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We provide conditions under which a Riemann surface $X$ is the asymptotic boundary of a convex co-compact hyperbolic manifold, homeomorphic to a handlebody, of negative renormalized volume. We prove that this is the case when there are on…

We show that if $\Omega$ is an $m$-convex domain in $\mathbb R^n$ for some $2\le m<n$ whose boundary $b\Omega$ has a tubular neighbourhood of positive radius and is not $m$-flat near infinity, then $\Omega$ does not contain any immersed…

We prove that if $\Omega$ is a simply connected quadrature domain for a distribution with compact support and the infinity point belongs the boundary, then the boundary has an asymptotic curve that is either a straight line or a parabola or…

We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume…

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichm\"uller space, Hitchin representations and geodesic currents. We add to…

We construct an unbounded strictly pseudoconvex Kobayashi hyperbolic and complete domain in $\mathbb{C}^2$, which also possesses complete Bergman metric, but has no nonconstant bounded holomorphic functions.

We study the space of pseudo-holomorphic spheres in compact symplectic manifolds with convex boundary. We show that the theory of Gromov-Witten invariants can be extended to the class of semi-positive manifolds with convex boundary. This…

Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set,…

We interpret the Hilbert entropy of a convex projective structure on a closed higher-genus surface as the Hausdorff dimension of the non-differentiability points of the limit set in the full flag space $\mathcal F(\mathbb R^3)$.…

Let $(M, \partial M)$ be a compact 3-manifold with boundary which admits a complete, convex co-compact hyperbolic metric. For each hyperbolic metric $g$ on $M$ such that $\dr M$ is smooth and strictly convex, the induced metric on $\dr M$…

A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the…

We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the…

We prove that the sublevel set $\big\{z\in\mathbb D\colon k_{\mathbb D}\big(z,z_0\big)-k_{\mathbb D}\big(f(z),w_0\big)<\mu\big\}$, ${\mu\in\mathbb R}$, is geodesically convex with respect to the Poincar\'e distance $k_{\mathbb D}$ in the…

An n-dimensional strictly pseudoconvex Hartogs domain D_F can be equipped with a natural Kaehler metric g_F. In this paper we prove that if m_0g_F is balanced for a given positive integer m_0 then m_0>n and (D_F, g_F) is holomorphically…

The existence of a smooth complete strictly locally convex hypersurface with prescribed scalar curvature and asymptotic boundary at infinity in $\mathbb{H}^{3}$ is proved under the assumption that there exists a strictly locally convex…

We prove four results towards a description, in terms of the null support function, of the set of isometric embeddings of the hyperbolic plane into Minkowski 3-space. We show that for sufficiently tame null support function, the…

We prove the following for a bounded convex planar domain that is symmetric with respect to both coordinate axes. Consider a centered rectangle with sides parallel to the axes that strictly contains the domain. If the domain is not a…

The main objective of this paper is to show that balls under invariant metrics on hyperbolic planar domains are finitely-connected. As applications, we give new and transparent proofs of classical results on conformal mappings of planar…

Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width $D$.…

In this paper we give an local estimate for the Kobayashi distance on a bounded convex domain of finite type, which relates to a local pseudodistance near the boundary. The estimate is precise up to a bounded additive term. Also we conclude…