Convex solutions to the mean curvature flow

Abstract

In this paper we study the classification of ancient convex solutions to the mean curvature flow in Rn+1 . An open problem related to the classification of type II singularities is whether a convex translating solution is k-rotationally symmetric for some integer 2 ≤ k ≤ n, namely whether its level set is a sphere or cylinder S k−1 × R n−k . In this paper we give an affirmative answer for entire solutions in dimension 2. In high dimensions we prove that there exist nonrotationally symmetric, entire convex translating solutions, but the blow-down in space of any entire convex translating solution is k-rotationally symmetric. We also prove that the blow-down in space-time of an ancient convex solution which sweeps the whole space R n+1 is a shrinking sphere or cylinder.

Expansion of co-compact convex spacelike hypersurfaces in Minkowski space by their curvature

“…There is a great deal of literature concerning the evolution of hypersurfaces by geometric heat equations in Euclidean space, particularly concerning evolution by mean curvature [Hu1,Hu2,W2,W3,HS1,HS2,HS3,SW,A7,CM,W1], but also including many other evolution equations in which the speed is a nonlinear function of the principal curvatures [T,C,Ge1,U,A1,A3,A4,HI,A5,AMZ,ALM]. There is an analogous situation for spacelike hypersurfaces in Minkowski space, and in particular the evolution of spacelike hypersurfaces by mean curvature flow has received considerable attention (see, e.g., [E1, E2, E3]); see also [Ge2,Ge3,Ge4], where spacelike hypersurfaces are deformed by inverse mean curvature flows in more general Lorenzian background spaces.…”

Symmetry of Traveling Wave Solutions to the Allen–Cahn Equation in $${{\mathbb R^{2}}}$$

“…Recent studies on the translating mean curvature flow reveal very interesting properties of convex solutions. See [51], [50], [26] and references therein. In particular, it is shown in [50] that convex solutions to (1.9) must be rotationally symmetric for n ≤ 3.…”

“…See [51], [50], [26] and references therein. In particular, it is shown in [50] that convex solutions to (1.9) must be rotationally symmetric for n ≤ 3. It is then natural to ask whether a traveling wave solution to (1.1) with monotone (1.4) and limit condition (1.5) must be rotationally symmetric, or, in the terminology of this paper, axially symmetric after a proper translation in x variable.…”

“…It remains open whether all monotone traveling wave solutions with the limit condition (1.5) must be either axially symmetric or trivial extension of a axially symmetric solution in lower dimensional space. Due to possible existence of non-rotationally convex translating mean curvature flow in higher dimensions ( [51], [50]), the answer for the above question is probably not affirmative except for n = 3. The latter will be discussed in a forthcoming paper [24].…”

Solutions to the Allen Cahn Equation and Minimal Surfaces

“…Wang [36] has found examples for N ≥ 3 of convex, non-radial solutions. To overcome this difficulty, we need to improve the approximation:…”