Minimal Submanifolds of Higher Codimension

In this series of lectures we will introduce methods for handling problems in Riemannian geometry involving curvature. These methods are especially effective in handling positive curvature, but they also motivate questions for minimal submanifolds in euclidean space. The theory of minimal hypersurfaces is particularly important for positive scalar curvature including questions in General Relativity (see [25] for a survey of this topic). Much of this paper concerns the second variation and variational existence questions in arbitrary codimension. In Section 2 we introduce the basic ideas and consider questions involving sectional curvature and geodesics. We illustrate how lower estimates on the Morse index may be combined with existence theory to derive geometric conclusions. In addition to the case of closed geodesics and the fixed endpoint problem we also consider the free boundary problem for geodesics and some of its consequences. In Section 3 we consider mainly the case of surfaces in arbitrary manifolds and show how the conditions of positive complex sectional curvature and PIC arise naturally from the second variation in complex form. We note that after these lectures were given, S. Brendle and the author ([4], [5]) were able to show that PIC is preserved by the Ricci flow and as a consequence to show that positive pointwise quarter-pinched manifolds are diffeomorphic to spherical space