In joint work with Blaine Lawson, I proved several results about ends of n-manifolds with scalar curvature bounded below by n(n-2)/2. We showed in particular that such an end cannot be a "bad" end. For complete minimal surfaces of finite (Morse) index in Euclidean three-space, this implies that the surface has quadratic area growth, finite total curvature and finite topological type . I also proved a converse, verifying a conjecture of Doris Fischer-Colbrie: a complete immersed minimal surface has finite index if and only if it has finite total curvature . Fischer-Colbrie proved this result independently in Inventiones Math. 82, 121-132.
. The structure of a stable minimal hypersurface near a
singularity (with Blaine Lawson), Proc. Symp. in Pure
Math. 44, 213-237 (1986).
. Index and total curvature of complete minimal surfaces, Proc. Symp. in Pure Math. 44, 207-211 (1986).