We show that for every simple closed curve , the extremal length and the hyperbolic length of  are quasi-convex functions along any Teichm¨uller geodesic. As a corollary, we conclude that, in Teichm¨uller space equipped with the Teichm¨uller metric, balls are quasi-convex.

In this paper we start the classification of strongly bracket-generated sub-symmetric spaces. We prove a structure theorem and analyze the nilpotent case.

Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann–Petty centroid inequality.

Let f : X .仺 Y be a holomorphic map of complex manifolds, which is proper, K丯ahler, and surjective with connected fibers, and which is smooth over Y \ Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X. In our previous paper, we discussed the Nakano semi-positivity of Rqf(KX/Y . E) for q . 0 with respect to the so-called Hodge metric, when the map f is smooth. Here we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(Rqf(KX/Y . E)) 乬over Y \ Z乭 as a singular Hermitian metric with semi-positive curvature 乬over Y 乭. As a particular consequence, if Y is projective, Rqf(KX/Y . E) is weakly positive over Y \ Z in the sense of Viehweg.

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.