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

TL;DR

This paper develops geometric inequalities for Riemannian submanifolds using monotonicity identities derived from Rauch's comparison theorem, leading to diameter, Sobolev, and isoperimetric bounds that are intrinsic and applicable to varifolds.

Contribution

It introduces a general Li-Yau inequality and related geometric bounds for submanifolds in Riemannian manifolds with bounded curvature, extending previous results to a broader setting.

Findings

Established monotonicity inequalities for Riemannian submanifolds.

Derived diameter bounds for minimal submanifolds.

Obtained Sobolev and isoperimetric inequalities for small-volume submanifolds.

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.