The Gap of the Consecutive Eigenvalues of the Drifting Laplacian on Metric Measure Spaces

TL;DR

This paper derives bounds on the gaps between consecutive eigenvalues of the drifting Laplacian on various metric measure spaces, with applications to Ricci solitons, self-shrinkers, and product manifolds, proposing a related conjecture.

Contribution

It establishes general formulas for eigenvalue gaps of the drifting Laplacian and provides explicit upper bounds in specific geometric contexts, advancing understanding of spectral properties.

Findings

Derived sharp upper bounds for eigenvalue gaps on metric measure spaces.

Applied bounds to Ricci solitons, self-shrinkers, and product manifolds.

Proposed a conjecture relating eigenvalue gaps to the first two eigenvalues and dimension.

Abstract

In this paper, we investigate eigenvalues of the Dirichlet problem and the closed eigenvalue problem of drifting Laplacian on the complete metric measure spaces and establish the corresponding general formulas. By using those general formulas, we give some upper bounds of consecutive gap of the eigenvalues of the eigenvalue problems, which is sharp in the sense of the order of the eigenvalues. As some interesting applications, we study the eigenvalue of drifting Laplacian on Ricci solitons, self-shrinkers and product Riemannian manifolds. We give the explicit upper bounds of the gap of the consecutive eigenvalues of the drifting Laplacian. Since eigenvalues is invariant in the sense of isometry, by the classifications of Ricci solitons and self-shrinkers, we give the explicit upper bounds for the consecutive eigenvalues of the drifting Laplacian on a large class metric measure spaces. In addition, we also consider the case of product Riemannian manifolds with certain curvature conditions and some upper bounds are obtained. Basing on the case of Laplace operator, we also present a conjecture as follows: all of the eigenvalues of the Dirichlet problem of drifting Laplacian on metric measure spaces satisfy: $$\lambda_{k+1}-\lambda_{k}\leq(\lambda_{2}-\lambda_{1})k^{\frac{1}{n}}.$$We note the conjecture is true in some special cases.