arXiv subjects
Publications and source records attributed to William P. Minicozzi II.
We give examples of proper minimal immersions in Euclidean space with very rapid area growth. The first is a proper embedding into $\bf{R}^4$ that yields a stable minimal surface, while the second is a proper immersion into $\bf{R}^3$. These results are motivated by [CM1] that shows that proper minimal submanifolds confined in space satisfy strong structural constraints.
We survey a circle of recent results showing that a minimal submanifold obeys a dichotomy: either it fills up space, spreading out like a space-filling curve, or it is confined, and confinement forces quantitative restrictions. On the confined side these restrictions are both geometric and function-theoretic: Euclidean volume growth, an optimal rate of convergence of the density, complex-curve rigidity for stable surfaces in $\bf{R}^4$, and a Liouville theorem forcing slowly growing harmonic functions on minimal disks to be constant. A prototypical restriction is Euclidean volume growth, which in these results is forced by the geometry rather than assumed. The mechanism is a volume doubling theorem that converts geometric confinement into quantitative rigidity: a stationary integral varifold trapped in a thin slab at a given scale cannot double its volume by more than a universal factor. We explain how this principle, and the height-excess bounds behind it, produce Euclidean volume growth for submanifolds of sublinearly growing height in every dimension and codimension, the optimal density rate in a slab, the complex-curve and Liouville rigidity above, a higher-codimension Bernstein theorem for disks, one-sided volume bounds, and an optimal stable Bernstein theorem in all dimensions generalizing Moser, Bombieri-De Giorgi-Miranda, Caffarelli-Nirenberg-Spruck and Ecker-Huisken. We also explain how this entire picture emerges from the structure theory of embedded minimal disks in $\bf{R}^3$, built on the one-sided curvature estimate and reflected globally in the half-space theorem, and how it extends as a weak analogue of that theory to all dimensions.
Already in $\bf{R}^4$, there are many minimal hypersurfaces, yet few structural results. We show that minimal submanifolds, of any dimension and codimension, that are confined in space are very restricted. It is well-known that the half-space theorem fails already for hypersurfaces in $\bf{R}^4$, where there are many examples contained in a slab. In $\bf{R}^3$ the height of the catenoid grows at a logarithmic rate, whereas in higher dimensions the height of the catenoid remains bounded. We will see that even in high dimensions, minimal submanifolds that are confined in space must satisfy strong structural restrictions. We show that any proper minimal immersion whose height grows sublinearly must have Euclidean volume growth. A consequence is an optimal Bernstein theorem in any dimension for stable hypersurfaces with sublinearly growing height that generalizes results of Moser, Bombieri-De Giorgi-Miranda, Trudinger, Caffarelli-Nirenberg-Spruck and Ecker-Huisken. Euclidean volume growth is a powerful property and there are many other consequences.
For a proper immersed minimal disk in $\bf{R}^N$ with quadratic area growth, we show that any harmonic function whose negative part grows at a slow sub-linear rate is constant. This leads to a higher codimensional Bernstein theorem for minimal disks contained in a sub-linearly growing cone. The catenoid, helicoid and Enneper's family of surfaces together show that this result is optimal. We also show uniform Hölder regularity of harmonic functions.
We will show that the distance between two minimal hypersurfaces is a Lipschitz continuous supersolution, in the viscosity sense, of a natural elliptic partial differential equation. This not only recovers several well-known properties of minimal hypersurfaces, but also encodes substantially richer information. Moreover, if the reference hypersurface is allowed to evolve by mean curvature flow, one obtains comparably strong estimates for a corresponding parabolic PDE, leading in particular to local Harnack inequalities for the distance. There is even a fully parabolic extension in which both hypersurfaces evolve. The problem of tracking the distance between two evolving hypersurfaces arises naturally in a wide range of settings.
There is a long history of parabolic monotonicity formulas that developed independently from several different fields and a much more recent elliptic theory. The elliptic theory can be localized and there are additional monotone quantities. There is also a surprising link: Taking a high-dimensional limit of the right elliptic monotonicity can give a parabolic one as a limit. Poincaré was the first to observe such a connection. We introduce two deficit functions, one elliptic and one parabolic, then show that the parabolic deficit is pointwise the limit of the elliptic and, that the elliptic satisfies an equation that converges to the equation for the parabolic. These pointwise quantities and their equations recover the monotonicities and leads to an elliptic proof of the log Sobolev inequality as well as new concentration of measure phenomena.
Comparing and recognizing metrics can be extraordinarily difficult because of the group of diffeomorphisms. Two metrics, that could even be the same, could look completely different in different coordinates. This is the gauge problem. The general gauge problem is extremely subtle for non-compact spaces. Often it can be avoided if one uses some additional structure of the particular situation. However, in many problems there is no additional structure. Instead we solve the gauge problem directly in great generality. The techniques and ideas apply to many problems. We use them to solve a well-known open problem in Ricci flow: Strong rigidity of cylinders. Strong rigidity is an illustration of a {\it shrinker principle} that uniqueness radiates out from a compact set. It implies that if one tangent flow at a future singular point is a cylinder, then all tangent flows are. We solve the gauge problem by solving a nonlinear system of PDEs. The PDE produces a diffeomorphism that fixes an appropriate gauge in the spirit of the slice theorem for group actions. We then show optimal bounds for the displacement function of the diffeomorphism. Strong rigidity relies on gauge fixing and several other new ideas. One of these is "propagation of almost splitting", another is quadratic rigidity in the right gauge, and a third is an optimal polynomial growth bound for PDEs that holds in great generality.
We show how to use the arguments of [CM2] to get a stronger effective version of uniqueness of blowups that has a number of consequences.
