Stanford University Geometry Seminar: Fall 2006 Unless otherwise noted, all
seminars are on |

**Fall
Quarter **

**Jan. 31 Simon Brendle (Stanford)**

**Jan. 24 Guofang Wei **

**Jan. 17 Tobias Lam **

**Jan. 10 Nicos Kapouleas (Stanford,
Brown) **

**Dec. 6 Sophie Chen (MSRI & UC
Berkeley)**

**Title: **Generalized Yamabe
problem on manifolds with boundary

**Nov. 22 - no seminar for Thanksgiving **

**Nov. 14 (Tuesday seminar!) Danny
Calegari (Caltech) **

**Time: 2:45 – 3:45**

**Room 380-380W **

**Title: Length and stable length Abstract:
Margulis' lemma says that there is a "universal" sense of
what it means for a closed geodesic in a hyperbolic manifold to be
short. We study stable commutator length (roughly, "rational
bounding genus") as a function from conjugacy classes in a group
to real numbers, and show that there is a "universal" sense
of what it means for a conjugacy class in a hyperbolic group to
have a small stable commutator length. There are related applications
to mapping class groups and groups acting on trees. (This is partly
joint work with Koji Fujiwara.) **

**November 8,2006 : Larry Gruth (Stanford University)**

**Title: Volumes of balls in large Riemannian manifolds
Abstract: In the 80's, Gromov made several conjectures about the
volumes of balls in Riemannian manifolds. The spirit of the
conjectures is that if a Riemannian manifold is "large",
then it should contain a unit ball whose volume is not too small.
For example, if you take the standard metric on the n-sphere and
increase it pointwise to form a new metric, then Gromov's conjecture
implies that the new metric should contain a unit ball whose volume
is bounded below by a constant c(n). I proved some of the
conjectures, including this one. I will explain the conjectures
and give some context, and then I will try to say something about the
proof. **

**November 1, 2006 : Jan Metzger
(Stanford University & Max-Planck-Institut f\"ur
Gravitationsphysik)**

Title: Stable marginally trapped surfaces and dynamical horizons.Abstract: Recently Andersson, Mars and Simon have proposed a concept of stability for marginally trapped surfaces. This concept generalizes the stability ofminimal surfaces and has many powerful applications. I will review the local existence result for dynamical horizons of Andersson, Mars and Simon.Furthermore I will present joint work with Andersson, in which we were able to prove curvature estimates for such surfaces, which in turn imply compactnessof a certain class of marginally trapped surfaces.

**October 25, 2006: Gerhard
Huisken(Max-Planck-Institut f\"ur Gravitationsphysik)**

**Title: An isoperimetric concept for the quasi-local
mass**

**October 18, 2006: Bianca Santoro (MSRI
)**

**Titile: On the asymptotic behavior of
complete Kahler Ricci-flat metrics on quasi-projective manifolds**

**Abstract: In 1990, Tian and Yau settled the
non-compact version of the Calabi Conjecture, by proving the
existence of complete Kahler Ricci-flat metrics on open complex
manifolds that can be compactified by adding a smooth, ample divisor
of multiplicity one. In this talk, we will discuss some results on
the asymptotic behavior of those metrics, therefore refining Tian and
Yau's results.**

October 11, 2006: Tristan Riviere ( ETH Zurich ) **Title
: Conservation laws for conformally invariant lagrangians and
solutions to Schroedinger systems with antisymmetric potentials
Abstract : We will explain how to write 2-dimensional conformally
invariant non-linear elliptic PDE (harmonic map equation, prescribed
mean curvature equations...etc) in divergence form. These
divergence-free quantities generalize to target manifolds without
symmetries the well known conservation laws for weakly harmonic maps
into homogeneous spaces. From this form we can recover, without the
use of moving frame, all the classical regularity results known
for 2-dimensional conformally invariant non-linear elliptic PDE . It
enables us also to establish new results. In particular we solve a
conjecture by E. Heinz asserting that the solutions to the
prescribed bounded mean curvature equation in arbitrary
manifolds are continuous. Our approach permits also to prove a more
general conjecture by S.Hildebrandt claiming that critical points of
continuously differentiable second order elliptic conformally
invariant Lagrangian in two dimensions are continuous. We will
explain how these results are deduced from a more general one on
solutions to Schroedinger systems with antisymmetric potentials.**

October 4, 2006: David Fisher (Indiana University, Bloomington)
**Title: Existence of equivariant harmonic maps without
reductivity.**

Abstract: Let M be compact manifold with fundamental group G and universal cover \tilde M. If G acts on a a nonpositively curved space X, then one would like to understand the action by finding a G equivariant harmonic map from \tilde M to X. In many settings, one can prove that such a map exists provided the action of G on X satisfies some sort of reductivity assumption, for example, provided there is no fixed point at infinity. I will discuss joint work with T.Hitchman in which we prove the existence of harmonic maps for certain choices of M,G and X without assuming anything about the action of G on X. Previous work based on the zero vanishing principal of Korevaar-Schoen had done this provided M and X were such that any equivariant harmonic map from \tilde M to X was necessarily constant. The novelty in our work is to show that harmonic maps exist in settings where non-constant harmonic maps exist.

