Gauss Curvature Flow Research Articles

Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space H n + 1 ( n ≥ 2 ) \mathbb {H}^{n+1} (n\geq 2) with the speed given by arbitrary positive power α \alpha of the Gauss curvature. We prove that if the initial hypersurface is convex, then the smooth solution of the flow remains convex and exists for all positive time t ∈ [ 0 , ∞ ) t\in [0,\infty ) . Moreover, we apply a result of Kohlmann which characterises the geodesic ball using the hyperbolic curvature measures and an argument of Alexandrov reflection to prove that the flow converges to a geodesic sphere exponentially in the smooth topology. This can be viewed as the first result for non-local type volume preserving curvature flows for hypersurfaces in the hyperbolic space with only convexity required on the initial data.

A Gauss curvature flow approach to the torsional Minkowski problem

This paper is devoted to reconsidering the Minkowski problem for torsional rigidity introduced by Colesanti-Fimiani [9]. Employing a Gauss curvature flow, we establish the new existence of smooth solutions of the torsional Minkowski problem under the premise of giving an initial convex body.

Existence of convex hypersurfaces with prescribed centroaffine curvature

In this paper, we study the existence of solutions to the centroaffine Minkowski problem, namely the existence of closed convex hypersurfaces in the Euclidean space R n + 1 \mathbb {R}^{n+1} with prescribed centroaffine curvature. This problem can be described as a variational problem with a functional resembling Kantorovich’s dual functional in optimal transportation. By using the Gauss curvature flow, we obtain new conditions for the existence of solutions to the problem.

A variant prescribed curvature flow on closed surfaces with negative Euler characteristic

On a closed Riemannian surface (M,{bar{g}}) with negative Euler characteristic, we study the problem of finding conformal metrics with prescribed volume A>0 and the property that their Gauss curvatures f_lambda = f + lambda are given as the sum of a prescribed function f in C^infty (M) and an additive constant lambda . Our main tool in this study is a new variant of the prescribed Gauss curvature flow, for which we establish local well-posedness and global compactness results. In contrast to previous work, our approach does not require any sign conditions on f. Moreover, we exhibit conditions under which the function f_lambda is sign changing and the standard prescribed Gauss curvature flow is not applicable.

Gauss curvature flow with shrinking obstacle

Gauss curvature flow with shrinking obstacle

Uniqueness of solutions to a class of isotropic curvature problems

Employing a local version of the Brunn-Minkowski inequality, we give a new and simple proof of a result due to Andrews, Choi and Daskalopoulos that the origin-centred balls are the only closed, self-similar solutions of the Gauss curvature flow. Extensions to various nonlinearities are obtained, assuming the centroid of the enclosed convex body is at the origin. By applying our method to the Alexandrov-Fenchel inequality, we also show that origin-centred balls are the only solutions to a large class of even Christoffel-Minkowski type problems.

Prescribed Gauss curvature flow on surfaces with negative Euler characteristic

Prescribed Gauss curvature flow on surfaces with negative Euler characteristic

Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure

Abstract In this article, we study the Musielak-Orlicz-Gauss image problem based on the Gauss curvature flow in Li et al. We deal with some cases in which there is no uniform estimate for the Gauss curvature flow. By the use of the topological method in Guang et al., a special initial condition is chosen such that the Gauss curvature flow converges to a solution of the Musielak-Orlicz-Gauss image problem.

Uniqueness of solutions to the logarithmic Minkowski problem in [formula omitted

In this paper, we prove the uniqueness of solutions to the logarithmic Minkowski problem in R3 without symmetry condition, provided the density of the measure is close to 1 in Cα norm. This result also implies the uniqueness of self-similar solutions to the anisotropic Gauss curvature flow in R3 when the speed function is Cα close to a positive constant.

Volume preserving flows for convex curves and surfaces in the hyperbolic space

In this paper, we study the curvature flows of convex curves and surfaces in the hyperbolic space. In the first part, we consider the area preserving and length preserving κα-type curvature flows of smooth closed convex curves in the hyperbolic plane H2 and show that these two types of flows evolve convex curve to a geodesic circle exponentially in C∞ topology. In the second part, we study the volume preserving Gauss curvature flow of smooth closed convex surfaces in H3 and show that the solution remains convex and converges to a geodesic sphere exponentially in C∞ topology.

