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

TL;DR

This paper investigates elliptic operators on planar tessellations, proving the absence of compactly supported eigenfunctions under nonpositive curvature and classifying certain repetitive tessellations with this curvature condition.

Contribution

It establishes a unique continuation property for eigenfunctions on tessellations with nonpositive curvature and classifies specific regular tessellations under these conditions.

Findings

No compactly supported eigenfunctions exist under nonpositive curvature.

Only three types of regular tessellations are geometrically finite and repetitive with nonpositive curvature.

Provides a classification of tessellations based on curvature and geometric finiteness.

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.