Extending minimal varieties

Abstract

Introduction The purpose of this paper is to study the general problem of extending minimal varieties across closed subsets in riemannian manifolds. The analogous theory in the complex analytic case falls into two categories: extension theorems for functions (Hartog's phenomenon) and extension theorems for varieties (the RemmertStein-Shiffman Theorem). The discussion here splits along similar lines. In w 1 we analyze the Bers, deGiorgi-Stampacchia results on removing singularities of solutions to the non-parametric minimal surface equation in codimension one. This theorem does not carry over directly to higher codimension. However, certain parts of their argument can be generalized and lead to a strong reflection principle for minimal submanifolds and a uniqueness result for minimal cones. Also a theorem for surfaces in general codimension is proved. The remainder of the paper is devoted to proving extension theorems of the following type for varieties. Let M be a riemannian manifold and A ~ M a compact subset of sufficient smoothness and appropriate Hausdorff dimension. Then any k-dimensional minimal variety which is stationary (or area minimizing) in M A extends to a minimal variety which is stationary (or, resp., area minimizing) in M. The proof falls into two parts. The first is to show that the mass of the variety is finite across A so that an extension exists (w 3). The second is to prove the properties of minimality for the extension. The main conclusions are stated precisely in w 5.