Good shadows, dynamics and convex hulls of complete submanifolds

Any non-empty open convex subset of Rn is the convex hull of a complete submanifold M , of any codimension, but there are obstructions if the geometry of M is, a priori, suitably controlled at infinity. In this paper we develop tools to explore the geometry of ∂[Conv(M)] when the Grassmanian-valued Gauss map of M is uniformly continuous, a condition that, in the C2 case, is weaker than bounding the second fundamental form of M . Our proofs are based on the Ekeland variational principle, and on a conceptual refinement of the Omori-Yau asymptotic maximum principle that is of interest in its own right. If the Ricci (sectional) curvature of M is bounded below and f is a C2 function on M that is bounded above, then not only there exists some maximizing sequence for f with good properties, as predicted by the Yau (Omori) principle but, in fact, every maximizing sequence for f can be shadowed by a maximizing sequence that has good properties. This abundance of good shadows allows for information to be localized at infinity, revealing in our geometric setting the relation between the asymptotic behavior of M and the supporting hyperplanes of ∂[Conv(M)] in general position that pass through some fixed boundary point. We also use ideas from dynamics to prove a special case of a conjecture meant to extend our refinement of the Yau maximum principle to manifolds that satisfy a property weaker than inf Ric > −∞. The authors expect that this new understanding of the Omori-Yau principle – in terms of good shadows and localization at infinity – will lead to applications in contexts other than convexity.