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Geodesics and nodal sets of Laplace eigenfunctions on hyperbolic manifolds

Authors: Chris Judge and Sugata Mondal
Journal: Proc. Amer. Math. Soc. 145 (2017), 4543-4550
MSC (2010): Primary 58J50
Published electronically: April 4, 2017
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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.

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Additional Information

Chris Judge
Affiliation: Department of Mathematics, Rawles Hall, Indiana University, Bloomington, Indiana 47405-7106

Sugata Mondal
Affiliation: Department of Mathematics, Rawles Hall, Indiana University, Bloomington, Indiana 47405-7106

Received by editor(s): January 21, 2016
Received by editor(s) in revised form: June 27, 2016, and October 19, 2016
Published electronically: April 4, 2017
Additional Notes: The work of the first author was supported in part by a Simons Collaboration Grant
Communicated by: Michael Wolf
Article copyright: © Copyright 2017 American Mathematical Society

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