Inverse Curvature Flow Research Articles

An inverse Gauss curvature flow to the Lp-Gauss Minkowski problem

In this paper, we confirm the existence of asymmetric smooth solutions to the Lp-Gauss Minkowski problem for p>0 by a class of inverse Gauss curvature flows. Furthermore, asymmetric weak solution for p>0 is obtained by a parabolic approximation method.

Stability of quermassintegral inequalities along inverse curvature flows

Stability of quermassintegral inequalities along inverse curvature flows

Two nonlocal inverse curvature flows of convex closed plane curves

In this paper, we introduce two [Formula: see text]-type ([Formula: see text]) curvature flows for closed convex planar curves. Along the flows the length of the curve is decreasing while the enclosed area is increasing. Finally, the evolving curves converge smoothly to a finite circle if they do not develop singularity during the evolution process.

A Study on Normal Motion of the Torus of Revolution in ℝ3

Abstract In the present research paper, we investigate the motion of surfaces in ℝ3 according to their curvatures. We study the motion of the torus of revolution via the normal velocity. We consider two cases: when the normal velocity is a function of both the time and the coordinates of the torus, and when it is a function of time only. We also study how the torus moves under different types of curvature flows, such as inverse mean curvature flow, inverse Gaussian curvature flow, and harmonic mean curvature flow. Moreover, we present some new applications of these flows.

Locally constrained inverse curvature flow and an Alexandrov-Fenchel inequality in de Sitter space

The long-time existence and convergence are shown for the locally constrained inverse curvature flow of compact spacelike convex hypersurfaces in de Sitter space. As an application, the Alexandrov-Fenchel inequality between the quermassintegrals A0 and An−1 is established in view of their opposite monotonicity along the considered flow. With the help of the dual relationship, a geometric inequality concerning quermassintegrals of convex hypersurfaces in a hyperbolic space of even dimension follows.

On an area-preserving locally constrained inverse curvature flow of convex curves

This paper will deal with a locally constrained inverse curvature flow which keeps the convexity and preserves the enclosed area of the evolving curve, and the limiting curve converges to a circle in the C∞ sense. As applications of this flow, an inequality for the integral of the power of support function for convex curves and the classical isoperimetric inequality are obtained.

Shifted inverse curvature flows in hyperbolic space

We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by $F^{-p}$ with positive power $p$ for a smooth, symmetric, strictly increasing and $1$-homogeneous curvature function $f$ of the shifted principal curvatures with some concavity properties. We study the maximal existence and asymptotical behavior of the flow for horo-convex hypersurfaces. In particular, for $0<p\leq 1$ we show that the limiting shape of the solution is always round as the maximal existence time is approached. This is in contrast to the asymptotical behavior of the (non-shifted) inverse curvature flow, as Hung and Wang [18] constructed a counterexample to show that the limiting shape of inverse curvature flow in hyperbolic space is not necessarily round.

The evolution of area-preserving and length-preserving inverse curvature flows for immersed locally convex closed plane curves

In this paper, we study an area-preserving inverse curvature flow and a length-preserving inverse curvature flow for immersed locally convex closed plane curves with rotation number m∈N+. The global-in-time flows are shown to converge smoothly to m-fold round circles as time goes to infinity. The sufficient conditions on initial curve are also found to guarantee the occurrence of the flow's singularity at finite time.

New weighted geometric inequalities for hypersurfaces in space forms

We prove a family of new sharp geometric inequalities involving weighted curvature integrals and quermassintegrals for smooth closed hypersurfaces in space forms. The tools we shall use are the inverse curvature flow by Gerhardt and Urbas and the locally constrained curvature flows introduced recently by Brendle, Guan and Li.