If time permits, I will discuss some motivations for this work from dynamical systems theory and the Zimmer program.

Past seminars**: Winter Quarter**

**February 15: Simon Brendle (Stanford University)**

**Title: Convergence of the Yamabe flow via test
function estimates**

**Abstract: **

**February 22: Bill Meeks (University of
Massachusetts, Amherst)**

**Title: The Dynamics Theorem for embedded
minimal surfaces.**

**Abstract: First I will give a quick
survey of recent classification theorems for classical minimal
surfaces. These results include the uniqueness of the catenoid,
the helicoid, Scherk's singly and doubly periodic minimal surfaces,
Riemann's family of minimal surfaces and the Topological
Classification Theorem for properly embedded minimal surfaces in R^3.
Next I will present and prove some recent results on minimal
laminations of 3-manifolds. These results include the Minimal
Lamination Closure Theorem, the Lamination Metric Theorem, the Local
Removable Singularity Theorem, and the C^{1,1} Regularity Theorem for
a Colding-Minicozzi lamination and its converse. As a
consequence, I will then sketch the proofs of the following central
theorems.**

**1. The Quadratic Decay of Curvature Theorem, which
states that a complete embedded minimal surface in R^3 has quadratic
decay of curvature (in terms or the distance R from the origin) if
and only if it has finite total curvature**

**2. The Dynamics Theorem for a properly embedded
minimal surface M in R^3. This theorem demonstrates that if M does
not have finite total curvature, then its space D(M) of nonflat
divergent dilation limits always contains a nonempty minimal dilation
invariant subset, in the sense of dynamics.**

**3. The Bounded Topology Theorem, which gives
bounds on the topology of a complete embedded minimal surface of
finite topology in terms of a bound on its genus. In particular, a
complete embedded minimal surface of finite index of stability and
fixed genus must have a bound on its index that only depends on its
genus.**

**March 8: Matthias Weber (Indiana University,
Bloomington)**

**Title: Bending the Helicoid**

**Abstract: We construct
Colding-Minicozzi limit minimal laminations in open domains in $R^3 $
with the singular set of $C^1$-convergence being any
properly embedded $C^{1,1}$-curve. By Meeks' $C^{1,1}$-regularity
theorem, the singular set of convergence of a
Colding-Minicozzi limit minimal lamination${\cal L}$ is a locally
finite collection $S({\cal L})$ of $C^{1,1}$-curves that are
orthogonal tothe leaves of the lamination. Thus, our existence
theorem gives a complete answer as to which curves appear as
the singular set of a Colding-Minicozzi limit minimal lamination. In
the case the curve is the unit circle $S^1$ in the $(x_1, x_2)$-
plane, the classical Bj\"orling theorem produces an
infinite sequence of complete minimalannuli $H_n$ of finite total
curvature which contain the circle. The complete minimal surfaces
$H_n$contain embedded compact minimal annuli $\overline{H}_n$ in
closed compact neighborhoods $N_n$ ofthe circle that converge as $n
\to \infty$ to $R^3 - x_3$-axis. In this case, we prove that
the $\overline{H}_n$ converge on compact sets to the
foliation of $R^3 - x_3$-axis by vertical half planes with
boundary the $x_3$-axis and with $S^1$ as the singular set of
$C^1$-convergence. The $\overline{H}_n$ have the appearance of
highly spinning helicoids with the circleas their axis and are named
{\em bent helicoids}.**

**March 15: Leon Simon (Stanford)**

**Title: An interior gradient estimate for the
symmetric minimal surface equation**

**Abstract: The symmetric minimal surface equation
(SME) on a domain $\Omega\subset\R^{n}$ is the equation
{\cal{}M}(u)={m-1\over{}u\sqrt{1+|Du|^{2}}}$, where $m$ is an integer
$\ge 2$ and ${\cal{}M}$ is the mean curvature operator.
Geometrically, for positive solutions, the SME says that the
symmetric graph $S(u)=\{(x,u(x)\xi):x\in\Omega,\, \xi \in S^{m-1}\}$
is a minimal (i.e.\ zero mean curvature) hypersurface in
$\Omega\times\R^{m}\subset\R^{n+m}$. For non-negative $u$ which are
locally uniform limits of positive solutions of the SME in $\Omega$
the symmetric graph $S(u)$ is still a stationary point for the area
functional in $\Omega\times \R^{m}$, and it has singular set exactly
$u^{-1}\{0\}\times \{0\}$. The talk will discuss the proof of
the interior gradient bound $|Du(x_{0})|\le C(n,m,\rho^{-1}M)$, where
$M=\sup_{B_{\rho}(x_{0})}u$, assuming $\overline
B_{\rho}(x_{0})\subset \Omega$.**

**March 22: Vincent Moncrief (Yale)**

**Title: Relativistic Teichmu"ller Theory—a
Hamilton-Jacobi approach to 2 + 1 dimensional gravity**

**Abstract: **

**March 29: Peter Petersen (UCLA)**

**Title: Exotic spheres with positive
curvature?**

**Abstract: I'll survey old and new work that
relates to whether or not certain exotic spheres admit metrics with
positive curvature. I'll also address what it takes to possibly
create a metric with positive curvature on the Gromoll-Meyer sphere.**

