Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces

TL;DR

This paper constructs specific metrics on the unit disc where the second eigenfunction of the Laplacian has a closed nodal line and attains its maximum inside, providing a counterexample to the hot spots conjecture.

Contribution

It introduces a family of S^{1}-invariant metrics on the disc with controlled eigenvalues, showing the existence of eigenfunctions with closed nodal lines and interior maxima.

Findings

Second eigenfunction has a closed nodal line.

Eigenfunction attains maximum at an interior point under Neumann conditions.

Any finite set of invariant eigenvalues can be made arbitrarily small within the family.

Abstract

We build a one-parameter family of S^{1}-invariant metrics on the unit disc with fixed total area for which the second eigenvalue of the Laplace operator in the case of both Neumann and Dirichlet boundary conditions is simple and has an eigenfunction with a closed nodal line. In the case of Neumann boundary conditions, we also prove that this eigenfunction attains its maximum at an interior point, and thus provide a counterexample to the hot spots conjecture on a simply connected surface. This is a consequence of the stronger result that within this family of metrics any given (finite) number of S^{1}-invariant eigenvalues can be made to be arbitrarily small, while the non-invariant spectrum becomes arbitrarily large.