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

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 and thus are valid for general varifolds.

Li–Yau inequalities for the Helfrich functional and applications

“… Assume , satisfy Hypothesis 2.2 . By [ 39 , Theorem 3.6], we find that the density exists and is finite for all . Hence there exist and such that This immediately implies for all .…”

“…Notice that for . In view of [ 39 , Lemma 2.3], there holds Moreover, by [ 18 , 2.4.18] and the area formula (cf. [ 39 , Lemma 2.3]), we have whenever is a nonnegative Borel function.…”

Li-Yau inequalities for the Helfrich functional and applications

“…Since (M, g F ) is flat, we have div g F X * = (div X) • F . Hence, by the divergence theorem for Riemannian manifolds (see [33,Theorem 5.11(2)]) and [36,Lemma 2.3],…”

“…This inequality holds true for all 2-varifolds in R 3 with generalized perpendicular mean curvature, finite Willmore energy, and whose weight measure is finite and has connected support (see [36,Theorem 1.5]). Hence, by (2.5) we obtain…”

“…for all ρ > 0. Hence, since θ 2 (µ V , x 0 ) ≥ 1 by [36,Theorem 3.6] in combination with Remark 2.3, we can let σ go to zero in Lemma 4.1 and use (4.13), (4.14) to infer…”