No 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.

It is shown that if a C 2 surface M ⊂ R 3 has negative curvature on the complement of a point q ∈ M , then the Z/2-valued Poincaré-Hopf index at q of either distribution of principal directions on M − {q} is non-positive. Conversely, any nonpositive half-integer arises in this fashion. The proof of the index estimate is based on geometric-topological arguments, an index theorem for symmetric tensors on Riemannian surfaces, and some aspects of the classical Poincaré-Bendixson theory.

The Bieberbach estimate, a pivotal result in the classical theory of univalent functions, states that any injective holomorphic function f on the open unit disc D satisfies |f ′′ (0)| ≤ 4|f ′ (0)|. We generalize the Bieberbach estimate by proving a version of the inequality that applies to all injective smooth conformal immersions f : D → R n , n ≥ 2. The new estimate involves two correction terms. The first one is geometric, coming from the second fundamental form of the image surface f (D). The second term is of a dynamical nature, and involves certain Riemannian quantities associated to conformal attractors. Our results are partly motivated by a conjecture in the theory of embedded minimal surfaces.

Abstract. Let M n be an entire graph in the Euclidean (n + 1)-space R n+1 . Denote by H, R and |A|, respectively, the mean curvature, the scalar curvature and the length of the second fundamental form of M n . We prove that if the mean curvature H of M n is bounded, then inf M |R| = 0, improving results of Elbert and Hasanis-Vlachos. We also prove that if the Ricci curvature of M n is negative, then inf M |A| = 0. The latter improves a result of Chern as well as gives a partial answer to a question raised by Smith-Xavier. Our technique is to estimate inf |H|, inf |R| and inf |A| for graphs in R n+1 of C 2 real-valued functions defined on closed balls in R n .