Weighted Cheeger constant and first eigenvalue lower bound estimates on smooth metric measure spaces

Abimbola Abolarinwa,Akram Ali,Ali Alkhadi

doi:10.1186/s13662-021-03431-8

Abimbola Abolarinwa, Akram Ali + Show 1 more

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https://doi.org/10.1186/s13662-021-03431-8

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Abstract

We establish new eigenvalue inequalities in terms of the weighted Cheeger constant for drifting p-Laplacian on smooth metric measure spaces with or without boundary. The weighted Cheeger constant is bounded from below by a geometric constant involving the divergence of suitable vector fields. On the other hand, we establish a weighted form of Escobar–Lichnerowicz–Reilly lower bound estimates on the first nonzero eigenvalue of the drifting bi-Laplacian on weighted manifolds. As an application, we prove buckling eigenvalue lower bound estimates, first, on the weighted geodesic balls and then on submanifolds having bounded weighted mean curvature.

Highlights

The first aim of this paper is to answer in the affirmative that there exists a weighted form of Cheeger constant for domains in complete smooth metric measure spaces
The determination of Cheeger constant on a domain aids the process of solving isoperimetric problems and the classical Cheeger problems
We show that the weighted Cheeger constant is bounded from below by some other geometric constant involving the divergence of suitable vector fields

Summary

Introduction

The first aim of this paper is to answer in the affirmative that there exists a weighted form of Cheeger constant for domains in complete smooth metric measure spaces. Futaki, Li, and Li [20] obtained a lower bound on the first eigenvalue of drifting Laplacian on a compact Riemannian manifold without boundary under the assumption that the generalized Ricci curvature is bounded from below.

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