Inverse Gauss Curvature Flows and Orlicz Minkowski Problem

Abstract Liu and Lu [27] investigated a generalized Gauss curvature flow and obtained an even solution to the dual Orlicz-Minkowski problem under some appropriate assumptions. The present paper investigates a inverse Gauss curvature flow, and achieves the long-time existence and convergence of this flow via a different C 0-estimate technique under weaker conditions. As an application of this inverse Gauss curvature flow, the present paper first arrives at a non-even smooth solution to the Orlicz Minkowski problem.

Anisotropic Gauss curvature flows and their associated Dual Orlicz-Minkowski problems

In this paper we study a normalized anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Ampère type equation and we obtain a new existence result of solutions to the Dual Orlicz-Minkowski problem for smooth measures, especially for even smooth measures.

Gauss curvature flow with an obstacle

Gauss curvature flow with an obstacle

Translating solutions to the Gauss curvature flow with flat sides

We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.

Connection between translating solutions for a generalized Gauss curvature flow in a cylinder

We study the motion of a hypersurface in a cylinder driven by its (generalized) Gauss curvature. Some special cases of the problem can be used to model the abrasion of a stick on its ends. First we consider the case where the slope of the hypersurface on the cylinder boundary is a positive constant h, and seek for a radially symmetric translating solution. Then we consider the case where the boundary slope of the hypersurface is equal to h(u), with u representing the hypersurface function and h being a positive function satisfying h(u)→h±>0 as u→±∞. We construct the unique entire solution to this problem and show that it connects a translating solution with boundary slope h− at t=−∞ and another translating solution with boundary slope h+ at t=+∞.

Nonhomogeneous inverse Gauss curvature flow in ℍ³

In this paper, we consider nonhomogeneous inverse Gauss curvature flows in hyperbolic space H 3 \mathbb {H}^{3} . We prove that if the initial hypersurface Σ 0 \Sigma _0 has nonnegative scalar curvature, then the evolving surface Σ t \Sigma _{t} has nonnegative scalar curvature along the flow for t > 0 t>0 . The solution of the flow exists for all time and becomes more and more umbilic as t → ∞ t\rightarrow \infty .

A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow

AbstractWe define a formal Riemannian metric on a given conformal class of metrics with signed curvature on a closed Riemann surface. As it turns out this metric is the well-known Mabuchi-Semmes-Donaldson metric of Kähler geometry in a different guise. The metric has many interesting properties, and in particular we show that the classical Liouville energy is geodesically convex. This suggests a different approach to the uniformization theorem by studying the negative gradient flow of the normalized Liouville energy with respect to this metric, a new geometric flow whose principal term is the inverse of the Gauss curvature. We prove long time existence of solutions with arbitrary initial data and weak convergence to constant scalar curvature metrics by exploiting the metric space structure.

Uniqueness of closed self-similar solutions to $$\varvec{\sigma _k^{\alpha }}$$-curvature flow

By adapting the test functions introduced by Choi–Daskaspoulos (Uniqueness of closed self-similar solutions to the Gauss curvature flow. arXiv:1609.05487, 2016) and Brendle–Choi–Daskaspoulos (Acta Math 219(1):1–16, 2017) and exploring the properties of the k-th elementary symmetric function \(\sigma _{k}\) intensively, we show that for any fixed k with \(1\le k\le n-1\), any strictly convex closed hypersurface in \({\mathbb {R}}^{n+1}\) satisfying \(\sigma _{k}^{\alpha }=\langle X,\nu \rangle \), with \(\alpha \ge \frac{1}{k}\), must be a round sphere. In fact, we prove a uniqueness result for any strictly convex closed hypersurface in \({\mathbb {R}}^{n+1}\) satisfying \(F+C=\langle X,\nu \rangle \), where F is a positive homogeneous smooth symmetric function of the principal curvatures and C is a constant.

Entropy and a convergence theorem for Gauss curvature flow in high dimension

We prove uniform regularity estimates for the normalized Gauss curvature flow in higher dimensions. The convergence of solutions in C^\infty -topology to a smooth strictly convex soliton as t goes to infinity is obtained as a consequence of these estimates together with an earlier result of Andrews. The estimates are established via the study of an entropy functional for convex bodies.

Deforming a Convex Hypersurface with Low Entropy by Its Gauss Curvature

We prove the asymptotic roundness under normalized Gauss curvature flow provided entropy is initially small enough.