We study sharp estimates for the $p$-capacity on complete non-compact Riemannian manifolds under lower Ricci curvature bounds. First, we establish sharp comparison inequalities for the $p$-capacity of bounded smooth domains in manifolds satisfying $\operatorname{Ric}\ge -ng.$ The estimates are expressed in terms of the boundary mean curvature and correspond to natural warped-product model ends. We characterize all equality cases and show that equality forces the exterior region to be isometric to the corresponding warped product. We also obtain an analogous sharp estimate under nonnegative Ricci curvature, whose equality case is described by an asymptotically flat model end. Second, we investigate normalized lower bounds for the relative $p$-capacity of condensers. We introduce scale-invariant quantities involving the volume of the inner set and the diameter of the ambient domain, establish uniform positive lower bounds, and determine the optimal ranges of the normalization parameters.
We establish a weighted extension of the Bishop-Gromov volume comparison theorem for complete Riemannian manifolds with Ricci curvature bounded below. Given a point $p$ and positive radial weights $f$ and $h$, we consider the weighted volume of a geodesic ball and a corresponding model weighted volume in the simply con...
Let $(M,g)$ be a connected, compact, $n$-dimensional Riemannian manifold with $\operatorname{Ric}(M,g)\geq-(n-1)\kappa g$. We introduce a weighted combinatorial Laplacian on $\varepsilon$-discretizations of $M$ and prove a spectral comparison theorem between the weighted graph Laplacian and the Laplace-Beltrami operato...
We explore fast convergence of the Riemannian Frank--Wolfe method for smooth geodesically convex optimization over compact feasible sets. Hadamard manifolds are the main setting. On general complete manifolds, the analysis accounts for all feasible minimizing geodesics. We consider the open loop step-size $\eta_k=a/(k+...
Let $n\geq3$ and $0\leq\tau\leq2$. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive $\tau$-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindric...
The goal of this work is to introduce a notion of mean curvature for level sets of functions in non-smooth spaces with Ricci curvature bounded below, and to prove that it satisfies sharp geometric inequalities. More precisely, we define a suitable Willmore functional $\mathcal{W}$ on Sobolev functions, whose domain of...
Let $S$ be an oriented surface, possibly of infinite type, endowed with a complete Riemannian metric with pinched negative curvature. We prove that the area form defines a non-trivial class in the second bounded cohomology group of $S$, unless $S$ is diffeomorphic to the disc or the cylinder. This is in sharp contrast...
R. Frigerio, Ervin Hadžiosmanović· 0 citations
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