Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

2019/07/18 by Hurtado, Ana, Palmer, Vicente, Rosales, César · 1 citation
#31C12 #35J25 #53C42 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1907.07920

Abstract

Let (M,g) be a complete non-compact Riemannian manifold together with a function e^h, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of M to deduce comparisons for the weighted isoperimetric quotient and the weighted capacity of metric balls in M centered at a point o∈ M. As a consequence, we obtain parabolicity and hyperbolicity criteria for weighted manifolds generalizing previous ones. A basic tool in our study is the analysis of the weighted Laplacian of the distance function from o. The technique extends to non-compact submanifolds properly immersed in M under certain control on their weighted mean curvature.

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