[Paper Review] Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces

Kwok‐Kun Kwong, Hojoo Lee+1 more|arXiv (Cornell University)|Aug 31, 2016

Geometric Analysis and Curvature Flows32 references3 citations

TL;DR

This paper establishes new rigidity results for closed embedded hypersurfaces in warped product manifolds and Euclidean space using weighted Hsiung-Minkowski formulas and Brendle's inequality. It proves that under certain monotonicity and curvature conditions on higher-order mean curvatures, hypersurfaces must be totally umbilical—specifically, round spheres—extending classical Alexandrov-type theorems beyond the scope of the Alexandrov reflection principle.

ABSTRACT

We use the weighted Hsiung-Minkowski integral formulas and Brendle's inequality to show new rigidity results. First, we prove Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of Riemannian warped product manifolds, including the Schwarzschild and Reissner-Nordström spaces, where the Alexandrov reflection principle is not available. Second, we prove that, in Euclidean space, the only closed immersed self-expanding solitons to the weighted generalized inverse curvature flow of codimension one are round hyperspheres.

Motivation & Objective

• To extend Alexandrov-type rigidity theorems to Riemannian warped product manifolds, such as Schwarzschild and Reissner-Nordström spaces, where the Alexandrov reflection principle is inapplicable.
• To establish new rigidity conditions for closed, $k$-convex hypersurfaces with radially symmetric higher-order mean curvatures in such ambient spaces.
• To prove that in Euclidean space, the only closed immersed self-expanding solitons to the weighted generalized inverse curvature flow are round hyperspheres.
• To weaken assumptions on ambient geometry while strengthening the rigidity conclusions via integral inequalities and monotonicity conditions on curvature functions.

Proposed method

• Derive and apply weighted Hsiung-Minkowski integral formulas in warped product manifolds with radial symmetry.
• Utilize Brendle’s inequality to control curvature evolution and derive comparison estimates for higher-order mean curvatures.
• Impose monotonicity and positivity conditions on coefficient functions $b_j(r), c_j(r)$, and $ heta(r)$ to constrain curvature behavior.
• Apply the Hsiung–Minkowski formulas in Euclidean space to relate integrals of mean curvatures to support functions and derive contradiction if non-spherical.
• Use Newton-Maclaurin inequalities to compare ratios of mean curvatures and deduce equality cases implying umbilicity.
• Analyze the weighted generalized inverse curvature flow in $\mathbb{R}^n$, defining self-expanders via a condition linking curvature ratios and the support function $\mathbf{p} = \langle X, \nu \rangle$.

Experimental results

Research questions

• RQ1 Can rigidity results for hypersurfaces with constant higher-order mean curvature be extended to warped product manifolds where the Alexandrov reflection principle fails?
• RQ2 What conditions on the warping function $h(r)$ and curvature functions ensure that a $k$-convex hypersurface in a warped product space is totally umbilical?
• RQ3 Under what conditions on the weight functions and curvature ratios does a self-expanding soliton to the weighted generalized inverse curvature flow in $\mathbb{R}^n$ become a round sphere?
• RQ4 How do monotonicity assumptions on coefficient functions $a_i(r), b_j(r)$ and $\eta(r)$ affect the rigidity of hypersurfaces in the absence of star-shapedness?
• RQ5 Can the classical Hsiung–Minkowski formulas be adapted to prove rigidity in non-constant curvature settings via weighted integral identities?

Key findings

• In warped product manifolds satisfying (H1)–(H4), a closed $k$-convex hypersurface with $\sum_{j=1}^k (b_j(r)H_j + c_j(r)H_1H_{j-1}) = \eta(r)$, where $\eta(r)$ is positive and radially decreasing, must be umbilical.
• For $k \geq 2$, under (H1)–(H4), any star-shaped, $k$-convex hypersurface satisfying the same curvature condition must be a slice $N^{n-1} \times \{r_0\}$, i.e., a totally umbilical hypersurface.
• In Euclidean space, the only closed immersed self-expanding solitons to the weighted generalized inverse curvature flow are round hyperspheres, as shown via equality in Hsiung–Minkowski and Newton-Maclaurin inequalities.
• The monotonicity assumptions on coefficient functions $b_j(r), c_j(r), \eta(r)$, and $a_i(r), b_j(r)$ are essential—counterexamples exist without them, such as thin tori with monotone increasing mean curvature depending only on radial distance.
• The results generalize prior work by Montiel, Brendle, and others, extending rigidity to non-star-shaped and non-constant curvature settings via integral identities.
• The equality case in the Hsiung–Minkowski formulas and Newton-Maclaurin inequalities forces $\mu = 1$, leading to $\mathbf{p} = 1$ and thus umbilicity, proving the hypersurface is a round sphere centered at the origin.