Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds

Abstract

Let (M,g) denote a smooth (say C 3 ) compact two-dimensional manifold, equipped with some Riemannian metric g.Then, as is well-known, M admits a metric g c of constant Gaussian curvature c in fact the metrics g and g c can be chosen to be conformally equivalent.Here, we determine sufficient conditions for a given non-simply connected manifold M to admit a Riemannian structure g (conformally equivalent to g) with arbitrarily prescribed (Holder continuous) Gaussian curvature K(x).If the Euler-Poincare characteristic χ(M) of M is negative, the sufficient condition we obtain is that K(x) < 0 over M. Note that this condition is independent of g, and this result is obtained by solving an isoperimetric variational problem for g.If K(x) is of variable sign for χ(M) < 0, or if χ(M) > 0, then the desired critical point may not be an absolute minimum and our methods do not succeed.If χ(M) = 0, our methods apply when K(x) satisfies an integral condition with respect to the given metric g (see § 3) this result is perhaps not unreasonable since, for χ(M) < 0, distinct Riemannian structures on M need not be conformally equivalent.Setting γ' = γ exp 2σ, in place of γ in (1), we obtain the desired equation Communicated by I. M. Singer,

Locations

• Journal of Differential Geometry - View - PDF