Isoperimetric inequalities for submanifolds with bounded mean curvature

Abstract

In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature.

Lpharmonic 1-forms and first eigenvalue of a stable minimal hypersurface

“…Proof. As mentioned in the introduction, one sees that the lower bound of λ 1 (M) is given as −K 2 (n − 1) 2 /4 from inequality (1) [Bessa and Montenegro 2003;Seo 2012]. Namely, the first eigenvalue of an n-dimensional minimal hypersurface in a complete simply connected Riemannian manifold with sectional curvature bounded above by a negative constant K 2 is bounded below by −K 2 (n − 1) 2 /4.…”

“…Theorem [Bessa and Montenegro 2003;Seo 2012]. Let N be an n-dimensional complete simply connected Riemannian manifold with sectional curvature K N satisfying K N ≤ −a 2 < 0 for a positive constant a > 0.…”

Optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space

“…See [1,7,15,20,25,27] for isoperimetric inequalities involving mean curvature in the more general setting of arbitrary submanifolds. In this paper we obtain optimal isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space H n .…”

“…See [1,6,14,19,24,26] for isoperimetric inequalities involving mean curvature in the more general setting of arbitrary submanifolds.…”

Isoperimetric inequalities and monotonicity formulas for submanifolds in warped products manifolds

“…Remark 1.4. It is possible to obtain another proof of Corollary 1.2 from the proofs of Theorem 6 (a), p. 185, of [7], for H = 0 and Corollary 3.4, p.533, of [16], for arbitrary H.…”