1 Preliminaries 3 1.1 Linear algebra 3 1.1. 1 Vectors and matrices 3 1.1. 2 Symmetric bilinear forms 5 1.1. 3 Vector subspaces 6 1.1. 4 Linear maps from Rn to Rn 7 1.1. 5 A convention 9 1.2 Vector calculus 9

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasilinear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial.

In a series of works, one of the authors has developed with J.- M. Hwang a geometric theory of uniruled projective manifolds, especially those of Picard number 1, basing on the study of varieties of minimal rational tangents. A fundamental result in this theory is a principle of analytic continuation under very mild assumptions,called Cartan-Fubini extension, of biholomorphisms between connected open subsets of two Fano manifolds of Picard number 1 which preserve varieties of minimal rational tangents. In this article we develop a generalization of Cartan-Fubini extension for non-equidimensional holomorphic immersions from a connected open subset of a Fano manifold of Picard number 1 into a uniruled projective manifold, under the assumptions that the map sends varieties of minimal rational tangents onto linear sections of varieties of minimal rational tangents and that it satisfies a mild geometric condition formulated in terms of second fundamental forms on varieties of minimal rational tangents. Formerly such a result was known only in the very special case of irreducible Hermitian symmetric manifolds of rank at least two, and the proof relied on the existence of flattening coordinates, viz.,Harish-Chandra coordinates, with respect to which the varieties of minimal rational tangents form a constant family. The proof of the main result, which is based on the deformation theory of rational curves, is differential-geometric in nature and is applicable to the general situation of uniruled projective manifolds without any assumption on the existence of special coordinate systems. As an application, we give a characterization of standard embeddings for certain pairs of rational homogeneous manifolds in terms of embeddings of varieties of minimal rational tangents.

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In \cite{rigidity}, Luo introduced a $\psi_{\lambda}$ edge invariant which turns out to be a coordinate of the Teichm\"uller space of a surface with boundary. And he proved that for $\lambda \geq 0$, the image of the Teichm\"uller space under $\psi_{\lambda}$ edge invariant coordinate is an open cell. In this paper we verify his conjecture that for $\lambda<0$, the image of the Teichm\"uller space is a bounded convex polytope.