Let $M^n$ be a $C^2$ closed convex hypersurface in Euclidean space, and $\sigma_m$ be its $m$th mean curvature. We show that $M$ is symmetric with respect to a hyperplane orthogonal to a given direction $e$, if $\sigma_m(p)\leq\sigma_m(q)$ whenever $p-q$ is parallel to $e$. For convex hypersurfaces, this settles a conjecture of Li, extends the mean-curvature theorem of Li-Yan-Yao to all $\sigma_m$, and strengthens some earlier results of Li-Nirenberg by removing nondegeneracy assumptions. The proof is based on the theory of mixed volumes, specifically the rigidity of quermassintegrals under Steiner symmetrization.
Let $(M^{n}, g)$ be a complete, simply connected Riemannian manifold without boundary, of dimension $n\ge3$, with curvature operator at least that of the unit sphere. We prove that $$\int_M {\rm scal}(x)\ d {\rm Vol}_x\le n(n-1)\omega_n,$$ where $\omega_n$ is the volume of the unit $n$-sphere. Equality holds if and onl...
Let $X\subset \mathbb{O}^n$ be a nodal hypersurface of degree $d$ with an ordinary $m$-fold point. Let $\delta_X$ be the largest value of $\delta$ such that $X$ is contained in the closure of the Severi variety of degree $d$ hypersurface in $\mathbb{P}^n$ with $\delta$ nodes. After recalling how the semicontinuity of t...
Let $(M^3,g)$ be a complete Riemannian manifold diffeomorphic to $\R^3\setminus\{0\}$, with nonnegative scalar curvature. Assume that a distinguished end is asymptotically flat, with ADM mass $m_+$. For each $p\in(1,3)$, define $c_{O,p}$ as the infimum of the Schwarzschild-normalized $p$-capacity over outward-minimizin...
For every $n\ge4$ and $L>0$, we construct a smooth $4$-PIC metric on $S^n$ with Urysohn $1$-width at least $L$ and an embedded stable minimal disk of intrinsic inradius at least $L$. These examples disprove the proposed width and stable-disk radius bounds under a positive lower bound for isotropic curvature. On closed...
Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption an...
This is the second article of a sequence of research on the local rigidity of constant mean curvature (CMC) hypersurfaces in space forms. In the previous one, we studied the local rigidity of CMC hypersurfaces whose the number of the distinct principal curvatures satisfies $g\leq 3$. In this paper, we study the local r...
Xing Cheng, Ya-Yun Chen, Tong-Zhu Li· 1 citation
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