Geodesics and nodal sets of Laplace eigenfunctions on hyperbolic manifolds

Judge, C. and Mondal, S. ORCID: https://orcid.org/0000-0002-2236-971X (2017) Geodesics and nodal sets of Laplace eigenfunctions on hyperbolic manifolds. Proceedings of the American Mathematical Society, 145 (10). pp. 4543-4550. ISSN 1088-6826 doi: 10.1090/proc/13544

Abstract/Summary

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.