arXiv subjects
Publications and source records attributed to William P. Minicozzi II.
We show that all closed $2$-dimensional singularities for higher codimension mean curvature flow that cannot be perturbed away have uniform entropy bounds and lie in a linear subspace of small dimension. The entropy and dimension of the subspace are both $\leq C\,(1+γ)$ for some universal constant $C$ and genus $γ$. These are the first general bounds on generic singularities in arbitrary codimension.
We will look for stable structures in four situations and discuss what is known and unknown.
The classical Liouville theorem states that a bounded harmonic function on all of $\RR^n$ must be constant. In the early 1970s, S.T. Yau vastly generalized this, showing that it holds for manifolds with nonnegative Ricci curvature. Moreover, he conjectured a stronger Liouville property that has generated many significant developments. We will first discuss this conjecture and some of the ideas that went into its proof. We will also discuss two recent areas where this circle of ideas has played a major role. One is Kleiner's new proof of Gromov's classification of groups of polynomial growth and the developments this generated. Another is to understanding singularities of mean curvature flow in high codimension. We will see that some of the ideas discussed in this survey naturally lead to a new approach to studying and classifying singularities of mean curvature flow in higher codimension. This is a subject that has been notoriously difficult and where much less is known than for hypersurfaces.
Parabolic geometric flows are smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of [CM6] and here is that, by bringing in the dynamical properties of the flow, we obtain also smoothing for large time for generic initial conditions. When combined with [CM1], this shows, in an important special case, the singularities are the simplest possible. The question of the dynamics of a singularity has two parts. One is: What are the dynamics near a singularity? The second is: What is the long time behavior? That is, if the flow leaves a neighborhood of a singularity, can it return at a much later time? The first question was addressed in [CM6] and the second here. Combined with [CM1], [CM6], we show that all other closed singularities than the (round) sphere have a neighborhood where `nearly every' closed hypersurface leaves under the flow and never returns, even to a dilated, rotated or translated copy of the singularity. In other words, it wanders off. In contrast, by Huisken, any closed hypersurface near a sphere remains close to a dilated or translated copy of the sphere at each time.
Parabolic geometric flows have the property of smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of this paper is that, by bringing in the dynamical properties of the flow, we obtain also smoothing for long time for generic initial conditions. When combined with our earlier paper this allows us to show that, in an important special case, the singularities are the simplest possible. We take here the first steps towards understanding the dynamics of the flow. The question of the dynamics of a singularity has two parts. One is: What are the dynamics near a singularity? The second is: What is the long time behavior of the flow of things close to the singularity. That is, if the flow leaves a neighborhood of a singularity, is it possible for the flow to re-enter the same neighborhood at a much later time? The first part is addressed in this paper, while the second will be addressed elsewhere.
By a classical result, solutions of analytic elliptic PDEs, like the Laplace equation, are analytic. In many instances, the properties that come from being analytic are more important than analyticity itself. Many important equations are degenerate elliptic and solutions have much lower regularity. Still, one may hope that solutions share properties of analytic functions. These properties are closely connected to important open problems. In this survey, we will explain why solutions of an important degenerate elliptic equation have analytic properties even though the solutions are not even $C^3$.
We prove conjectures of Rene Thom and Vladimir Arnold for C^2 solutions to the degenerate elliptic equation that is the level set equation for motion by mean curvature. We believe these results are the first instances of a general principle: Solutions of many degenerate equations behave as if they are analytic, even when they are not. If so, this would explain various conjectured phenomena.
We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein-Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.
Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second order differential equations on Euclidean space. One naturally wonders "what is the regularity of solutions?" A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. Without deeply understanding the underlying geometry, it is impossible to prove fine analytical properties.
We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closed $C^1$ manifold with cylindrical singularities.
This paper deals with interactions between committee members as they rank a large list of applicants for a given position and eventually reach consensus. We will see that for a natural deterministic model the ranking can be described by solutions of a discrete quasilinear heat equation with time dependent coefficients on a graph. We show first that over time consensus emerges exponentially fast. Second, if there are clusters of members whose views are closer than those of the rest of the committee, then over time the clusters' views become closer at a faster exponential rate than the views of the entire committee. We will also show that the variance of the rankings decays a definite amount, independent of the initial variance, when the influence of the members does not decay too quickly as opinions differ. When the influence between members is exactly a negative power, then the variance is convex and satisfies a three circles theorem.
