Soliton solutions of the mean curvature flow and minimal hypersurfaces

Hungerbühler, Norbert; Mettler, Thomas

doi:10.1090/s0002-9939-2011-11205-x

Cited by 13 publications

(21 citation statements)

References 9 publications

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“…We observed in §3.1 that there is a one-to-one correspondence between Curve Shortening on the sphere S n−1 and certain shrinking solutions of Curve Shortening in R n . Under this correspondence some of the rotating shrinking solitons we study in this paper correspond to purely rotating solitons on S n−1 as studied by Hungerbühler and Smoczyk in [10] (e.g. see Figure 6 in [10]).…”

Section: Rotating Shrinking Solitons In Rmentioning

confidence: 73%

“…Under this correspondence some of the rotating shrinking solitons we study in this paper correspond to purely rotating solitons on S n−1 as studied by Hungerbühler and Smoczyk in [10] (e.g. see Figure 6 in [10]). In particular, every rotating soliton on S n−1 is also a shrinking rotating soliton for Curve Shortening on R n .…”

Section: Rotating Shrinking Solitons In Rmentioning

confidence: 73%

“…Recall that we had chosen coordinates in which the matrix of M is given by (10). The exponential e M therefore acts on R n by rotating the (x 2j−1 , x 2j ) component by an angle ω j for j = 1, .…”

Section: Circles and Shrinking Helicesmentioning

confidence: 99%

“…These were studied by Hungerbühler and Smoczyk in [10] (in [10] solitons on other surfaces were also considered). The connection is explained in §3.1 and §6.3.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The zoo of solitons for curve shortening in $\mathbb{R}^n$

Altschuler¹,

Wu²,

Altschuler³

et al. 2013

Nonlinearity

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Abstract. We provide a detailed description of solutions of Curve Shortening in R n that are invariant under some one-parameter symmetry group of the equation, paying particular attention to geometric properties of the curves, and the asymptotic properties of their ends. We find generalized helices, and a connection with curve shortening on the unit sphere S n−1 . Expanding rotating solitons turn out to be asymptotic to generalized logarithmic spirals. In terms of asymptotic properties of their ends the rotating shrinking solitons are most complicated. We find that almost all of these solitons are asymptotic to circles.Many of the curve shortening solitons we discuss here are either space curves, or evolving space curves. In addition to the figures in this paper, we have prepared a number of animations of the solitons, which can be viewed at

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Section: Rotating Shrinking Solitons In Rmentioning

confidence: 73%

Section: Rotating Shrinking Solitons In Rmentioning

confidence: 73%

Section: Circles and Shrinking Helicesmentioning

confidence: 99%

“…These were studied by Hungerbühler and Smoczyk in [10] (in [10] solitons on other surfaces were also considered). The connection is explained in §3.1 and §6.3.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The zoo of solitons for curve shortening in $\mathbb{R}^n$

Altschuler¹,

Wu²,

Altschuler³

et al. 2013

Nonlinearity

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“…where H is the mean curvature of M and ν its unit normal vector field. A proof can be found for example in [16]. Special solitons are the translators in the Euclidean space: V is a constant vector field and therefore M evolves translating with constant speed in the direction of V .…”

Section: Introductionmentioning

confidence: 99%

Invariant Translators of the Heisenberg Group

Pipoli

2020

J Geom Anal

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We classify all the translating solitons to the mean curvature flow in the threedimensional Heisenberg group that are invariant under the action of some oneparameter group of isometries of the ambient manifold. We provide a complete classification for any canonical deformation of the standard Riemannian metric of the Heisenberg group. We highlight similarities and differences with the analogous Euclidean translators: we mention in particular that we describe the analogous of the tilted grim reaper cylinders, of the bowl solution and of translating catenoids, but some of them are not convex in contrast with a recent result of Spruck and Xiao (Complete translating solitons to the mean curvature flow in R 3 with non-negative mean curvature, arXiv:1703.01003v2, 2017) in the Euclidean space. Moreover we also prove some non-existence results. Finally we study the convergence of these surfaces as the ambient metric converges to the standard sub-Riemannian metric on the Heisenberg group.

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Solitons of Discrete Curve Shortening

Rademacher

2016

Results Math

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Abstract. For a polygon x = (xj) j∈Z in R n we consider the polygon (T (x))j = {xj−1 + 2xj + xj+1} /4 . This transformation is obtained by applying the midpoints polygon construction twice. For a closed polygon or a polygon with finite vertices this is a curve shortening process. We call a polygon x a soliton of the transformation T if the polygon T (x) is an affine image of x. We describe a large class of solitons for the transformation T by considering smooth curves c which are solutions of the differential equationc(t) = Bc(t)+d for a real matrix B and a vector d. The solutions of this differential equation can be written in terms of power series in the matrix B. For a solution c and for any s > 0, a ∈ R the polygon x(a, s) = (xj(a, s)j) j∈Z ; xj(a, s) = c(a + sj) is a soliton of T. For example we obtain solitons lying on spiral curves which under the transformation T rotate and shrink.

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Soliton solutions of the mean curvature flow and minimal hypersurfaces

Cited by 13 publications

References 9 publications

The zoo of solitons for curve shortening in $\mathbb{R}^n$

The zoo of solitons for curve shortening in $\mathbb{R}^n$

Invariant Translators of the Heisenberg Group

Solitons of Discrete Curve Shortening

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