We prove that the quotient space of a variationally complete group action is a good Riemannian orbifold. The result is generalized to singular Riemannian foliations without horizontal conjugate points.

We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmuller space (due to Wolpert (1987)). We then estimate this Hessian from below in terms of local quantities and distance along the geodesic. The formula extends to proper arcs on punctured hyperbolic surfaces, and the estimate to laminations. Wolpert’s result that the Thurston metric is a multiple of the Weil-Petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves. We give further applications to upper bounds of the Hessian, especially near pinching loci, recover through a geometric argumentWolpert’s result on the convexity of length to the half-power, and give a lower bound for growth of length in terms of twist.

Let X be a compact complex, not necessarily K¨ahler, manifold of dimension n. We characterize the volume of any holomorphic line bundle L → X as the supremum R of the Monge-Amp`ere masses X Tn ac over all closed positive currents T in the first Chern class of L, where Tac is the absolutely continuous part of T in its Lebesgue decomposition. This result, new in the non-K¨ahler context, can be seen as holomorphic Morse inequalities for the cohomology of high tensor powers of line bundles endowed with arbitrarily singular Hermitian metrics. It gives, in particular, a new bigness criterion for line bundles in terms of existence of singular Hermitian metrics satisfying positivity conditions. The proof is based on the construction of a new regularization for closed (1, 1)-currents with a control of the Monge-Amp`ere masses of the approximating sequence. To this end, we prove a potential-theoretic result in one complex variable and study the growth of multiplier ideal sheaves associated with increasingly singular metrics.

In this paper we prove that given a point p ∈ Mn, where Mn is a closed Riemannian manifold of dimension n, the length of a shortest geodesic loop lp(Mn) at this point is bounded above by 2nd, where d is the diameter of Mn. Moreover, we show that on a closed simply connected Riemannian manifold Mn with a nontrivial second homotopy group there either exist at least three geodesic loops of length less than or equal to 2d at each point of Mn, or the length of a shortest closed geodesic on Mn is bounded from above by 4d.

We study the rigidity results for self-shrinkers in Euclidean space by restriction of the image under the Gauss map. The geometric properties of the target manifolds carry into effect. In the self-shrinking hypersurface situation Theorem 3.1and Theorem 3.2 not only improve the previous results, but also are optimal. In higher codimensional case, using geometric properties of the Grassmannian manifolds (the target manifolds of the Gauss map) we give a rigidity theorem for self-shrinking graphs.