Self-similar solutions to the curvature flow and its inverse on the 2-dimensional light cone

We consider a novel curvature flow for curves on the 2-dimensional light cone, contained in the 3-dimensional Minkowski space. We show that the solutions to the curvature flow (CF) for such curves are in correspondence with the solutions to the inverse curvature flow (ICF). We prove that the ellipses and the hyperbolas are the only curves that evolve under homotheties. The ellipses are the only closed ones and they are ancient solutions. We show that a spacelike curve on the cone is a self-similar solution to the CF (resp. (ICF)) if, only if, its curvature (resp. inverse of its curvature) differs by a constant c from being the inner product between its tangent vector field and a fixed vector v of the 3-dimensional Minkowski space. The curve is a soliton solution when \(c=0\). We prove that for each vector v there exists a 2-parameter family of self-similar solutions to the CF and to the ICF, on the light cone. Considering non-trivial solutions to the CF, we prove that the corresponding solutions to the ICF may have at most three connected components. Moreover, at each end of such a curve the curvature is either unbounded or it tends to 0 or to the constant c. Explicitly given soliton solutions are included and some self-similar solutions on the light cone are visualized.

A two-point function, non-collapsing property and pinching estimates for inverse curvature flow in space forms

A two-point function, non-collapsing property and pinching estimates for inverse curvature flow in space forms

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.

On an inverse curvature flow in two-dimensional space forms

We study the evolution of compact convex curves in two-dimensional space forms. The normal speed is given by the difference of the weighted inverse curvature with the support function, and in the case where the ambient space is the Euclidean plane, is equivalent to the standard inverse curvature flow. We prove that solutions exist for all time and converge exponentially fast in the smooth topology to a standard round geodesic circle. This has a number of consequences: first, to prove the isoperimetric inequality; second, to establish a range of weighted geometric inequalities; and third, to give a counterexample to the $$n=2$$ case of a conjecture of Girão–Pinheiro.

On an area-preserving inverse curvature flow of convex closed plane curves

This paper deals with the 1/κα-type area-preserving nonlocal flow of smooth convex closed plane curves for all constant α>0. Under this flow, the convexity of the evolving curve is preserved. Due to the existence of finite time curvature blow-up examples, it is shown that, if the curvature κ will not blow up in finite time, the evolving curve will converge smoothly to a circle as t→∞.

Self-expanders to inverse curvature flows by homogeneous functions

In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete non-compact self-expanders to any of these flows with asymptotically cylindrical ends must be rotationally symmetric. Thirdly, we show that when such a flow is uniformly parabolic, there exist complete rotationally symmetric self-expanders which are asymptotic to two round cylinders with different radii. These extend some earlier results of inverse mean curvature flow to a wider class of flows.

Minkowski inequalities and constrained inverse curvature flows in warped spaces

Abstract This paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows, we prove long-time existence and smooth convergence to a radial coordinate slice. In the case of two-dimensional surfaces and a suitable speed, these flows enjoy two monotone quantities. In such cases, new Minkowski type inequalities are the consequence. In higher dimensions, we use the inverse mean curvature flow to obtain new Minkowski inequalities when the ambient radial Ricci curvature is constantly negative.

Uniqueness theorems of self-conformal solutions to inverse curvature flows

It is known from the literature that round spheres are the only closed homothetic self-similar solutions to the inverse mean curvature flow and parabolic curvature flows by degree − 1 -1 homogeneous functions of principal curvatures in the Euclidean space. In this article, we prove that the round sphere is rigid in a stronger sense: under some natural conditions such as star-shapedness, round spheres are the only closed solutions to the above-mentioned flows which evolve by diffeomorphisms generated by conformal Killing fields.

On a length-preserving inverse curvature flow of convex closed plane curves

This paper deals with a 1/κα-type length-preserving nonlocal flow of convex closed plane curves for all α>0. Under this flow, the convexity of the evolving curve is preserved. For a global flow, it is shown that the evolving curve converges smoothly to a circle as t→∞. Some numerical blow-up examples and a sufficient condition leading to the global existence of the flow are also constructed.

Locally constrained inverse curvature flows

We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order—namely, the radial coordinate and the generalized support function. Under various assumptions we prove longtime existence and smooth convergence to a coordinate slice. We apply this result to deduce a new Minkowski-type inequality in the anti–de Sitter Schwarzschild manifolds and a weighted isoperimetric-type inequality in hyperbolic space.

On a family of inverse curvature flows for closed convex plane curves

In this note we introduce a family of flows for closed convex curves in the plane. Along the flows the enclosed area of the curve is increasing, and the curve remains convex and converges to a circle. The flows include various flows studied by various authors as special cases.