Elliptic operators on planar graphs: Unique continuation for eigenfunctions and nonpositive curvature

S. Klassert, D. Lenz, N. Peyerimhoff, P. Stollmann

Published 2004-10-07Version 1

This paper is concerned with elliptic operators on plane tessellations. We show that such an operator does not admit a compactly supported eigenfunction, if the combinatorial curvature of the tessellation is nonpositive. Furthermore, we show that the only geometrically finite, repetitive plane tessellations with nonpositive curvature are the regular $(3,6), (4,4)$ and $(6,3)$ tilings.