Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds. (English) Zbl 1448.53044
Summary: 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 \(o \in M\). 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.
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
Keywords:
weighted Laplacian; weighted sectional curvature; weighted capacities and parabolicity; isoperimetric quotient; capacity; parabolicity; submanifolds; smooth metric measure spaceReferences:
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