Characterization of facet breaking for nonsmooth mean curvature flow in the convex case

Abstract A translating soliton is a hypersurface $M$ in ${\mathbb{R}}^{n+1}$ such that the family $M_t= M- t \,\mathbf e_{n+1}$ is a mean curvature flow, that is, such that normal component of the velocity at each point is equal to the mean curvature at that point $\mathbf{H}=\mathbf e_{n+1}^{\perp }.$ In this paper we obtain a characterization of hyperplanes that are parallel to the velocity and the family of tilted grim reaper cylinders as the only translating solitons in $\mathbb{R}^{n+1}$ that are $C^1$-asymptotic to two half-hyperplanes outside a non-vertical cylinder. This result was proven for translators in $\mathbb{R}^3$ by the 2nd author, Perez-Garcia, Savas-Halilaj, and Smoczyk under the additional hypotheses that the genus of the surface was locally bounded and the cylinder was perpendicular to the translating velocity.

Download Full-text

The dynamics of drops and attached interfaces for the constrained Allen–Cahn equation

European Journal of Applied Mathematics ◽

10.1017/s0956792501004272 ◽

2001 ◽

Vol 12 (1) ◽

pp. 1-24 ◽

Cited By ~ 10

Author(s):

D. STAFFORD ◽

M. J. WARD ◽

B. WETTON

Keyword(s):

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow ◽

Asymptotic Result ◽

Interface Motion ◽

Front Tracking Method ◽

Cahn Equation ◽

Area Preserving ◽

Dimensional Domain

The motion of interfaces for a mass-conserving Allen–Cahn equation that are attached to the boundary of a two-dimensional domain is studied. In the limit of thin interfaces, the interface motion for this problem is known to be governed by an area-preserving mean curvature flow. A numerical front-tracking method, that allows for a numerical solution of this type of curvature flow, is used to compute the motion of interfaces that are attached orthogonally to the boundary. Results obtained from these computations are favourably compared with a previously-derived asymptotic result for the motion of attached interfaces that enclose a small area. The area-preserving mean curvature flow predicts that a semi-circular interface is stationary when it is attached to a flat segment of the boundary. For this case, the interface motion is shown to be metastable and an explicit characterization of the metastability is given.

Download Full-text

Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

Open Mathematics ◽

10.1515/math-2020-0090 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1518-1530

Author(s):

Xuesen Qi ◽

Ximin Liu

Keyword(s):

Laplace Operator ◽

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow ◽

Second Fundamental Form ◽

Coefficient Function ◽

First Eigenvalue ◽

Initial Hypersurface ◽

The Mean ◽

The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

Download Full-text

Mean curvature flow with free boundary outside a hypersphere

Transactions of the American Mathematical Society ◽

10.1090/tran/7305 ◽

2017 ◽

Vol 369 (12) ◽

pp. 8319-8342 ◽

Cited By ~ 1

Author(s):

Glen Wheeler ◽

Valentina-Mira Wheeler

Keyword(s):

Free Boundary ◽

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow

Download Full-text

Ancient solutions for Andrews’ hypersurface flow

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2020-0020 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Peng Lu ◽

Jiuru Zhou

Keyword(s):

Ricci Flow ◽

Mean Curvature ◽

Mean Curvature Flow ◽

Curvature Flow ◽

Euclidean Spaces ◽

Round Cylinder ◽

Ancient Solutions ◽

Singular Time

AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.

Download Full-text