Solutions Of Mean Curvature Flow Research Articles

BV solutions for mean curvature flow with constant angle: Allen-Cahn approximation and weak-strong uniqueness

BV solutions for mean curvature flow with constant angle: Allen-Cahn approximation and weak-strong uniqueness

Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

Given a double cone $${\mathcal {C}}$$ with entropy at most two that is symmetric across some hyperplane, we show that any integral Brakke flow coming out of the cone must inherit the reflection symmetry for all time, provided the flow is smooth for a short time. As a corollary we prove that any such flow coming out of a rotationally symmetric double cone must stay rotationally symmetric for all time. We also show the existence of a non-self-similar flow coming out of a double cone with entropy at most two, and give an example of such a flow with a finite time singularity. Additionally, we show the existence of self-expanders with triple junctions, which are exceptions to our main theorem.

Singularity models of pinched solutions of mean curvature flow in higher codimension

Abstract We consider ancient solutions to the mean curvature flow in R n + 1 \mathbb{R}^{n+1} ( n ≥ 3 n\geq 3 ) that are weakly convex, uniformly two-convex, and satisfy two pointwise derivative estimates | ∇ ⁡ A | ≤ γ 1 ⁢ | H | 2 \lvert\nabla A\rvert\leq\gamma_{1}\lvert H\rvert^{2} , | ∇ 2 ⁡ A | ≤ γ 2 ⁢ | H | 3 \lvert\nabla^{2}A\rvert\leq\gamma_{2}\lvert H\rvert^{3} . We show that such solutions are noncollapsed. As an application, in arbitrary codimension, we consider compact 𝑛-dimensional ( n ≥ 5 n\geq 5 ) solutions to the mean curvature flow in R N \mathbb{R}^{N} that satisfy the pinching condition | A | 2 < c ⁢ | H | 2 \lvert A\rvert^{2}<c\lvert H\rvert^{2} for a suitable constant c = c ⁢ ( n ) c=c(n) . We conclude that any blow-up model at the first singular time must be a codimension one shrinking sphere, shrinking cylinder, or translating bowl soliton.

Uniqueness of convex ancient solutions to hypersurface flows

Abstract We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which are convex or concave in the second fundamental form. Neither of these results assumes interior noncollapsing.

A planarity estimate for pinched solutions of mean curvature flow

We show that the blow-ups of compact solutions to the mean curvature flow in $\mathbb{R}^N$ initially satisfying the pinching condition $|A|^2 < c |H|^2$ for a suitable constant $c = c(n)$ must be codimension one.

Collapsing ancient solutions of mean curvature flow

We construct a compact, convex ancient solution of mean curvature flow in $\mathbb{R}^{n+1}$ with $O(1) \times O(n)$ symmetry that lies in a slab of width $\pi$. We provide detailed asymptotics for this solution and show that, up to rigid motions, it is the only compact, convex, $O(n)$-invariant ancient solution that lies in a slab of width $\pi$ and in no smaller slab.

Ancient solution of mean curvature flow in space forms

In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere. Then we show that under certain pointwise curvature pinching condition the ancient solution in a hyperbolic space is a family of shrinking spheres. We also obtain a rigidity result for ancient solutions in a nonnegatively curved space form under an asymptotic integral curvature pinching condition.

On the existence of translating solutions of mean curvature flow in slab regions

We prove, in all dimensions n≥2, that there exists a convex translator lying in a slab of width πsec𝜃 in ℝn+1 (and in no smaller slab) if and only if 𝜃∈[0,π2]. We also obtain convexity and regularity results for translators which admit appropriate symmetries and study the asymptotics and reflection symmetry of translators lying in slab regions.

