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

Abstract

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 ) (3,6), (4,4) and ( 6 , 3 ) (6,3) tilings.