Geodesics and nodal sets of Laplace eigenfunctions on hyperbolic manifolds

TL;DR

This paper investigates the properties of Laplace eigenfunctions on hyperbolic manifolds, proving finiteness results for certain geodesic hypersurfaces and providing constructions of specific nodal geodesics.

Contribution

It establishes the finiteness of totally geodesic hypersurfaces in the zero set of eigenfunctions and constructs examples of nodal geodesics without symmetry.

Findings

Finiteness of totally geodesic hypersurfaces in eigenfunction zero sets.

Bound on the number of such hypersurfaces in hyperbolic surfaces based on area.

Existence of geodesics in nodal sets lacking symmetry.

Abstract

Let X be a manifold equipped with a complete Riemannian metric of constant negative curvature and finite volume. We demonstrate the finiteness of the collection of totally geodesic immersed hypersurfaces in X that lie in the zero-level set of some Laplace eigenfunction. For surfaces, we show that the number can be bounded just in terms of the area of the surface. We also provide constructions of geodesics in hyperbolic surfaces that lie in a nodal set but that do not lie in the fixed point set of a reflection symmetry.