Uniformization of metric surfaces using isothermal coordinates

We establish a uniformization result for metric surfaces – metric spaces that are topological surfaces with locally finite Hausdorff 2-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be covered by quasiconformal images of Euclidean domains is quasiconformally equivalent to a Riemannian surface. To prove this, we construct an atlas of suitable isothermal coordinates.

Mean field equations on a closed Riemannian surface with the action of an isometric group

Let [Formula: see text] be a closed Riemannian surface, [Formula: see text] be an isometric group acting on it. Denote a positive integer [Formula: see text], where [Formula: see text] is the number of all distinct points of the set [Formula: see text]. A sufficient condition for existence of solutions to the mean field equation [Formula: see text] is given. This recovers results of Ding–Jost–Li–Wang, Asian J. Math. (1997) 230–248 when [Formula: see text] or equivalently [Formula: see text], where Id is the identity map.

Geometric Study of Marginally Trapped Surfaces in Space Forms and Robertson-Walker Spacetimes—An Overview

A marginally trapped surface in a spacetime is a Riemannian surface whose mean curvature vector is lightlike at every point. In this paper we give an up-to-date overview of the differential geometric study of these surfaces in Minkowski, de Sitter, anti-de Sitter and Robertson-Walker spacetimes. We give the general local descriptions proven by Anciaux and his coworkers as well as the known classifications of marginally trapped surfaces satisfying one of the following additional geometric conditions: having positive relative nullity, having parallel mean curvature vector field, having finite type Gauss map, being invariant under a one-parameter group of ambient isometries, being isotropic, being pseudo-umbilical. Finally, we provide examples of constant Gaussian curvature marginally trapped surfaces and state some open questions.

Disjoint superheavy subsets and fragmentation norms

We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding [Formula: see text] with respect to the fragmentation norm on the group [Formula: see text] of Hamiltonian diffeomorphisms of a symplectic manifold [Formula: see text]. As an application, we provide an answer to Brandenbursky’s question on fragmentation norms on [Formula: see text], where [Formula: see text] is a closed Riemannian surface of genus [Formula: see text].

Short loops in surfaces with a circular boundary component

It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if [Formula: see text] is a Riemannian surface with connected boundary in [Formula: see text], such that the boundary curve is a standard unit circle, then the length of the shortest non-contractible loop in [Formula: see text] is bounded in terms of the area of [Formula: see text].

Rigidity of Nonnegatively Curved Surfaces Relative to a Curve

We prove that any properly oriented $\mathcal{C}^{2,1}$ isometric immersion of a positively curved Riemannian surface $M$ into Euclidean 3-space is uniquely determined, up to a rigid motion, by its values on any curve segment in $M$. A generalization of this result to nonnegatively curved surfaces is presented as well under suitable conditions on their parabolic points. Thus, we obtain a local version of Cohn-Vossen’s rigidity theorem for convex surfaces subject to a Dirichlet condition. The proof employs in part Hormander’s unique continuation principle for elliptic partial differential equations. Our approach also yields a short proof of Cohn-Vossen’s theorem.