Professor **William P. Minicozzi II**

Department of Mathematics,
Johns Hopkins University

Office: (410) 516 6656;
Fax: (410) 516 5549

__Fall 2011 Course__: --- 110.744

Office hours: Mondays 11 to 12

Slides (missing figures) from
ICM talk, Madrid 2006.

Slides from talk on The rate of change of width under flows, 2008.

Spring 2010 graduate
student and post-doc workshop at Johns Hopkins, March 15 to 17, 2010; part
of NSF FRG grant with Toby
Colding and David Gabai.

I am an editor of the
following journals:

American Journal of Mathematics

Journal of
Topology and Analysis

__Links to two books and some papers:__

2011 textbook: A Course in
Minimal Surfaces, available from
the AMS.

Survey on minimal surfaces and mean curvature flow
(with Colding) in honor of Rick Schoen.

Survey on
geometric analsysis for Surveys in Differential
Geometry IX (with Colding); LANL link.

__Geometric evolution equations:__

Sharp estimates for mean
curvature flow of graphs -

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.

Estimates for the extinction
time for the Ricci flow on certain 3-manifolds and a question of Perelman
(joint with Colding; Journal of the AMS, volume 3,
2005; link);
LANL link.

Width and mean curvature flow
(joint with Colding); 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).

Width and finite
extinction time of Ricci flow (joint with Colding); LANL
link.

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.

Smooth compactness of self-shrinkers (joint with Colding).

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.

Generic mean curvature flow I; generic
singularities (joint with Colding).

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.

__Analysis
of Schrodinger operators:__

Three circles theorems for Schrodinger
operators on cylindrical ends and geometric applications (joint with Colding and Camillo De Lellis).

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:__

Shapes of embedded minimal
surfaces - (joint with Colding, Proceedings of
the National Academy of Science): LANL link.

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.

Embedded minimal surfaces (ICM
2006 Proceedings)-

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.

Minimal submanifolds:
Encyclopedia Entry -

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

Fixed genus (Embedded minimal surfaces
V) -

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.

The Calabi--Yau conjectures for embedded surfaces -

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)

Embedded minimal disks:
Proper versus nonproper - global versus local -

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.

Minimal disks that are double spiral
staircases -

Survey intended for general
audiences (joint with Colding, Notices of the
AMS, 2003); figures added to 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.

Locally simply connected (Embedded minimal
surfaces IV) -

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.

Planar domains (Embedded minimal surfaces
III) -

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.

On the structure of embedded minimal annuli-

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).

Multi-valued minimal graphs and properness
of disks-

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).

MSRI
lectures (July 16-20, 2001): I,
II,
III,
IV,
and V-

Pdf files of 5 lectures on joint work
with Toby Colding given at the Clay
Summer School on Minimal Surfaces, July 2001.

Minimal surfaces- Courant Lecture
Notes by Colding and Minicozzi.

Estimates off the axis for disks
(Embedded minimal surfaces I)-

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.

Multi-valued graphs in disks (Embedded minimal
surfaces II)-

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.

Minimal annuli with and without slits-

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).

Removable singularities for minimal limit
laminations - announcement; (joint with Colding; C.R.A.S.
2000).

Estimates for elliptic integrands-

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).

Convergence of embedded minimal surfaces without area bounds- announcement; (joint with Colding; C.R.A.S. 1998) --- httplink.

__Complete properly embedded minimal surfaces in Euclidean
space__-

__http://www.dukeupress.edu/dmj/2001/abstracts/articles/DMJ10702_6.pdf__

__Examples of embedded minimal tori with unbounded area__-

__http://imrn.hindawi.com/volume-1999/S1073792899000604.html__

__Embedded minimal surfaces without area bounds in 3-folds__-

__Function
theory, I; special cases (linear growth or Euclidean volume growth):__

__Linear growth harmonic functions__-

__http://math.ucsd.edu/~mrl/TOC/vol3/v32toc.pdf__

__Harmonic functions with polynomial growth __-

__http://www.intlpress.com/JDG/archive/vol.46/1_1.pdf__

__http://muse.jhu.edu/journals/american_journal_of_mathematics/toc/ajm119.6.html__

__Function
theory, II; full generality:__

__http://muse.jhu.edu/journals/american_journal_of_mathematics/toc/ajm119.6.html__

__Generalized Liouville properties__-
announcement; (joint with Colding; __MRL 1996__).

__http://math.ucsd.edu/~mrl/TOC/vol3/v36toc.pdf__

__Harmonic functions on manifolds__-

__http://www.math.princeton.edu/~annals/issues/1997/146_3.html__

__http://link.springer.de/link/service/journals/00222/bibs/8131002/81310257.htm__

__Liouville theorems for harmonic sections__-

__http://www3.interscience.wiley.com/cgi-bin/abstract/29247/START__

__Volumes for sublevel sets of eigensections__-
(joint with Colding; Geom. Ded., 2003).

Lower bounds for nodal
sets of eigenfunctions
(joint with Colding, CMP, 2011).

__Maximal functions and oscillatory integrals__-

Nikodym
and Kakeya type maximal functions on Riemannian manifolds (joint with C.D.
Sogge; __MRL 1997__).

__http://math.ucsd.edu/~mrl/TOC/vol4/v42_43toc.pdf__

__Geometry
of Lagrangian surfaces:__

__Lagrangian variational problems__-

__webshare/wwwroot/billcv8o03.pdf____ __

__JHU Math____The American Journal of Mathematics__- Analysis and Geometry at JHU:
Matthew Blair,
__Daniela De Silva__,__Jian Song__,__Chika Mese__,__Bill Minicozzi__,__Bernie Shiffman__,__Chris Sogge__,__Joel Spruck__,__Richard Wentworth__,__Steve Zelditch__,__Steve Zucker__