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.

If M is a hyperbolic 3-manifold with a quasigeodesic flow, then we show that 1(M) acts in a natural way on a closed disc by homeomorphisms. Consequently, such a flow either has a closed orbit or the action on the boundary circle is M¨obius-like but not conjugate into PSL(2,R).We conjecture that the latter possibility cannot occur.

We consider complex projective space with its Fubini–Study metric and the X-ray transform defined by integration over its geodesics. We identify the kernel of this transform acting on symmetric tensor fields.

We classify simply connected compact Sasaki manifolds of dimension 2 n+ 1 with positive transverse bisectional curvature. In particular, the Khler cone corresponding to such manifolds must be bi-holomorphic to C n+ 1\{0}. As an application we recover the theorem of Mori and SiuYau on the Frankel conjecture and extend it to certain orbifold version. The main idea is to deform such Sasaki manifolds to the standard round sphere in two steps, both fixing the complex structure on the Khler cone. First, we deform the metric along the SasakiRicci flow and obtain a limit SasakiRicci soliton with positive transverse bisectional curvature. Then by varying the Reeb vector field which essentially decreases the volume functional, we deform the SasakiRicci soliton to a SasakiEinstein metric with positive transverse bisectional curvature, ie a round sphere. The second deformation is only possible when one treats

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