In this paper we study the solvability of the rotationally symmetric centroaffine Minkowski problem. By delicate blow-up analyses, we remove a technical condition in the existence result obtained by Lu and Wang [30].

In this paper, we solve a Monge-Ampre-type equation by the continuity method. This equation is motivated by the superstring theory.

We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal Weil-Petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface is comparable to $-1$, independently of the genus. This provides a counterexample to some suggestions that the Weil-Petersson metric becomes asymptotically flat, as the genus $g$ goes to infinity, in the thick loci in \tec space. Adopting a different perspective on curvature, we also show that the minimal (most negative) eigenvalue of the curvature operator at any point in the \tec space $\Teich(S_g)$ of a closed surface $S_g$ of genus $g$ is uniformly bounded away from zero. Restricting to a thick part of $\Teich(S_g)$, we show that the minimal eigenvalue is uniformly bounded below by an explicit constant which does not depend on the topology of the surface but only on the given bound on injectivity radius.

In this paper the centroaffine Minkowski problem, a critical case of the L p-Minkowski problem in the n+ 1 dimensional Euclidean space, is studied. By its variational structure and the method of blow-up analyses, we obtain two sufficient conditions for the existence of solutions, for a generalized rotationally symmetric case of the problem.

The purpose of this paper is to study the monodromy groups associated to the quasi-bounded holomorphic quadratic forms on punctured surfaces. As a consequence, we obtain a natural family of symplectic structures on the Teichm/iller space Tg,, for n> 0. As another consequence, we show that the projective monodromy map from a class of Fuchsian equations to the representation variety is generically a local diffeomorphism.