Abstract. We study the Berger-Nirenberg problem on surfaces with conical singularities, i.e. we discuss conditions under which a function on a Riemann surface is the Gaussian curvature of some conformai metric with a prescribed set of singularities of conical types.

Let f : X → CP 1 be a meromorphic function of degree N with simple poles and simple critical points on a compact Riemann surface X of genus g and let m be the standard round metric of curvature 1 on the Riemann sphere CP 1. Then the pullback f * m of m under f is a metric of curvature 1 with conical singularities of conical angles 4π at the critical points of f. We study the ζ-regularized determinant of the Laplace operator on X corresponding to the metric f * m as a functional on the moduli space of the pairs (X, f) (i.e. on the Hurwitz space H g,N (1,. .. , 1)) and derive an explicit formula for the functional.

ABSTRACT. We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold (M, g) and the L q,p -cohomology of that manifold.The L q,p -cohomology of (M, g) is defined to be the quotient of the space of closed differential forms in L p (M) modulo the exact forms which are exterior differentials of forms in L q (M).

For every Finsler metric F we associate a Riemannian metric g F (called the Binet-Legendre metric). The Riemannian metric g F behaves nicely under conformal deformation of the Finsler metric F , which makes it a powerful tool in Finsler geometry. We illustrate that by solving a number of named Finslerian geometric problems. We also generalize and give new and shorter proofs of a number of known results. In particular we answer a question of M. Matsumoto about local conformal mapping between two Minkowski spaces, we describe all possible conformal self maps and all self similarities on a Finsler manifold. We also classify all compact conformally flat Finsler manifolds. We solve a conjecture of S. Deng and Z. Hou on the Berwaldian character of locally symmetric Finsler spaces, and extend a classic result by H.C.Wang about the maximal dimension of the isometry groups of Finsler manifolds to manifolds of all dimensions.Most proofs in this paper go along the following scheme: using the correspondence F → g F we reduce the Finslerian problem to a similar problem for the Binet-Legendre metric, which is easier and is already solved in most cases we consider. The solution of the Riemannian problem provides us with the additional information that helps to solve the initial Finslerian problem.Our methods apply even in the absence of the strong convexity assumption usually assumed in Finsler geometry. The smoothness hypothesis can also be replaced by the weaker notion of partial smoothness, a notion we introduce in the paper. Our results apply therefore to a vast class of Finsler metrics not usually considered in the Finsler literature.

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Abstract. We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we show that these are given by the so-called Funk weak metric. We conclude the paper with a discussion of geodesics, of metric balls and of convexity properties of the Funk weak metric.

We develop an axiomatic approach to the theory of Sobolev spaces on metric measure spaces and we show that this axiomatic construction covers the main known examples (Hajtasz Sobolev spaces, weighted Sobolev spaces, Upper-gradients, etc). We then introduce the notion of variational p-capacity and discuss its relation with the geometric properties of the metric space. The notions of p-parabolic and p-hyperbolic spaces are then discussed. infinitesimal stretching constant lu(y)-u(z)l L~,(x) := lim sup r-tO d(y,x)

Abstract. The goal of this paper is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces.