The "Hot Spots" Conjecture on Homogeneous Hierarchical Gaskets

Abstract

In this paper, using spectral decimation, we prove that the "hot spots" conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly, i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian (introduced by Kigami) attains its maximum and minimum on the boundary.