General estimate of the first eigenvalue on manifolds

TL;DR

This paper reviews and compares ten sharp lower bounds for the first non-trivial Laplacian eigenvalue on compact Riemannian manifolds, introducing improved formulas and providing a comprehensive overview of current estimates.

Contribution

It introduces an improved variational formula, a general estimate, and a new sharp estimate, updating the best known lower bounds for eigenvalues on manifolds.

Findings

Updated the best lower estimates for eigenvalues.

Provided a global perspective on eigenvalue bounds.

Compared ten sharp estimates and introduced new ones.

Abstract

Ten sharp lower estimates of the first non-trivial eigenvalue of Laplacian on compact Riemannian manifolds are reviewed and compared. An improved variational formula, a general common estimate, and a new sharp one are added. The best lower estimates are now updated. The new estimates provide a global picture of what one can expect by our approach.