+1})$ is realized by either a multiplicity one two sided index one minimal hypersurface $\Sigma$ or a multiplicity two one sided minimal hypersurface $\hat{\Sigma}$. As usual, $\Sigma$ and $\hat{\Sigma}$ are smooth away for a closed set of Hausdorff dimension $n-7$. By the result of do Carmo, Ritoré, and Ros [10], the only possibility for the former is a Clifford hypersurface; the proof still holds in the singular case by the cut-off argument of Morgan and Ritoré[23] discussed in the previous section. In the latter case we have $|\hat{\Sigma}|\geq\mathbb{|}\mathbb{RP}^{n}|$. The claim now follows from the inequalities:

[TABLE]

To prove these inequalities, we argue as in Stone [36]. We consider the function $f(p)=|\mathbb{S}^{p}||\mathbb{S}^{n-p}|(\frac{p}{n})^{\frac{p}{2}}(\frac{n-p}{n})^{\frac{n-p}{2}}$. Recall the standard formula for the volume of spheres in terms of the Gamma function $\Gamma$:

[TABLE]

Note that $f(x)=f(n-x)$ and $\lim_{x\rightarrow 0}f(x)=2|\mathbb{S}^{n}|$. Our goal is to show that $f(x)$ is concave up on the interval $[0,n]$. It is convenient to consider the function $\overline{f}(x)=\ln(f(x))$ instead. A computation gives

[TABLE]

Using the identity $(\Gamma^{\prime}(t)/\Gamma(t))^{\prime}=\sum_{l=0}^{\infty}(t+l)^{-2}$, we compute:

[TABLE]

As observed in [36], we have that $\sum_{l=0}^{\infty}\frac{1}{(x+\frac{1}{2}+l)^{2}}<\frac{1}{x}$. Applying this inequality to the expression above, we obtain $\overline{f}^{\prime\prime}(x)>0$. This implies $f^{\prime\prime}(x)>0$ and consequently (5.6). ∎