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 establish a Bochner-type formula on Alexandrov spaces with Ricci curvature bounded below. Yau’s gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

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We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds$M$with the property that$M$×$R$^{2}has non-negative isotropic curvature.

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