Level sets of Neumann eigenfunctions Rodrigo BanuelosMichael Pang 35P0535J2535J05Neumann eigenfunctionslevel setsconnected components In this paper we prove that the level sets of the first non--constant eigenfunction of the Neumann Laplacian on a convex planar domain have only finitely many connected components. This problem is motivated, in part, by the "hot spots" conjecture of J. Rauch. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2808 10.1512/iumj.2006.55.2808 en Indiana Univ. Math. J. 55 (2006) 923 - 940 state-of-the-art mathematics http://iumj.org/access/