We consider inverse curvature ows in Hn+1 with star-shaped initial hypersurfaces and prove that the ows exist for all time, and that the leaves converge to innity, become strongly convex exponentially fast and also more and more totally umbilic. Afteran appropriate rescaling the leaves converge in C1 to a sphere.

We show that the attenuated geodesic ray transform on two dimensional simple surfaces is injective. Moreover, we give a stability estimate and develop a reconstruction procedure.

By using Yau's Schwarz lemma and the Quillen determinant line bundles, several results about fibered algebraic surfaces and the moduli spaces of curves are improved and reproved.

We prove that a compact Riemannian 3-symmetric space is globally symmetric if every Jacobi field along a geodesic vanishing at two points is the restriction to that geodesic of a Killing field induced by the isotropy action or, in particular, if the isotropy action is variationally complete.

We introduce and study an approximate solution of the p-Laplace equation, and a linearlization L_{ε} of a perturbed p-Laplace operator. By deriving an L_{ε}-type Bochner's formula and Kato type inequalities, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p-harmonic functions with finite q-energy on Riemannian manifolds. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type Theorem for p-harmonic morphisms.