Mathematics – Differential Geometry
Scientific paper
2010-12-02
Mathematics
Differential Geometry
13 pages
Scientific paper
We obtain in this paper bounds for the capacity of a compact set $K$ with smooth boundary. If $K$ is contained in an $(n+1)$-dimensional Cartan-Hadamard manifold and the principal curvatures of $\partial K$ are larger than or equal to $H_0>0$, then ${\rm Cap(K)}\geq (n-1)\,H_0 {\rm vol}(\partial K)$. When $K$ is contained in an $(n+1)$-dimensional manifold with non-negative Ricci curvature and the mean curvature of $\partial K$ is smaller than or equal to $H_0$, we prove the inequality ${\rm Cap}(K)\leq (n-1)\,H_0 {\rm vol}(\partial K)$. In both cases we are able to characterize the equality case. Finally, if $K$ is a convex set in Euclidean space $\mathbb{R}^{n+1}$ which admits a supporting sphere of radius $H_0^{-1}$ at any boundary point, then we prove ${\rm Cap}(K)\geq (n-1)\,H_0 {\rm vol}(\partial K)$ and that equality holds for the round sphere of radius $H_0^{-1}$.
Hurtado Ana
Palmer Vicente
Ritoré Manuel
No associations
LandOfFree
Comparison results for capacity does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Comparison results for capacity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Comparison results for capacity will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-604318