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We establish an interior C 2 estimate for k+ 1 convex solutions to Dirichlet problems of k-Hessian equations. We also use such estimate to obtain a rigidity theorem for k+ 1 convex entire solutions of k-Hessian equations in Euclidean space.

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 → Rn, 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.

In this paper, we observe that the brane functional studied in hep-th/9910245 can be used to generalize some of the works that Schoen and I [4] did many years ago. The key idea is that if a three dimensional manifold M has a boundary with strongly positive mean curvature, the effect of this mean curvature can influence the internal geometry of M. For example, if the scalar curvature of M is greater than certain constant related to this boundary effect, no incompressible surface of higher genus can exist.

This is the first of a series of three papers which prove the fact that a K-stable Fano manifold admits a Khler-Einstein metric. The main result of this paper is that a Khler-Einstein metric with cone singularities along a divisor can be approximated by a sequence of smooth Khler metrics with controlled geometry in the Gromov-Hausdorff sense.