Spectral and Nodal Properties of Static Manifolds with Boundary

Abstract

We study compact static manifolds with boundary from a spectral point of view, with particular emphasis on the relation between Dirichlet, Neumann, and Robin spectra and the nodal structure of static potentials. We establish comparison inequalities linking Dirichlet and Neumann eigenvalues with the scalar curvature and derive sufficient spectral conditions ensuring that a static potential changes sign. We also obtain bounds on the number of connected components of the zero set, sufficient conditions for its connectedness, and sufficient conditions ensuring that its components intersect the boundary. Finally, we study variational properties of Robin static triples and characterize compact Robin static triples whose metrics are critical for the first mixed Robin--Dirichlet eigenvalue under admissible volume-preserving deformations.