Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

TL;DR

This paper establishes conditions under which eigenfunction boundary data can reach maximal sup norms on manifolds with concave boundaries, linking geometric properties like self-focal points to eigenfunction behavior.

Contribution

It proves that maximal sup norm bounds of eigenfunction boundary data occur only at self-focal points, and shows such bounds are not achieved in manifolds with non-positive curvature.

Findings

Maximal sup norms occur only at self-focal boundary points.

In non-positively curved manifolds with concave boundary, eigenfunctions do not reach maximal sup bounds.

Boundary traces of eigenfunctions are constrained by geometric properties.

Abstract

We prove that the Cauchy data of Dirichlet or Neumann $\Delta$- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e. a point at which a positive measure of geodesics leaving the point return to the point. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds.