Mixed volume preserving curvature flows

McCoy, James

doi:10.1007/s00526-004-0316-3

Cited by 86 publications

(115 citation statements)

References 24 publications

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“…In order to achieve this, using a trick of Tso [26] for the Gauß curvature flow, see also [1], [7] and [21], we study the evolution under (1.1) of the function…”

Section: The Long Time Existencementioning

confidence: 99%

Contraction of Horosphere-Convex Hypersurfaces by Powers of the Mean Curvature in the Hyperbolic Space

Guo¹,

Li²,

Wu³

2013

Journal of the Korean Mathematical Society

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Abstract. This paper concerns the evolution of a closed hypersurface of the hyperbolic space, convex by horospheres, in direction of its inner unit normal vector, where the speed equals a positive power β of the positive mean curvature. It is shown that the flow exists on a finite maximal interval, convexity by horospheres is preserved and the hypersurfaces shrink down to a single point as the final time is approached.

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“…In order to achieve this, using a trick of Tso [26] for the Gauß curvature flow, see also [1], [7] and [21], we study the evolution under (1.1) of the function…”

Section: The Long Time Existencementioning

confidence: 99%

Contraction of Horosphere-Convex Hypersurfaces by Powers of the Mean Curvature in the Hyperbolic Space

Guo¹,

Li²,

Wu³

2013

Journal of the Korean Mathematical Society

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“…In this article, we remove that restriction in the case of pure Neumann boundary conditions by consideration of an appropriate pinching function of the principal curvatures of the evolving hypersurface. Our idea is based upon earlier constructions for evolving convex surfaces by Andrews [An2,An4] of preserved pinching functions, constructions which were also used by the first author [Mc1,Mc2]. Very recently, the first author, together with Andrews and Langford, controlled a pinching function under fully nonlinear curvature flow of nonconvex surfaces [ALM1].…”

Section: Introductionmentioning

confidence: 99%

Fully nonlinear curvature flow of axially symmetric hypersurfaces

McCoy

Mofarreh

Wheeler

2014

Nonlinear Differ. Equ. Appl.

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Recently, fully nonlinear curvature flow of a certain class of axially symmetric hypersurfaces with boundary conditions time of existence was obtained, in the case of convex speeds ( J. A. McCoy et al., Annali di Matematica Pura ed Applicata 1-13, 2013). In this paper we remove the convexity condition on the speed in the case it is homogeneous of degree one in the principal curvatures and the boundary conditions are pure Neumann. Moreover, we classify the singularities of the flow of a larger class of axially symmetric hypersurfaces as Type I. Our approach to remove the convexity requirement on the speed is based upon earlier work of Andrews for evolving convex surfaces (B. H. Andrews, Invent Math 138(1):151-161, 1999; Calc Var Partial Differ Equ 39(3-4):649-657, 2010); these arguments for obtaining a 'curvature pinching estimate' may be adapted to this setting due to axial symmetry. As further applications of curvature pinching in this setting, we show that closed, convex, axially symmetric hypersurfaces contract under the flow to round points, and hypersurfaces contracting self-similarly are necessarily spheres. These results are new for n ≥ 3. Abstract. Recently, fully nonlinear curvature flow of a certain class of axially symmetric hypersurfaces with boundary conditions was considered and a partial characterisation of the finite maximal time of existence was obtained, in the case of convex speeds [MMW]. In this paper we remove the convexity condition on the speed in the case it is homogeneous of degree one in the principal curvatures and the boundary conditions are pure Neumann. Moreover, we classify the singularities of the flow of a larger class of axially symmetric hypersurfaces as Type I. Our approach to remove the convexity requirement on the speed is based upon earlier work of Andrews for evolving convex surfaces [An2, An4]; these arguments for obtaining a 'curvature pinching estimate' may be adapted to this setting due to axial symmetry. As further applications of curvature pinching in this setting, we show that closed, convex, axially symmetric hypersurfaces contract under the flow to round points, and hypersurfaces contracting self-similarly are necessarily spheres. These results are new for n ≥ 3.Mathematics Subject Classification (2010). Primary 35K55, 35R35, 53C44; Secondary 35K60.

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“…We then discuss flow-dependent results, and prove, in Lemma 2.4, that each of the ancillary conditions (viii)-(x) implies condition (v). We follow the conventions of [Andrews et al 2013b;Andrews 2007;McCoy 2005], where proofs or references for much of this section may be found. Many of the results can also be found in the book [Gerhardt 2006].…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

Convexity estimates for hypersurfaces moving by convex curvature functions

2014

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We consider the evolution of compact hypersurfaces by fully non-linear, parabolic curvature ows for which the normal speed is given by a smooth, convex, degree one homoge-neous function of the principal curvatures. We prove that solution hypersurfaces on which the speed is initially positive become weakly convex at a singularity of the ow. The result extends the convexity estimate [HS99b] of Huisken and Sinestrari for the mean curvature ow to a large class of speeds, and leads to an analogous description of `type-II' singularities. We remark that many of the speeds considered are positive on larger cones than the positive mean half-space, so that the result in those cases also applies to non-mean-convex initial data. We consider the evolution of compact hypersurfaces by fully nonlinear, parabolic curvature flows for which the normal speed is given by a smooth, convex, degree-one homogeneous function of the principal curvatures. We prove that solution hypersurfaces on which the speed is initially positive become weakly convex at a singularity of the flow. The result extends the convexity estimate of Huisken and Sinestrari [Acta Math. 183:1 (1999), 45-70] for the mean curvature flow to a large class of speeds, and leads to an analogous description of "type-II" singularities. We remark that many of the speeds considered are positive on larger cones than the positive mean half-space, so that the result in those cases also applies to non-mean-convex initial data.

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Mixed volume preserving curvature flows

Cited by 86 publications

References 24 publications

Contraction of Horosphere-Convex Hypersurfaces by Powers of the Mean Curvature in the Hyperbolic Space

Contraction of Horosphere-Convex Hypersurfaces by Powers of the Mean Curvature in the Hyperbolic Space

Fully nonlinear curvature flow of axially symmetric hypersurfaces

Convexity estimates for hypersurfaces moving by convex curvature functions

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