Evolving plane curves by curvature in relative geometries II

Gage, Michael E.; Li, Yi

doi:10.1215/s0012-7094-94-07503-0

Cited by 63 publications

(44 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The basic reason is that such ideas crucially relied on the fact that, in the convex case, the normal velocity is always directed inward (which is clearly false for non-convex droplets, at points where the curvature is negative). Secondly, proving existence and regularity of solution requires very different analytic and geometric arguments in the non-convex case with respect to the convex one (there, we were able to use ideas from [15,13,14]). At any rate, our previous result [20] is important in Section 6, where the evolution of the droplet boundary is controlled by locally comparing it with that of a suitable convex droplet.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Heat Equation Shrinks Ising Droplets to Points

Lacoin

Simenhaus

Toninelli

2014

Comm Pure Appl Math

View full text Add to dashboard Cite

Abstract. Let D be a bounded, smooth enough domain of R 2 . For L > 0 consider the continuous time, zero-temperature heat bath dynamics for the nearest-neighbor Ising model on (Z/L) 2 (the square lattice with lattice spacing 1/L) with initial condition such that σx = −1 if x ∈ D and σx = +1 otherwise. We prove the following classical conjecture [24,5] due to H. Spohn: In the diffusive limit where time is rescaled by L 2 and L → ∞, the boundary of the droplet of "−" spins follows a deterministic anisotropic curve-shortening flow, such that the normal velocity is given by the local curvature times an explicit function of the local slope. Locally, in a suitable reference frame, the evolution of the droplet boundary follows the onedimensional heat equation.To our knowledge, this is the first proof of mean curvature-type droplet shrinking for a lattice model with genuine microscopic dynamics.An important ingredient is our recent work [20], where the case of convex D was solved. The other crucial point in the proof is obtaining precise regularity estimates on the deterministic curve shortening flow. This builds on geometric and analytic ideas of Grayson [16], GageHamilton [15], Gage-Li [13,14], Chou-Zhu [6] and others.

show abstract

Section: Introductionmentioning

confidence: 99%

“…The other crucial point in the proof is obtaining precise regularity estimates on the deterministic curve shortening flow. This builds on geometric and analytic ideas of Grayson [16], GageHamilton [15], Gage-Li [13,14], Chou-Zhu [6] and others. …”

mentioning

confidence: 99%

The Heat Equation Shrinks Ising Droplets to Points

Lacoin

Simenhaus

Toninelli

2014

Comm Pure Appl Math

View full text Add to dashboard Cite

show abstract

“…This problem has been extensively studied in the last two decades. See [1][2][3][4][5][6][7]15,16,18,19,[21][22][23][24][25]32]. Assuming that γ (·, t) is convex, then we can use the normal angle θ to parameterize γ , and Eq.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

2π-periodic self-similar solutions for the anisotropic affine curve shortening problem

2010

View full text Add to dashboard Cite

show abstract

“…For example, the solution to the even L p -Minkowski problem was one of the critical ingredients needed to obtain sharp affine Sobolev inequalities [13], [12] and, for p = 2, it has implications to the Cramer-Rao inequality, one of the basic inequalities in information theory [14]. Recent progress has been made in a variety of cases depending on p and n with a plethora of methods mostly from PDE, [1], [2], [3], [4], [5], [6], [7], [8], [9], [11], [15], [18], [19], [20] to cite just a few. Yet, much of the problem still presents a real challenge when p < 1.…”

mentioning

confidence: 99%

“…The uniqueness was established only for the even problem, [6]. In fact, even more interesting is the fact that for any p ≥ 1, and any n ≥ 1, the discrete L pMinkowski problem has a unique solution independent of the structure of the sets U ⊂ S n and , [9], where γ i corresponds now to the surface area of the i-th facet of outer normal → u i .…”

mentioning

confidence: 99%

The necessary condition for the discrete L0-Minkowski problem in $${\mathbb{R}}^{2}$$

Stancu

2008

J. geom.

View full text Add to dashboard Cite

We prove that the sufficiency condition employed to show the existence and, in certain cases the uniqueness, of solutions to the discrete, planar L 0 -Minkowski problem with data containing, at least, a pair of opposite vectors is also a necessary condition. (2000): 28A75, 52A10. Mathematics Subject ClassificationThe classical Minkowski problem deals with the existence, uniqueness, regularity and stability of closed, convex hypersurfaces whose Gauss curvature, viewed as a function on the unit sphere, is preassigned. For an atomic measure on the unit sphere, the question concerns the existence and uniqueness of polytopes with facets of fixed normal directions and fixed surface areas. In the planar setting, the Minkowski problem consists of a sufficient and necessary condition for the existence of a convex polygon whose sides have preassigned lengths and orientations:an ordered set of directions in S 1 , not all in a half-disk, and let L = {l 1 , ..., l N } be an ordered set of strictly positive numbers. There exists a convex N-gon whose i-th side has outer normal → u i and, respectively, length l i if and only ifThis is the simplest and the trivial case of Minkowski's problem as the aforementioned condition represents simply the closure of a polygonal line with the desired properties. One should note that the convex polygon so obtained is unique up to translation. See [16] for a detailed discussion on the full extent of Minkowski's problem.Due to Lutwak [10], a significantly more difficult question is whether a measure on the unit sphere S n can be realized as the L p -surface area measure of a convex body, where p = n is some fixed real number. If so, is this body unique? Lutwak showed within the Brunn-Minkowski-Firey theory that the classical problem, corresponding to p = 1, generalizes naturally to the L p -Minkowski problem stated above. In [10] a solution to the even L p -Minkowski problem in R n+1 was given for all p ≥ 1 except for p = n when it was shown that no solution is possible. The problem is called even if the measure takes equal values on opposite directions of S n . It is conjectured that in the even case the convex body, if it exists, is centrally symmetric, [10].162

show abstract

Evolving plane curves by curvature in relative geometries II

Cited by 63 publications

References 11 publications

The Heat Equation Shrinks Ising Droplets to Points

The Heat Equation Shrinks Ising Droplets to Points

2π-periodic self-similar solutions for the anisotropic affine curve shortening problem

The necessary condition for the discrete L0-Minkowski problem in $${\mathbb{R}}^{2}$$

Contact Info

Product

Resources

About