Related papers: $\beta$-Uniform Convexity and Divisible Domains
In this paper we study the flag curvature of a particular class of Finsler metrics called general $(\alpha,\beta)$-metrics, which are defined by a Riemannian metric $\alpha$ and a $1$-form $\beta$. The classification of such metrics with…
We prove the following localized version of a classical ellipsoid characterization: Let $B\subset\mathbb R^3$ be convex body with a smooth strictly convex boundary and 0 in the interior, and suppose that there is an open set of planes…
J. Mather characterized uniform hyperbolicity of a discrete dynamical system as equivalent to invertibility of an operator on the set of all sequences bounded in norm in the tangent bundle of an orbit. We develop a similar characterization…
We prove that a bounded domain in $\mathbb{C}^n$ admitting a complete K\"ahler metric with negatively pinched holomorphic bisectional curvature near the boundary, admits a complete K\"ahler metric with negatively pinched holomorphic…
Heinrich Tietze has shown that for a closed connected subset of euclidean space being convex is a local property. We generalize this to CAT(0)-spaces and locally compact CAT(\kappa) spaces. As an application we give a construction of…
We study dentable maps from a closed convex subset of a Banach space into a metric space as an attempt of generalize the Radon-Nikod\'ym property to a "less linear" frame. We note that a certain part of the theory can be developed in rather…
On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local CR invariant of the boundary. For a…
In this paper we consider the following question: For bounded domains with smooth boundary, can strong pseudoconvexity be characterized in terms of the intrinsic complex geometry of the domain? Our approach to answering this question is…
For a strictly convex set $K\subset \mathbb{R}^2$ of class $C^2$ we consider its associated sub-Finsler $K$-perimeter $|\partial E|_K$ in $\mathbb{H}^1$ and the prescribed mean curvature functional $|\partial E|_K-\int_E f$ associated to a…
When a single time-like vector is distinguished geometrically to present the only preferred direction in extending the pseudoeuclidean geometry, the hyperboloid may not be regarded as an exact carrier of the unit-vector image. So under…
We introduce the notion of domains with uniform squeezing property, study various analytic and geometric properties of such domains and show that they cover many interesting examples, including Teichmuller spaces and Hermitian symmetric…
A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly).…
Recently, Lima and Randrianarivony pointed out the role of the property $(\beta)$ of Rolewicz in nonlinear quotient problems, and answered a ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. In the present…
Let $\gamma$ be a smooth, non-closed, simple curve whose image is symmetric with respect to the $y$-axis, and let $D$ be a planar domain consisting of the points on one side of $\gamma$, within a suitable distance $\delta$ of $\gamma$.…
A conjecture of Fuglede states that a bounded measurable set $\Omega$ in space, of measure 1, can tile space by translations if and only if the Hilbert space $L^2(\Omega)$ has an orthonormal basis consisting of exponentials. If $\Omega$ has…
We construct a differentiable locally Lipschitz function $f$ in $\mathbb{R}^{N}$ with the property that for every convex body $K\subset \mathbb{R}^N$ there exists $\bar x \in \mathbb{R}^N$ such that $K$ coincides with the set $\partial_L…
For $f$ an entire transcendental map with a univalent Baker domain $U$ of hyperbolic type I, we study pinching deformations in $U$, the support of this deformation being certain laminations in the grand orbit of $U$. We show that pinching…
Among all plastic deformations of the gravitational Lorentz vacuum \cite{wr1} a particular role is being played by conformal deformations. These are conveniently described by using the homogeneous space for the conformal group…
Given a pseudoconvex domain U with C^1-boundary in P^n, n>2, we show that if H^{2n-2}_\dR}(U)\not=0, then there is a strictly psh function in a neighborhood of boundary U. We also solve the \dbar-equation in X=P^n\ U, for data smooth (0,1)…
Given a subspace $U\subset\mathbb{C}[x_1,\dots,x_n]_d$ we consider the closure of the image of the rational map $\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{\dim U-1}$ given by $U$. Its coordinate ring is isomorphic to $\bigoplus_{i\ge 0}…