We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved...
Shi-Bing Chen, M. Ghomi, Peng Wang· 3 citations· ⚡1
We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $n$-manifolds of nonpositive sectional curvature, $3\leq n\leq9$, which establishes the Cartan-Hadamard conjecture in these dimensions. The main step is a sharp inequality for constant-mean-curvature hype...
We establish a sharp lower bound for the total Gauss-Kronecker curvature of convex hypersurfaces in Cartan-Hadamard manifolds with pinched negative curvature. The bound holds in all dimensions when the diameter is small relative to the curvature scale, and in dimensions 4 and 5 without any restriction on the diameter,...
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 conj...
M. Ghomi· 0 citations
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