In real algebraic geometry, Lojasiewicz's theorem asserts that any integral curve of the gradient flow of an analytic function that has an accumulation point has a unique limit. Lojasiewicz proved this result in the early 1960s as a consequence of his gradient inequality. Many problems in calculus of variations are questions about critical points or gradient flow lines of an infinite dimensional functional. Perhaps surprisingly, even blowups at singularities of many nonlinear PDE's can, in a certain sense, be thought of as limits of infinite dimensional gradient flows of analytic functionals. Thus, the question of uniqueness of blowups can be approached as an infinite dimensional version of Lojasiewicz's theorem. The question of uniqueness of blowups is perhaps the most fundamental question about singularities. This approach to uniqueness was pioneered by Leon Simon thirty years ago for the area functional and many related functionals using an elaborate reduction to a finite dimensional setting where Lojasiewicz's arguments applied. Recently, the authors proved two new infinite dimensional Lojasiewicz inequalities at noncompact singularities where it was well-known that a reduction to Lojasiewicz's arguments is not possible, but instead entirely new techniques are required. As a consequence, the authors settled a major long-standing open question about uniqueness of blowups for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities. Using this, the authors have obtained a rather complete description of the space-time singular set for MCF with generic singularities. In particular, the singular set of a MCF in $\bf{R}^{n+1}$ with only generic singularities is contained in finitely many compact Lipschitz submanifolds of dimension at most $n-1$ together with a set of dimension at most $n-2$.
A mean curvature flow starting from a closed embedded hypersurface in $R^{n+1}$ must develop singularities. We show that if the flow has only generic singularities, then the space-time singular set is contained in finitely many compact embedded $(n-1)$-dimensional Lipschitz submanifolds plus a set of dimension at most $n-2$. If the initial hypersurface is mean convex, then all singularities are generic and the results apply. In $R^3$ and $R^4$, we show that for almost all times the evolving hypersurface is completely smooth and any connected component of the singular set is entirely contained in a time-slice. For $2$ or $3$-convex hypersurfaces in all dimensions, the same arguments lead to the same conclusion: the flow is completely smooth at almost all times and connected components of the singular set are contained in time-slices. A key technical point is a strong {\emph{parabolic}} Reifenberg property that we show in all dimensions and for all flows with only generic singularities. We also show that the entire flow clears out very rapidly after a generic singularity. These results are essentially optimal.
For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the critical set has finite codimensional two Hausdorff measure. For a monotonically advancing front, the arrival time is equivalent to the level set method; a priori not even differentiable but only satisfies the equation in the viscosity sense. Using that it is twice differentiable and that we can identify the Hessian at critical points, we show that it satisfies the equation in the classical sense. The arrival time has a game theoretic interpretation. For the linear heat equation, there is a game theoretic interpretation that relates to Black-Scholes option pricing. From variations of the Sard and Lojasiewicz theorems, we relate differentiability to whether or not singularities all occur at only finitely many times for flows.
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, [CM1] showed that the only generic are round cylinders $\SS^k\times \RR^{n-k}$. 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. To our knowledge, this is the first general rigidity theorem for singularities of a nonlinear geometric flow. We expect that the techniques and ideas developed here have applications to other flows. Our results hold in all dimensions and do not require any a priori smoothness.
Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow, by the combined work of Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave open the possibility that a minimal variety or a mean curvature flow looked at under a microscope will resemble one blowup, but under higher magnification, it might (as far as anyone knows) resemble a completely different blowup. Whether this ever happens is perhaps the most fundamental question about singularities. It is this long standing open question that we settle here for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities.
This paper is the fifth and final in a series on embedded minimal surfaces. Following our earlier papers on disks, we prove here two main structure theorems for non-simply connected embedded minimal surfaces of any given fixed genus. The first of these asserts that any such surface without small necks can be obtained by gluing together two oppositely--oriented double spiral staircases. The second gives a pair of pants decomposition of any such surface when there are small necks, cutting the surface along a collection of short curves. After the cutting, we are left with graphical pieces that are defined over a disk with either one or two sub--disks removed (a topological disk with two sub--disks removed is called a pair of pants). Both of these structures occur as different extremes in the two-parameter family of minimal surfaces known as the Riemann examples. The results of [CM3]--[CM6] have already been used by many authors; see, e.g., the surveys [MeP], [P], [Ro] and the introduction in [CM6] for some of these applications. There is much current research on minimal surfaces with infinite topology. Some of the results of the present paper were announced previously and have already been widely used to study infinite topology minimal surfaces; see, e.g., [MeP], [MePRs1], [MePRs2], [MePRs3], and [P].
In this paper we generalize the monotonicity formulas of [C] for manifolds with nonnegative Ricci curvature. Monotone quantities play a key role in analysis and geometry; see, e.g., [A], [CM1] and [GL] for applications of monotonicity to uniqueness. Among the applications here is that level sets of Green's function on open manifolds with nonnegative Ricci curvature are asymptotically umbilic.