La g\'eom\'etrie de Bakry-\'Emery et l'\'ecart fondamental

TL;DR

This paper explores the relationship between Bakry-Emery geometry and the fundamental gap, highlighting recent proofs and results for various geometric shapes, emphasizing the interplay between eigenvalues and geometric properties.

Contribution

It provides a concise overview of the connections between Bakry-Emery geometry and the fundamental gap, including recent proof techniques and specific results for triangles and simplices.

Findings

Connections between Laplacian eigenvalues under different boundary conditions.

Key ideas from Andrews and Clutterbuck's proof of the fundamental gap conjecture.

Results on the fundamental gap for triangles and simplices.

Abstract

This article is a brief presentation of results surrounding the fundamental gap. We begin by recalling Bakry-Emery geometry and demonstrate connections between eigenvalues of the Laplacian with the Dirichlet and Neumann boundary conditions. We then show a connection between the fundamental gap and Bakry-Emery geometry, concluding with a presentation of the key ideas in Andrews's and Clutterbuck's proof of the fundamental gap conjecture. We conclude with a presentation of results for the fundamental gap of triangles and simplices.