Isoperimetric inequalities on curved surfaces

Abstract

In this paper we extend the solutions of Lord Rayleigh’s and St. Venant’s conjectures to bounded simply connected domains on curved 2-dimensional Riemannian manifolds. The conjectures investigate the relation of the fundamental tone of a vibrating membrane with fixed boundary and the torsional rigidity of cylindrical beams to the respective areas of the membrane and the cross section of the beam. Both problems are related to the isoperimetric inequality relating the area of a bounded domain to the length of its boundary, and indeed the isoperimetric inequality will be the starting point of our work. To state our results we require some definitions. M will denote a 2-dimensional manifold with complete Ck, K > 2, Riemannian metric, {In, ,..., Q,} will be a collection of pairwise disjoint bounded simply connected domains in M, such that for each j = l,..., m the boundary of Qj , r, , is a simply closed continuous, piecewise regular curve in M (by regular we mean Ck, K > 1, and of maximal rank). We let Q = UE, szj , and F = uj”=, I’j be the boundary of 8. Let A denote the area of B and L the length of r with respect to the given Riemannian metric. We denote the Gauss curvature function of the Riemannian metric by K: M + R.