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.

We prove that a constrained Willmore immersion of a 2–torus into the conformal 4–sphere S4 is of “finite type”, that is, has a spectral curve of finite genus, or of “holomorphic type” which means that it is super conformal or Euclidean minimal with planar ends in R4 = S4\{1} for some point 1 2 S4 at infinity. This implies that all constrained Willmore tori in S4 can be constructed rather explicitly by methods of complex algebraic geometry. The proof uses quaternionic holomorphic geometry in combination with integrable systems methods similar to those of Hitchin’s approach [19] to the study of harmonic tori in S3.

We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future asymptotically flat null cone, and show that the solution exists uniformly around the null cone for general such initial data. Therefore, the solution contains a piece of the future null infinity. The initial data are not required to be small and the decaying condition is consistent with those in the works of [8] and [11].

We show an example of a non-archimedean version of the existence part of the Calabi-Yau theorem in complex geometry. Precisely, we study totally degenerate abelian varieties and certain probability measures on their associated analytic spaces in the sense of Berkovich.

A horospherical torus about a cusp of a hyperbolic manifold inherits a Euclidean similarity structure, called a cusp shape. We bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp. The bounds are in terms of the change in structure in a neighborhood of the singular locus alone. We then apply this result to provide information on the cusp shape of many hyperbolic knots, given only a diagram of the knot. Specifically, we show there is a universal constant C such that if a knot admits a prime, twist reduced diagram with at least C crossings per twist region, then the length of the second shortest curve on the cusp torus is bounded. The bound is linear in the number of twist regions of the diagram.