Stability of translating solutions to mean curvature flow

Clutterbuck, Julie; Schnürer, Oliver C.; Schulze, Felix

doi:10.1007/s00526-006-0033-1

Cited by 161 publications

(156 citation statements)

References 10 publications

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“…Uniqueness of solutions to Ricci flow with bounded curvature on non-compact manifolds is discussed in [2,8]. J. Clutterbuck and the first two authors proved stability of convex rotationally symmetric translating solutions to mean curvature flow in [3]. Short time existence results for C 0 -metrics were shown in [14] using similar techniques to this paper.…”

Section: Introductionmentioning

confidence: 73%

Stability of Euclidean space under Ricci flow

Schnürer¹,

Schulze²,

Simon³

2008

Communications in Analysis and Geometry

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Abstract. We study the Ricci flow for initial metrics which are C 0 small perturbations of the Euclidean metric on R n . In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In proving this stability result, we introduce a monotone integral quantity which measures the deviation of the evolving metric from the Euclidean metric. We also investigate the convergence of the diffeomorphisms relating Ricci harmonic map heat flow to Ricci flow.

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Section: Introductionmentioning

confidence: 73%

Stability of Euclidean space under Ricci flow

Schnürer¹,

Schulze²,

Simon³

2008

Communications in Analysis and Geometry

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“…it is well-known [14] that there exists a one-dimensional travelling front v(t, z) = φ 0 (z+c 0 t) solution to (4) with N = 1. The speed c 0 is unique and strictly positive by [iv] while the profile φ 0 is unique up to translations.…”

Section: Connection With Reaction Diffusion Equationsmentioning

confidence: 99%

“…In this case, c 0 = 0 and the forced mean curvature equation is replaced by the mean curvature equation. Chen, Guo, Hamel, Ninomiya and Roquejoffre [3] proved that there exist cylindrically symmetric traveling waves with paraboloid like interfaces solutions to (4) in dimension N ≥ 3. Precisely, they proved that those solutions' level sets are asymptotically given by the equation z = c 2(N −1) |x| 2 .…”

Section: Connection With Reaction Diffusion Equationsmentioning

confidence: 99%

“…Precisely, they proved that those solutions' level sets are asymptotically given by the equation z = c 2(N −1) |x| 2 . On the other hand, Clutterbuck, Schnürer and Schulze [4] proved that there exists a unique rotationally symmetric, strictly convex, translating graph u(t, x) = −ct + φ(r) to the mean curvature motion (3) with c 0 = 0 and whose asymptotics is given by…”

Section: Connection With Reaction Diffusion Equationsmentioning

confidence: 99%

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Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

Monneau¹,

Roquejoffre²,

Roussier-Michon³

2013

Ann. Sci. École Norm. Sup.

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36 pages, 6 figuresInternational audienceWe construct travelling wave graphs of the form z = -ct phi(x), phi : x is an element of RN-1 -> phi(x) is an element of R, N >= 2, solutions to the N-dimensional forced mean curvature motion V-n = -co + k (c >= co) with prescribed asymptotics. For any 1-homogeneous function phi(infinity) , viscosity solution to the eikonal equation vertical bar D phi(infinity)vertical bar =root(c/c0)(2) - 1, we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by phi(infinity). We also describe phi(infinity) in terms of a probability measure on SN-2

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“…• Altschuler and Wu [3], Clutterbuck, Schnürer and Schulze [11]: A unique radially symmetric solution (for c = 1, N ≥ 2)…”

Section: The Allen Cahn Equation and Minimal Surfaces 61mentioning

confidence: 99%

Solutions to the Allen Cahn Equation and Minimal Surfaces

Pino

Wei

2011

Milan J. Math.

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Abstract. We discuss and outline proofs of some recent results on application of singular perturbation techniques for solutions in entire space of the Allen-Cahn equation Δu + u − u 3 = 0. In particular, we consider a minimal surface Γ in R 9 which is the graph of a nonlinear entire function x 9 = F (x 1 , . . . , x 8 ), found by Bombieri, De Giorgi and Giusti, the BDG surface. We sketch a construction of a solution to the Allen Cahn equation in R 9 which is monotone in the x 9 direction whose zero level set lies close to a large dilation of Γ, recently obtained by M. Kowalczyk and the authors. This answers a long standing question by De Giorgi in large dimensions (1978), whether a bounded solution should have planar level sets. We sketch two more applications of the BDG surface to related questions, respectively in overdetermined problems and in eternal solutions to the flow by mean curvature for graphs.

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Stability of translating solutions to mean curvature flow

Abstract: Abstract. We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates.

Cited by 161 publications

References 10 publications

Stability of Euclidean space under Ricci flow

Stability of Euclidean space under Ricci flow

Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

Solutions to the Allen Cahn Equation and Minimal Surfaces

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