Surfaces expanding by the inverse Gauß curvature flow

Schnürer, Oliver C.

doi:10.1515/crelle.2006.088

Cited by 24 publications

(20 citation statements)

References 19 publications

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“…The argument differs from that in previous work [3,27] in only a few points. As in other flows with speed growing super linearly in the curvature, the main difficulty is in the non-uniform parabolicity of the flow.…”

Section: Convergencecontrasting

confidence: 95%

“…There are many papers which consider the evolution of hypersurfaces under flows of this kind, beginning with the work of Huisken [21] on mean curvature flow, and including flow by powers of Gauss curvature [15,29], scalar curvature [16], and large classes of other examples [2,7]. Several previous papers have considered such flows in the special case of surfaces in three-dimensional space [4,5,8,[25][26][27][28], where the low-dimensional setting allows a more complete understanding of the equation for the evolution of the second fundamental form, and where there is also a more general regularity theory available [6]. In particular, in Euclidean space [4], compact convex surfaces moving by their Gauss curvature become spherical as they shrink to points, and the same is true if the speed of motion is an arbitrary monotone increasing homogeneous degree one function of the principal curvatures [8].…”

Section: Introductionmentioning

confidence: 99%

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Curvature flow in hyperbolic spaces

Andrews

Chen

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We study the evolution of compact convex hypersurfaces in hyperbolic space H nC1 , with normal speed governed by the curvature. We concentrate mostly on the case of surfaces, and show that under a large class of natural flows, any compact initial surface with Gauss curvature greater than 1 produces a solution which converges to a point in finite time, and becomes spherical as the final time is approached. We also consider the higher-dimensional case, and show that under the mean curvature flow a similar result holds if the initial hypersurface is compact with positive Ricci curvature.

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Section: Convergencecontrasting

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Curvature flow in hyperbolic spaces

Andrews

Chen

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…This technique was previously used in [12,30,31,32]. Expanding flows by Gauss curvature have been studied in [23,24,38,40,41]. Our estimates are also inspired by these works.…”

Section: By (23) and (42)mentioning

confidence: 99%

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

2019

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In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space R n+1 with speed f r α K, where K is the Gauss curvature, r is the distance from the hypersurface to the origin, and f is a positive and smooth function. If α ≥ n + 1, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if f ≡ 1. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang [29], for the case q < 0. If α < n+1, corresponding to the case q > 0, we also establish the same results for even function f and originsymmetric initial condition, but for non-symmetric f , counterexample is given for the above smooth convergence.2010 Mathematics Subject Classification. 35K96, 53C44.

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“…Finally, the case θ( ν) = −1 and ρ = −1 is also called the inverse Gauß curvature flow, see e.g. [37].…”

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confidence: 99%

Parametric Approximation of Willmore Flow and Related Geometric Evolution Equations

Barrett¹,

Garcke²,

Nürnberg³

2008

SIAM J. Sci. Comput.

125

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Abstract. We present various variational approximations of Willmore flow in R d , d = 2, 3. As well as the classic Willmore flow, we consider also variants that are (a) volume preserving and (b) volume and area preserving. The latter evolution law is the so-called Helfrich flow. In addition, we consider motion by Gauß curvature. The presented fully discrete schemes are easy to solve as they are linear at each time level, and they have good properties with respect to the distribution of mesh points. Finally, we present numerous numerical experiments, including simulations for energies appearing in the modelling of biological cell membranes.

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Surfaces expanding by the inverse Gauß curvature flow

Abstract: Abstract. We show that strictly convex surfaces expanding by the inverse Gauß curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.

Cited by 24 publications

References 19 publications

Curvature flow in hyperbolic spaces

Curvature flow in hyperbolic spaces

Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Parametric Approximation of Willmore Flow and Related Geometric Evolution Equations

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