# Sharp Estimates for the First Eigenvalues of the Bi-drifting Laplacian

## TL;DR

This paper derives sharp lower bounds for the first eigenvalues of the bi-drifting Laplacian operator on compact manifolds with boundary, considering weighted Ricci curvature bounds, advancing spectral geometry understanding.

## Contribution

It provides new sharp estimates for the first eigenvalues of the bi-drifting Laplacian on smooth metric measure spaces with boundary, under curvature constraints.

## Key findings

- Established sharp lower bounds for the first eigenvalue.
- Extended eigenvalue estimates to manifolds with boundary.
- Connected eigenvalue bounds to weighted Ricci curvature.

## Abstract

In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact manifolds with boundary (also called a smooth metric measure space) and weighted Ricci curvature bounded inferiorly.