On Plateau's Problem for Minimal Surfaces of Higher Genus in R by Friedrich Tomi and Anthony Tromba

The classical solution of the Plateau problem by Radó [10] and Douglas [3] shows that any rectifiable Jordan curve in R is spanned by a minimal surface of disc type. Under what conditions a minimal surface of a given higher genus exists, spanning a given Jordan curve in a Riemannian manifold N, seems to be a much more difficult problem. For compact minimal surfaces without boundary and in case N has sufficient topological complexity, the "incompressibility" method of Schoen and Yau gives a sufficient condition for existence. In [4] Douglas did develop a method to treat the problem of when a given contour is spanned by a surface of genus p. Douglas' condition, however, seems quite difficult to verify in concrete cases. In this paper we will give simple geometric and topological sufficient conditions. THEOREM. Let N be a solid torus of class C and genus g in R whose boundary has nonnegative inward mean curvature, and let 7 6 Ili(iV) denote the homotopy class of a rectifiable Jordan curve T in N. (a) If g = 2p and 7 = aia2aj~a^" • • • a2p-ia2pa2p_iU2p where 0 1 , . . . ,a2p is a basis for Ili(iV) thenT bounds an immersed oriented minimal surface of genus p. (b) If g = 1 and 7 = 2a for some a j=0 in Hi(N) then T bounds an immersed minimal surface of Mobius type.