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

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.

This result is used in the article arXiv:1401.4520 of the second author with J.Jung on counting nodal domains on surfaces of non-positive curvature and concave boundary

References in corpus (1)

• Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary

Cited by in corpus (1)

• Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary