We prove theorems on the structure of the fundamental group of a compact riemannian manifold of non-positive curvature. In particular, a conjecture of J. Wolf [<i>J. Differential Geometry</i>, 2, 421-446 (1968)] is proved.

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.

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.

In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a minimal submanifold $\sum$  is the graph of a (strictly) distance-decreasing map, then  $\sum$ is (strictly) stable. It is known that a minimal graph of codimension one is stable without assuming the distance-decreasing condition. We give another criterion for the stability in terms of the two-Jacobians of the map which in particular covers the codimension one case. All theorems are proved in the more general setting for minimal maps between Riemannian manifolds. The complete statements of the results appear in Theorem 3.1, Theorem 3.2, and Theorem 4.1.

The limiting behavior of the normalized KÄahler-Ricci °ow for manifolds with positive ¯rst Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi Kenergy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabuchi Kenergy is bounded from below and if the lowest positive eigenvalue of the ¹@y ¹@ operator on smooth vector ¯elds is bounded away from0 along the °ow, then the metrics converge exponentially fast in C1 to a KÄahler-Einstein metric.