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

Klassert, Steffen; Lenz, Daniel; Peyerimhoff, Norbert; Stollmann, Peter

doi:10.1090/S0002-9939-05-08103-7

Elliptic operators on planar graphs: Unique continuation for eigenfunctions and nonpositive curvature
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by Steffen Klassert, Daniel Lenz, Norbert Peyerimhoff and Peter Stollmann PDF

Proc. Amer. Math. Soc. 134 (2006), 1549-1559 Request permission

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

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