Calculus of Variations and Geometric Measure Theory
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C. Scharrer

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

created by scharrer on 01 Jun 2021

[BibTeX]

preprint

Inserted: 1 jun 2021

Year: 2021

ArXiv: 2105.13211 PDF

Abstract:

Using Rauch's comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li-Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch's comparison theorem, the proofs mainly rely on the first variation formula, thus are valid for general varifolds.

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