[Paper Review] Note on Brendle-Eichmair's paper "Isoperimetric and Weingarten surfaces in the Schwarchild manifold"

Haizhong Li, Yong Wei+1 more|arXiv (Cornell University)|Oct 21, 2012

Geometric Analysis and Curvature Flows18 references4 citations

TL;DR

This paper removes the convexity assumption from Brendle-Eichmair's Theorem 7 on isoperimetric and Weingarten surfaces in the Schwarzschild manifold. By proving the existence of an elliptic point on any star-shaped hypersurface and leveraging properties of elementary symmetric functions of principal curvatures, the authors show that constant $\sigma_p$ implies the hypersurface is a rotational slice $\mathbb{S}^{n-1} \times \{r\}$, even without convexity. The result extends to general Einstein base manifolds under similar geometric conditions.

ABSTRACT

In this short note, we show that the assumption "convex" in Theorem 7 of Brendle-Eichmair's paper \cite{BE} is unnecessary.

Motivation & Objective

• To remove the unnecessary convexity assumption in Brendle-Eichmair's Theorem 7 on $\sigma_p$-constant hypersurfaces in the Schwarzschild manifold.
• To establish that star-shaped hypersurfaces with constant $\sigma_p$ must be rotational slices $\mathbb{S}^{n-1} \times \{r\}$ without requiring convexity.
• To generalize the result to warped product manifolds $N \times [0,\bar{r})$ with Einstein base $N$ and suitable warping function $\lambda(r)$.
• To demonstrate that the existence of an elliptic point on any star-shaped hypersurface suffices to derive the rigidity result, even when principal curvatures are not all positive globally.

Proposed method

• Prove the existence of an elliptic point on any closed, embedded, star-shaped hypersurface $\Sigma$ by analyzing the height function $h = r$ and its Hessian at a maximum point.
• Use the conformal vector field $X = \lambda(r)\partial_r$ and its properties to derive curvature estimates and identities involving $\sigma_p$ and $\sigma_{p-1}$.
• Apply the Heintze-Karcher-type inequality and Minkowski-type inequality under the assumption of constant $\sigma_p > 0$, derived via integration of tensor fields on $\Sigma$.
• Leverage the positivity of $\sigma_j$ for $1 \leq j \leq p-1$ on $\Sigma$ by showing the set where $\sigma_j > 0$ is both open and closed, hence the full hypersurface lies in the positive cone $\Gamma_p^+$.
• Use the Ricci curvature expression in the warped product metric to control the sign of the divergence term in the Minkowski identity, ensuring non-negativity of the error term.
• Generalize the argument to manifolds $M = N \times [0,\bar{r})$ with Einstein base $N$ satisfying $Ric_N = (n-2)B g_N$, under modified conditions (H1')-(H4') on $\lambda(r)$.

Experimental results

Research questions

• RQ1 Can the convexity assumption in Brendle-Eichmair's rigidity theorem for $\sigma_p$-constant hypersurfaces in the Schwarzschild manifold be removed?
• RQ2 Does the existence of an elliptic point on a star-shaped hypersurface suffice to imply that constant $\sigma_p$ implies a rotational slice, even without global convexity?
• RQ3 Can the rigidity result be extended to warped product manifolds with general Einstein base manifolds $N$?
• RQ4 What geometric conditions on the warping function $\lambda(r)$ ensure the validity of the Minkowski-type inequality and Heintze-Karcher inequality in the generalized setting?

Key findings

• The convexity assumption in Theorem 7 of Brendle-Eichmair's paper is unnecessary; the rigidity conclusion holds for all star-shaped hypersurfaces with constant $\sigma_p$.
• Every closed, embedded, star-shaped hypersurface in $(M,\bar{g})$ with constant $\sigma_p$ must be a rotational slice $\mathbb{S}^{n-1} \times \{r\}$ for some $r \in (0,\bar{r})$.
• The existence of an elliptic point on any star-shaped hypersurface ensures that $\sigma_p > 0$ and that the principal curvatures lie in the positive cone $\Gamma_p^+$, enabling the use of Minkowski-type inequalities.
• The Minkowski-type inequality $p \int_\Sigma \langle X,\nu\rangle \sigma_p \geq (n-p)\int_\Sigma \lambda' \sigma_{p-1}$ holds under the given conditions, with the error term non-negative due to the Ricci curvature sign and $\sigma_{p-2;j} > 0$.
• The result generalizes to $M = N \times [0,\bar{r})$ with Einstein base $N$ satisfying $Ric_N = (n-2)B g_N$, under modified conditions (H1')-(H4') on $\lambda(r)$, yielding the same rigidity conclusion.
• The de Sitter-Schwarzschild manifold satisfies all conditions (H1)-(H4), so Corollary 3 confirms that $\sigma_p$-constant star-shaped hypersurfaces are slices in this physical spacetime model.