2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/57846This 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.Mathematical PhysicsFunctional Analysis81Q10; 35J10; 82B44text