Good Shadows, Dynamics and Convex Hulls of Complete Submanifolds

Abstract

Abstract. Any non-empty open convex subset of R n 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 C 2 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 C 2 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 introduce a new approach to asymptotic maximum principles, based on 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.

Some Remarks on the Pigola–rigoli–setti Version of the Omori–yau Maximum Principle

“…The proof below was inspired by the proof of a conceptual refinement of the OmoriYau maximum principle stated in [7]. P  T 1.2.…”

Finding umbilics on open convex surfaces

“…In [6] an application of the above lemma was given to the study of the convex hull of complete submanifolds. As a corollary, one obtains the sharp result that the boundary of the convex hull of a compact C 1 hypersurface in R n is itself of class C 1 .…”

“…Lemma 3.4 is the first order version of the main result of [6], where it is shown that in the well-known Omori-Yau maximum principle [1], [10], [17], one can find sequences of points resembling local maxima (in terms of properties of the gradient, Hessian and Laplacian) that shadow asymptotically any prescribed maximizing sequence. By way of comparison, the original Omori-Yau maximum principle predicts the existence of some maximizing sequence whose points resemble local maxima.…”

On Minimal Surfaces Satisfying the Omori–yau Principle

“…This principle turned out to be a powerful tool in geometric analysis (see, for instance, [3,5,11,14] for a recent application).…”