On the entropy of closed hypersurfaces and singular self-shrinkers

Self-shrinkers are the special solutions of mean curvature flow in $\mathbf{R}^{n+1}$ that evolve by shrinking homothetically; they serve as singularity models for the flow. The entropy of a hypersurface introduced by Colding–Minicozzi is a Lyapunov functional for the mean curvature flow, and is fundamental to their theory of generic mean curvature flow. In this paper we prove that a conjecture of Colding–Ilmanen–Minicozzi–White, namely that any closed hypersurface in $\mathbf{R}^{n+1}$ has entropy at least that of the round sphere, holds in any dimension $n$. This result had previously been established for the cases $n \leq 6$ by Bernstein–Wang using a carefully constructed weak flow. The main technical result of this paper is an extension of Colding–Minicozzi’s classification of entropy-stable self-shrinkers to the singular setting. In particular, we show that any entropystable self-shrinker whose singular set satisfies Wickramasekera’s α-structural hypothesis must be a round cylinder $\mathbf{S}^k (\sqrt{2k}) \times \mathbf{R}^{n-k}$.

Smooth Compactness for Spaces of Asymptotically Conical Self-Expanders of Mean Curvature Flow

AbstractWe show compactness in the locally smooth topology for certain natural families of asymptotically conical self-expanding solutions of mean curvature flow. Specifically, we show such compactness for the set of all 2D self-expanders of a fixed topological type and, in all dimensions, for the set of self-expanders of low entropy and for the set of mean convex self-expanders with strictly mean convex asymptotic cones. From this we deduce that the natural projection map from the space of parameterizations of asymptotically conical self-expanders to the space of parameterizations of the asymptotic cones is proper for these classes.

Uniqueness of convex ancient solutions to mean curvature flow in $${\mathbb {R}}^3$$ R 3

A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension 3 which have positive sectional curvature and are $$\kappa $$ -noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in $${\mathbb {R}}^3$$ , and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in $${\mathbb {R}}^3$$ which is strictly convex and noncollapsed.

Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures

We provide a uniqueness result for a class of viscosity solutions to sub-Riemannian mean curvature flows. In a sub-Riemannian setting, uniqueness cannot be deduced by the comparison principle, which is known only for graphs and for radially symmetry surfaces. Here we use a definition of continuous viscosity solutions of sub-Riemannian mean curvature flows motivated from a regularized Riemannian approximation of the flow. With this definition, we prove that any continuous viscosity solution of the equation is a limit of a sequence of solutions of Riemannian flow and obtain as a consequence uniqueness and the comparison principle. The results are provided in the settings of both 3-dimensional rototranslation group ${SE}(2)$ and Carnot groups of step 2, which are particularly important due to their relation to the surface completion problem of a model of the visual cortex.

A general pinching principle for mean curvature flow and applications

We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken (J Differ Geom 20(1):237–266, 1984), the convexity estimates of Huisken–Sinestrari (Acta Math 183(1):45–70, 1999) and the cylindrical estimate of Huisken–Sinestrari (Invent Math 175(1):137–221, 2009; see also Andrews and Langford in Anal PDE 7(5):1091–1107, 2014; Huisken and Sinestrari in J Differ Geom 101(2):267–287, 2015). Namely, we show that the curvature of the solution pinches onto the convex cone generated by the curvatures of any shrinking cylinder solutions admitted by the initial data. For example, if the initial data is \((m+1)\)-convex, then the curvature of the solution pinches onto the convex hull of the curvatures of the shrinking cylinders \(\mathbb {R}^m\times S^{n-m}_{\sqrt{2(n-m)(1-t)}}\), \(t<1\). In particular, this yields a sharp estimate for the largest principal curvature, which we use to obtain a new proof of a sharp estimate for the inscribed curvature for embedded solutions (Brendle in Invent Math 202(1):217–237, 2015; Haslhofer and Kleiner in Int Math Res Not 15:6558–6561, 2015; Langford in Proc Am Math Soc 143(12):5395–5398, 2015). Making use of a recent idea of Huisken–Sinestrari (2015), we then obtain a series of sharp estimates for ancient solutions. In particular, we obtain a convexity estimate for ancient solutions which allows us to strengthen recent characterizations of the shrinking sphere due to Huisken–Sinestrari (2015) and Haslhofer–Hershkovits (Commun Anal Geom 24(3):593–604, 2016).

