Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds

Abstract

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.

General Reilly-type inequalities for submanifolds of weighted Euclidean spaces

“…As a corollary, we obtain a similar inequality for submanifolds of the sphere S N which generalizes the corresponding inequality of [4] and [6] for the operator L T,f (see Corollary 4.4). We also prove a general non-weighted Reilly-type inequality (Theorem 5.1).…”

“…In [4], Batista, Cavalcante and Pyo proved the following upper bound for the first positive eigenvalue of ∆ f :…”

Reilly-Type Inequalities for Paneitz and Steklov Eigenvalues

“…Moreover, Reilly studied the equality cases and proved that equality in (1) as in (2) is attained if and only if X(M ) is a geodesic sphere. These inequalities have been generalized for other ambient spaces [18,20], other operators, in particular of Jacobi type [1,4,7], in the anisoptropic setting [32] or for weighted ambient spaces [8,17,33]. In particular, in [33], we prove the following general inequality…”

Local rigidity, bifurcation, and stability of $H_f$-hypersurfaces in weighted Killing warped products

“…So, let µ 1 be the first eigenvalue of ∆ f on Σ n . If µ = µ 1 , then the variational characterization of λ 1 (see, for instance, Section 1 of [7]) gives…”