Characterization of minimal submanifolds by total Gauss curvature

The main goal of this paper is to investigate necessary and sufficient conditions for finite total Gauss curvature of a complete, connected, embedded minimal submanifold of dimension two or greater in Euclidean N-space; such results may then be used to further characterize the submanifolds in question. We prove that any smooth, complete, real algebraic submanifold has finite total Gauss curvature. We also show that any smooth, complete, connected, orientable manifold which conformally embeds with smoothly embedded ends into some compact Riemannian manifold of the same dimension has finite total Gauss curvature.