arXiv research

Ancient solutions found for mean curvature flows in hyperbolic spaces.

A discrete method approximates hyperbolic curvature flow in the plane.

Curvature flows in hyperbolic space preserve positive sectional curvature and contract to a point.

Paper introduces new flows to find circle packings with specific curvature.

New flow expands hypersurfaces in hyperbolic space, showing round limiting shape for certain powers.

Inverse mean curvature flow converges to a disk in hyperbolic space.

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 solution and possesses the nonlinear stability defined on the Euclidean space with …

The paper solves curvature problems on infinite hyperbolic surfaces.

Study on stability of mean curvature flow in hyperbolic space.

The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.

Topological entropy decreases strictly along Ricci flow near hyperbolic metrics.

The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.

The paper examines flows that preserve area and length in hyperbolic geometry.

In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow…

Study of flow in hyperbolic space with capillary boundary.

Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.

Geometric inequalities for static convex domains in hyperbolic space proved.

The paper studies how convex hypersurfaces in hyperbolic space evolve under a specific curvature flow.

Classifies self-similar curve shortening flows in hyperbolic 2-space.

We show that there exists a suitable neighborhood of a constant curvature hyperbolic metric such that, for all initial data in this neighborhood, the corresponding solution to a normalized cross curvature flow exists for all time and converges to a hyperbolic metric. We show that the same technique proves an analogous …

The paper constructs Markov partitions for geodesic flow on hyperbolic surfaces.

The paper applies combinatorial Ricci flows to prove hyperbolic structures on 3-manifolds.

We consider the evolution of the asymptotically hyperbolic mass under the curvature-normalized Ricci flow of asymptotically hyperbolic, conformally compactifiable manifolds. In contrast to asymptotically flat manifolds, for which ADM mass is constant during Ricci flow, we show that the mass of an asymptotically hyperbo…

Study mean curvature flow in hyperbolic 3-manifolds, proving foliations and existence of minimal surfaces.

We consider the hyperbolic geometric flow $\frac{\partial^2}{\partial t^2}g(t)=-2Ric_{g(t)}$ introduced by Kong and Liu [KL]. When the Riemannian metric evolve, then so does its curvature. Using the techniques and ideas of S.Brendle [Br,BS], we derive evolution equations for the Levi-Civita connection and the curvature…

We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.

We consider inverse curvature flows in hyperbolic space with starshaped initial hypersurface, driven by positive powers of a homogeneous curvature function. The solutions exist for all time and, after rescaling, converge to a sphere.

Study classifies minimal surfaces and solitons in hyperbolic 3-space as translation surfaces.

In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a pinching condition in a hyperbolic space form to a round point in finite time.

In this paper we introduce and study a new kind of hyperbolic geometric flows --dissipative hyperbolic geometric flow. This kind of flow is defined by a system of quasilinear wave equations with dissipative terms. Some interesting exact solutions are given, in particular, a new concept-- hyperbolic Ricci soliton is int…

We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which are strictly convex by horospheres, we show the long time existence of mixed volu…

Proves heat kernel superconvexity in hyperbolic space.

The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.

The curvature and the reduced curvature are basic differential invariants of the pair (Hamiltonian system, Lagrange distribution) on the symplectic manifold. It is shown that the negativity of the reduced curvature implies the hyperbolicity of any compact invariant set of the Hamiltonian flow restricted to a prescribed…

Study on relative entropy for hypersurfaces in hyperbolic space.

New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.

In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian curvature by dividing the classical discrete Gauss curvature by an area element, which …

In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$ ($n\geqslant2$) is mean convex and star-shaped. Several interesting examples and some hyperbol…

The paper proves Michael-Simon inequalities in hyperbolic space using novel curvature flows.

New inequalities derived for hyperbolic space via specific flows.

Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metr…

Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic 3-manifolds…

We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $m\geq3$ starting from any smooth, conformally hyperbolic initial metric. We do not require initial completeness or curvature bounds. With the same methods, we show rigidity of hyperbolic space under the Yamabe…

Entropy for submanifolds in hyperbolic space defined.

Flow of convex hypersurfaces in hyperbolic space converges to geodesic spheres.

Develops Aleksandrov reflection for hyperbolic flows, proving convergence to umbilic surfaces.

We introduce a combinatorial curvature flow for PL metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow…

This note revisits the inverse mean curvature flow in the 3-dimensional hyperbolic space. In particular, we show that the limiting shape is not necessarily round after scaling, thus resolving an inconsistency in the literature.