A gradient estimate is a crucial tool used to control the rate of change of a function on a manifold, paving the way for deeper analysis of geometric properties. A celebrated result of Cheng and Yau gives gradient bounds on manifolds with Ricci curvature $\geq 0$. The Cheng-Yau bound is not sharp, but there is a gradient sharp estimate. To explain this, a Green's function $u$ on a manifold can be used to define a regularized distance $b= u^{\frac{1}{2-n}}$ to the pole. On $\bf{R}^n$, the level sets of $b$ are spheres and $|\nabla b|=1$. If $\text{Ric} \geq 0$, then [C3] proved the sharp gradient estimate $|\nabla b| \leq 1$. We show that the average of $|\nabla b|$ is $\leq 1$ on a three manifold with nonnegative scalar curvature. The average is over any level set of $b$ and if the average is one on even one level set, then $M=\bf{R}^3$.
We prove a strong Frankel theorem for mean curvature flow shrinkers in all dimensions: Any two shrinkers in a sufficiently large ball must intersect. In particular, the shrinker itself must be connected in all large balls. The key to the proof is a strong Bernstein theorem for incomplete stable Gaussian surfaces.
We will show that if a gradient shrinking Ricci soliton has an approximate symmetry on one scale, this symmetry propagates to larger scales. This is an example of the shrinker principle which roughly states that information radiates outwards for shrinking solitons.
We prove sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow. The modified Ricci flow is a system of coupled equations for a metric and weighted volume that plays an important role in Ricci flow. We will also show that there is a splitting theorem in the case of equality.
Comparing and recognizing metrics can be extraordinarily difficult because of the group of diffeomorphisms. Two metrics, that could even be the same, could look completely different in different coordinates. This is the gauge problem. The general gauge problem is extremely subtle, especially for non-compact spaces. Often it can be avoided if one uses some additional structure of the particular situation. However, in many problems there is no additional structure. Instead we solve the gauge problem directly in great generality. The techniques and ideas apply to many problems. We use them to solve a well-known open problem in Ricci flow. We solve the gauge problem by solving a nonlinear system of PDEs. The PDE produces a diffeomorphism that fixes an appropriate gauge in the spirit of the slice theorem for group actions. We then show optimal bounds for the displacement function of the diffeomorphism.
Analysis of non-compact manifolds almost always requires some controlled behavior at infinity. Without such, one neither can show, nor expect, strong properties. On the other hand, such assumptions restrict the possible applications and often too severely. In a wide range of areas non-compact spaces come with a Gaussian weight and a drift Laplacian. Eigenfunctions are $L^2$ in the weighted space allowing for extremely rapid growth. Rapid growth would be disastrous for many applications. Surprisingly, for very general tensors, manifolds and weights, we show the same polynomial growth bounds that Laplace and Hermite observed for functions on Euclidean space for the standard Gaussian. This covers all shrinkers for Ricci and mean curvature flows. These results open a door for understanding general non-compact spaces. It provides an analytic framework for doing nonlinear PDE on Gaussian spaces where previously the Gaussian weight allowed wild growth that made it impossible to approximate nonlinear by linear. It is key to bound the growth of diffeomorphisms of non-compact manifolds and is the key for solving the "gauge problem". The relative nature of the estimates and the slow growth in the bounds lead to "propagation of almost splitting" that is significantly stronger than pseudo locality and key for applications.
For any manifold with polynomial volume growth, we show: The dimension of the space of ancient caloric functions with polynomial growth is bounded by the degree of growth times the dimension of harmonic functions with the same growth. As a consequence, we get a sharp bound for the dimension of ancient caloric functions on any space where Yau's 1974 conjecture about polynomial growth harmonic functions holds.
We prove monotonicity of a parabolic frequency on manifolds. This is a parabolic analog of Almgren's frequency function. Remarkably we get monotonicity on all manifolds and no curvature assumption is needed. When the manifold is Euclidean space and the drift operator is the Ornstein-Uhlenbeck operator this can been seen to imply Poon's frequency monotonicity for the ordinary heat equation. Monotonicity of frequency is a parabolic analog of the 19th century Hadamard three circles theorem about log convexity of holomorphic functions on $\CC$. From the monotonicity, we get parabolic unique continuation and backward uniqueness.
We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the most prevalent in any codimension. Mean curvature flow in higher codimension is a nonlinear parabolic system where many of the methods used for hypersurfaces do not apply and uniqueness of cylindrical blowups remained a major open problem. Our results imply regularity of the singular set for the system.
We first bound the codimension of an ancient mean curvature flow by the entropy. As a consequence, all blowups lie in a Euclidean subspace whose dimension is bounded by the entropy and dimension of the evolving submanifolds. This drastically reduces the complexity of the system. Combined with \cite{CM12}, this gives the first general bounds on generic singularities of surfaces in arbitrary codimension. We also show sharp bounds for codimension in arguably some of the most important situations of ancient flows. Namely, we prove that in any dimension and codimension any ancient flow that is cylindrical at $-\infty$ must be a flow of hypersurfaces in a Euclidean subspace. This extends well-known classification results to higher codimension. The bound on the codimension in terms of the entropy is a special case of sharp bounds for spectral counting functions for shrinkers and, more generally, ancient flows. Shrinkers are solutions that evolve by scaling and are the singularity models for the flow. Finally, we show rigidity of cylinders as shrinkers in all dimension and all codimension in a very strong sense: Any shrinker, even in a large dimensional space, that is sufficiently close to a cylinder on a large enough, but compact, set is itself a cylinder. This is an important tool in the theory and is key for regularity; cf. \cite{CM8}.