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

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

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

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Description

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.