Extremal properties of the principal Dirichlet eigenvalue for regular polygons in the hyperbolic plane (Q1849628)

The authors show that, in the hyperbolic plane, the equilateral triangle is the unique triangle with minimal principal Dirichlet eigenvalue amongst the hyperbolic triangles of equal finite perimeter. The proof is based on a generalization of Steiner symmetrization to the hyperbolic plane and on the fact that the convex hull of a symmetrized hyperbolic triangle is still a triangle. Moreover, the paper describes the locus of all points \(P\) such that, for two given points \(P_1\) and \(P_2\) in the hyperbolic plane, the triangles with vertices, \(P_1, P_2\) and \(P\) have a prescribed area. Besides, the authors identify the unique hyperbolic quadrilateral with minimal principal Dirichlet eigenvalue amongst those in a given geodesic ball.