Eigenvalue estimates for a class of elliptic differential operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

TL;DR

This paper derives eigenvalue estimates for a broad class of elliptic operators on Riemannian manifolds immersed in Euclidean space, with applications to Gaussian shrinking solitons and insights into Pólya's conjecture.

Contribution

It extends eigenvalue estimates to a larger class of elliptic operators on immersed manifolds and explores their behavior in specific geometric contexts.

Findings

Eigenvalue bounds for elliptic operators in divergence form.

Application to Gaussian shrinking solitons.

Insights into Pólya's conjecture.

Abstract

In this paper, we obtain eigenvalue estimates for a larger class of elliptic differential operators in divergence form on a bounded domain in a complete Riemannian manifold isometrically immersed in Euclidean space. As an application, we give eigenvalue estimates in the Gaussian shrinking soliton, and we find a domain that makes the behavior of these estimates similar to the estimates for the case of the Laplacian. Moreover, we also give an answer to the generalized conjecture of P\'olya.