The rigidity of eigenfunctions' gradient estimates

TL;DR

This paper demonstrates that if a sharp gradient estimate for eigenfunctions holds at certain points on a compact Riemannian manifold with boundary, then the manifold must be isometric to a product space involving a lower-dimensional manifold and a circle or line segment.

Contribution

It establishes a rigidity result linking the equality case of gradient estimates to the geometric structure of the manifold.

Findings

Manifold is isometric to a product space under certain gradient estimate conditions.

The result applies to manifolds with non-negative Ricci curvature and convex boundary.

Gradient estimate equality implies strong geometric constraints.

Abstract

On compact Riemannian manifolds with non-negative Ricci curvature and smooth (possibly empty), convex (or mean convex) boundary, if the sharp Li-Yau type gradient estimate of an Neumann (or Dirichlet) eigenfunction holds at some non-critical points of the eigenfunction; we show that the manifold is isometric to the product of one lower dimension manifold and a round circle (or a line segment).