Related papers: Combinatorial p-th Calabi Flows for Total Geodesic…
In this paper, we study the area-preserving and length-preserving $\kappa^\alpha$-type curvature flows of smooth, closed, convex curves in the two-dimensional hyperbolic plane $\mathbb H^2$ for $\alpha<0$ and prove that convexity is…
We study the geometry of the foliation by constant Gaussian curvature surfaces $(\Sigma_k)_k$ of a hyperbolic end, and how it relates to the structures of its boundary at infinity and of its pleated boundary. First, we show that the…
This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time…
In this paper, we investigate the prescribed curvature problem associated with a special Lin-Lu-Yau curvature on finite graphs of girth at least 6. We define the corresponding Calabi flow for this curvature type, and establish an equivalent…
We show that the analog of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle…
We investigate the mean curvature flows in a class of warped product manifolds with closed hypersurfaces fibering over $\mathbb{R}$. In particular, we prove that under natural conditions on the warping function and Ricci curvature bound for…
The Koebe-Andreev-Thurston circle packing theorem, as well as its generalization to circle patterns due to Bobenko and Springborn, holds for Euclidean and hyperbolic metrics possibly with conical singularities, but fails for spherical…
Given a constant $k>1$ and a real valued function $K$ on the hyperbolic plane $\mathbb H^2$, we study the problem of finding, for any $\epsilon\approx 0$, a closed and embedded curve $u^\epsilon $ in $\mathbb H^2$ having geodesic curvature…
Given a non circular spacial closed curve whose total torsion is an integer multiple of $2\pi$, we construct a germ of a smooth surface that contains it as a hyperbolic principal cycle.
We examine a simple hard disc fluid with no long range interactions on the two dimensional space of constant negative Gaussian curvature, the hyperbolic plane. This geometry provides a natural mechanism by which global crystalline order is…
We consider the Gauss curvature type flow for uniformly convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}\ (n\geqslant 2)$. We prove that if the initial closed hypersurface is smooth and uniformly convex, then the smooth…
Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by…
In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations are strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth…
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 study constant mean curvature spacelike hypersurfaces and in particular maximal hypersurfaces immersed in pp-wave spacetimes satisfying the timelike convergence condition. We prove the non-existence of compact spacelike hypersurfaces…
We show that any immersion, which is not a covering of an embedded 2-orbifold, of a totally geodesic hyperbolic turnover in a complete orientable hyperbolic 3-orbifold is contained in a hyperbolic 3-suborbifold with totally geodesic…
We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by…
We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation…
Let $\Sigma$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $\Sigma$. A geodesic curve $\gamma$ in $\Sigma$ is filling, if it cuts the surface into topological disks and…