Rigidity of generic singularities of mean curvature flow

Shrinkers are special solutions of mean curvature flow (MCF) that evolve by rescaling and model the singularities. While there are infinitely many in each dimension, Colding and Minicozzi II (Ann. Math. 175(2):755–833, 2012) showed that the only generic are round cylinders Sk×Rn−k. We prove here that round cylinders are rigid in a very strong sense. Namely, any other shrinker that is sufficiently close to one of them on a large, but compact, set must itself be a round cylinder.

Foliations of a smooth metric measure space by hypersurfaces with constantf-mean curvature

We study smooth codimension-one foliations F of a smooth metric measure space whose leaves have the same constant f -mean curvature. Firstly, we show that all the leaves of F are f -minimal hypersurfaces when either the smooth metric measure space is compact and has nonnegative Bakry‐ Emery Ricci curvature, or the limit of the ratio of the weighted volume of a geodesic ball B and the weighted area of a geodesic sphere @B vanishes. Secondly, we prove that every leaf of F is strongly f -stable. Lastly, we show that there is no complete proper foliation of the Gaussian space whose leaves have the same constant f -mean curvature. In particular, there are no foliations of R nC1 whose leaves are complete proper self-similar solutions for mean curvature flow.

The extension of the H k mean curvature flow in Riemannian manifolds

In this paper, the authors consider a family of smooth immersions Ft: Mn → Nn+1 of closed hypersurfaces in Riemannian manifold Nn+1 with bounded geometry, moving by the Hk mean curvature flow. The authors show that if the second fundamental form stays bounded from below, then the Hk mean curvature flow solution with finite total mean curvature on a finite time interval [0, Tmax) can be extended over Tmax. This result generalizes the extension theorems in the paper of Li (see “On an extension of the Hk mean curvature flow, Sci. China Math., 55, 2012, 99–118”).

Martingales arising from minimal submanifolds and other geometric contexts

We consider a class of martingales on Cartan–Hadamard manifolds that includes Brownian motion on a minimal submanifold. We give sufficient conditions for such martingales to be transient, extending previous results on the transience of minimal submanifolds. We also give conditions for the almost sure convergence of the angular component (in polar coordinates) of a martingale in this class, including both the negatively pinched case (using earlier results on martingales of bounded dilation), and the radially symmetric case with quadratic decay of the upper curvature bound. Applied to minimal submanifolds, this gives curvature conditions on the ambient Cartan–Hadamard manifold under which any minimal submanifold admits a non-constant, bounded, harmonic function. Though our discussion is primarily motivated by minimal submanifolds, this class of martingales includes diffusions naturally associated to ancient solutions of mean curvature flow and to certain sub-Riemannian structures, and we briefly discuss these contexts as well. Our techniques are elementary, consisting mainly of comparison geometry and Ito’s rule.

A general regularity theory for weak mean curvature flow

We give a new proof of Brakke's partial regularity theorem up to C^{1,\varsigma} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard's regularity theorem in the sense that the special time-independent case reduces to Allard's theorem.

The extension for mean curvature flow with finite integral curvature in Riemannian manifolds

We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature at a finite time interval [0, T) can be extended over time T. Moreover, we show that the condition is optimal in some sense.

Extend Mean Curvature Flow with Finite Integral Curvature

In this note, we first prove that the solution of mean curvature flow on a finite time interval [0,T) can be extended over time T if the space-time integration of the norm of the second fundamental form is finite. Secondly, we prove that the solution of certain mean curvature flow on a finite time interval [0,T) can be extended over time T if the space-time integration of the mean curvature is finite. Moreover, we show that these conditions are optimal in some sense.