Two-dimensional Space Forms Research Articles

On an inverse curvature flow in two-dimensional space forms

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetric inequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the $$n=2$$ case of a conjecture of Girão–Pinheiro.

On reverse isoperimetric inequalities in two-dimensional space forms and related results

On reverse isoperimetric inequalities in two-dimensional space forms and related results

Area preserving transformations in two-dimensional space forms and classical differential geometry

We generalize Scheffers’ method to construct area preserving transformations in the Euclidean plane to Riemannian and Lorentzian two-dimensional space forms in a unified way. We review and extend the classical applications of such transformations in classical differential geometry. We introduce two classes of surfaces in Lorentzian 3-space that admit holomorphic representation and are analogous to the classical Appell and Bonnet surfaces.

Focal sets in two-dimensional space forms

Focal sets in two-dimensional space forms

Tiling the torus and other space forms

We consider graphs on two-dimensional space forms which are quotient graphs Γ/F, where Γ is an infinite, 3-connected, face, vertex, or edge transitive planar graph andF is a subgroup of Aut(Γ), all of whose elements act freely on Γ. The enumeration of quotient graphs with transitivity properties reduces to computing the normalizers in Aut(Γ) of the subgroupsF. Results include: all isogonal toriodal polyhedra belong to the two families found by Grunbaum and Shephard; there are no transitive graphs on the Mobius band; there is a graph on the Klein bottle whose automorphism group acts transitively on its faces, edges, and vertices.