William Minicozzi

Sharp estimates for mean curvature flow of graphs are shown - a gradient estimate and an area estimate - and examples are given to illustrate why these are sharp. The gradient estimate improves an earlier (non-sharp) estimate of Klaus Ecker and Gerhard Huisken (joint with Colding, Crelles Journal, volume 574, 2004); LANL link.

Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show:
Each curve in the tightened sweepout whose length is close to the length of the longest curve in the sweepout must itself be close to a closed geodesic. In particular, there are curves in the sweepout that are close to closed geodesics.
Finding closed geodesics on the 2-sphere by using sweepouts goes back to Birkhoff in 1917. As an application, we bound from above, by a negative constant, the rate of change of the width for a one-parameter family of convex hypersurfaces that flows by mean curvature. The width is loosely speaking up to a constant the square of the length of the shortest closed curve needed to ``pull over'' $M$. This estimate is sharp and leads to a sharp estimate for the extinction time; cf. above where a similar bound for the rate of change for the two dimensional width is shown for homotopy 3-spheres evolving by the Ricci flow (see also Perelman).

This is an expository article with complete proofs intended for a general non-specialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2-spheres. For instance, when $M$ is a homotopy 3-sphere, the width is loosely speaking the area of the smallest 2-sphere needed to ``pull over'' $M$. Second, we use this to conclude that Hamilton's Ricci flow becomes extinct in finite time on any homotopy 3-sphere. We have chosen to write this since the results and ideas given here are quite useful and seem to be of interest to a wide audience.

We prove a smooth compactness theorem for the space of embedded self-shrinkers in $\RR^3$. Since self-shrinkers model singularities in mean curvature flow, this theorem can be thought of as a compactness result for the space of all singularities and it plays an important role in studying generic mean curvature flow.

It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere.

The key in showing this conjecture is to show that shrinking spheres, cylinders and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every other singularity than spheres and cylinders can be perturbed away.

We show that for a Schrodinger operator with bounded potential on a manifold with cylindrical ends the space of solutions which grows at most exponentially at infinity is finite dimensional and, for a dense set of potentials (or, equivalently for a surface, for a fixed potential and a dense set of metrics), the constant function zero is the only solution that vanishes at infinity. Clearly, for general potentials there can be many solutions that vanish at infinity. These results follow from a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution to a Schrodinger equation on a product $N\times [0,T]$, where $N$ is a closed manifold with a certain spectral gap. Examples of such $N$'s are all (round) spheres $\SS^n$ for $n\geq 1$ and all Zoll surfaces. Finally, we discuss some examples arising in geometry of such manifolds and Schrodinger operators.

Minimal surfaces with uniform curvature (or area) bounds have been well understood and the regularity theory is complete, yet essentially nothing was known without such bounds. We discuss here the theory of embedded (i.e., without self-intersections) minimal surfaces in Euclidean 3-space without a priori bounds. The study is divided into three cases, depending on the topology of the surface. Case one is where the surface is a disk, in case two the surface is a planar domain (genus zero), and the third case is that of finite (non-zero) genus. The complete understanding of the disk case is applied in both cases two and three.
As we will see, the helicoid, which is a double spiral staircase, is the most important example of an embedded minimal disk. In fact, we will see that every such disk is either a graph of a function or part of a double spiral staircase. The helicoid was discovered to be a minimal surface by Meusnier in 1776.
For planar domains the fundamental examples are the catenoid, also discovered by Meusnier in 1776, and the Riemann examples discovered by Riemann in the beginning of the 1860s. Finally, for general fixed genus an important example is the recent example by Hoffman-Weber-Wolf of a genus one helicoid.
In the last section we discuss why embedded minimal surfaces are automatically proper. This was known as the Calabi-Yau conjectures for embedded surfaces. For immersed surfaces there are counter-examples by Jorge-Xavier and Nadirashvili.

The study of embedded minimal surfaces in $\RR^3$ is a classical problem, dating to the mid 1700's, and many people have made key contributions. We will survey a few recent advances, focusing on joint work with Tobias H. Colding of MIT and Courant, and taking the opportunity to focus on results that have not been highlighted elsewhere; LANL link.

This is an introduction to the basic results on minimal submanifolds, written for the Encyclopedia of Mathematical Physics (joint with Colding).

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. (joint with Colding); LANL link.

We give a quick tour through the field of minimal submanifolds. Starting at the definition and the classical results and ending up with current areas of research. Many references are given for further readings (joint with Colding; Bulletin of the London Math. Society); LANL link.

In this paper we will prove the Calabi-Yau conjectures for embedded surfaces. In fact, we will prove considerably more. The Calabi-Yau conjectures about surfaces date back to the 1960s. Much work has been done on them over the past four decades. In particular, examples of Jorge-Xavier from 1980 and Nadirashvili from 1996 showed that the immersed versions were false; we will show here that for embedded surfaces, i.e., injective immersions, they are in fact true. (Joint with Colding; LANL link. Annals of Mathematics 2008)

We construct a sequence of (compact) embedded minimal disks in a ball where the curvature blows up only at the center. This converges to a limit which is not smooth and not proper (joint with Colding, Trans. AMS, 2004); LANL link.

Survey of Embedded minimal surfaces I, II, and IV - intended also as a reader's guide (joint with Colding, The Proceedings of the Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces); LANL link.

We prove the one-sided estimate and the global compactness theorem (to a foliation) for embedded minimal disks (joint with Colding; Annals of Math. 2004); figures added to LANL link.

We prove that stable embedded minimal annuli are graphs away from their boundary (joint with Colding; Annals of Math. 2004); figures added to LANL link.

We give a decomposition of embedded minimal annuli which illustrates our pair of pants decomposition for embedded minimal planar domains (joint with Colding; IMRN 2002).

We prove estimates for multi-valued solutions of the minimal surface equation and apply these to prove properness of some embedded minimal disks (joint with Colding; IMRN 2002).

A multi-valued graph in an embedded minimal disk can be extended almost all the way to the boundary (joint with Colding; Annals of Math. 2004); figures added to LANL link.

Proves that an embedded minimal disk with large curvature contains a nearby multi-valued graph (joint with Colding; Annals of Math. 2004); figures added to LANL link.

Bounds the oscillation of the normal for minimal annuli with slits; applied in the proof of removable singularity theorem for minimal limit laminations (joint with Colding; J. Symplectic Geom. 2002).

Proves estimates on area and total curvature for intrinsic balls in two-sided stable minimal surfaces in three-manifolds; as consequences, we get Bernstein theorems and curvature estimates. In the case of area, this curvature estimate is due to Schoen. (joint with Colding; IMRN 2002).