**Spring Quarter**

**April 5: Gregoire Montcouquiol (Universite de
Paris Sud)**

**Title: Einstein deformations of hyperbolic
cone-manifolds.**

**Abstract: Hodgson and Kerckhoff showed that, for a
large class of 3- dimensional hyperbolic cone-manifolds, the space of
hyperbolic cone-metrics is locally parametrized by the p-tuple of
cone angles. We will generalize this property to higher dimensions,
replacing hyperbolic metrics by Einstein metrics. We obtain a
similar result, where infinitesimal Einstein deformations of a
Hyperbolic cone-manifold exactly correspond to variations of its cone
angles.**

**April 11(Tuesday): Ben Chow (UC San Diego)**

**Title: On the works of D. Glickenstein and F. Luo
on semi-discrete curvature flows**

**Abstract: **

**April 18 (Tuesday 4PM): Niall O'Murchadha
(University of Cork) ------NOTE: SPECIAL DATE & TIME!!!**

**Title: The Liu-Yau quasi-local mass in spherical
gravity.**

**Abstract: Liu and Yau recently introduced a new
quasi-local mass in GR. It is a function on a 2-surface in a
4-manifold. They showed that it was positive. It is the maximum of
the Brown-York energy over all 3-slicings containing the given
2-surface. The Liu-Yau mass has unpleasant features, it looks much
more like an energy than a mass. In particular it is unboundedly
large on any solution of the Einstein equations, including Minkowski
space! In spherical symmetry, however, it has a natural physical
interpretation. Consider a regular spherical 3-slice filling the
interior of the given2-slice. Take the integral of the energy density
of the interior. The Liu-Yau mass of the boundary is the minimum of
this total energy content over all possible interiors. No other
quasi-local mass gives such interior**

**April 26: Robin Graham (University of Washington)**

**Title: **

**Abstract: **

**May 3: Mei-Chi Shaw (University of Notre Dame)**

**Title: Bounded plurisubharmonic functions and the
$\bar\partial$-Cauchy problem in the complex projective spaces**

**Abstract: In this talk we will discuss bounded
plurisubharmonic functions on pseudoconvex domains in the complex
projective spaces. Such functions are used to study the function
theory via the $\bar\partial$-Cauchy problem. We also discuss
the application on the nonexistence of Lipschitz
Levi-flat hypersurfaces in the complex projective space of dimension
greater or equal to 3 (Joint work with Jianguo Cao).**

**May 10: Brian White (Stanford University)**

**Title: **

**Abstract: **

**May 17: Robert Hardt (Rice University)**

**Title: Rectifiable Scans in a Metric Space with
Coefficients in a Group**

**Abstract: In 1960 Federer and Fleming developed a
theory of rectifiable currents which allowed for a solution of
least-area Plateau problems in arbitrary dimension and codimension in
Euclidean space. To account for nonorientable chains like a minimal
Mobius band and other examples, Fleming introduced a theory of flat
and rectifiable chains with coefficients in any of the finite groups
Z/jZ. In 1999, B.White generalized this to essentially the largest
possible class of groups allowing compactness of rectifiable cycles
in Euclidean space. Also in 2000, L. Ambrosio and B.Kirchheim
generalized much of the Federer-Fleming work using their notion of
currents in a metric space. Last year T. Adams solved Plateau
problems in Banach spaces with chains with general group
coefficients Here we describe some new elementary definitions
and arguments that allows one to treat both metric spaces, other
coefficient groups, and rectifiability simultaneously. We use some of
the best ideas from the previous works.**

**May 24: Tobias Colding (MIT & NYU)**

**Title: A three circles theorem for Schr\"odinger
operators on manifolds with cylindrical ends and applications.**

**Abstract: **

**May 31: Bun Wong (UC Riverside)**

**Title: Higgs structure and local moduli group of
tangent bundle on complex surface of general type**

**Abstract: The purpose of this talk is to
demonstrate a relationship between the Higgs structure on certain
flat bundle of the ball quotient and the first cohomology group with
value in the endomorphism bundle on complex twofold of general type.**

**Past Seminars:**

**January 11: Harold Rosenberg (Université Denis Diderot -
Paris 7)**

**Title: Constant mean curvature H surfaces in 3-manifolds, for H
large.**

**Abstract: **

**January 13 (Friday): Harold Rosenberg (Université Denis
Diderot - Paris 7)**

**Title: Minimal and constant mean curvature surfaces in
homogeneous 3-manifolds.**

**Abstract: **

**January 18: Harold Rosenberg (Université Denis Diderot -
Paris 7)**

**Title: Holomorphic quadratic differentials associated to
surfaces in homogeneous 3-manifolds, of constant mean curvature, and
constant Gaussiann curvature.**

**Abstract:**

January 25: Albin, Pierre (MIT)

Title: Index theory on Poincare-Einstein and edge manifolds.

Abstract**: **

**Fall Quarter**

**September 21: Hugh Bray (Duke University) **

**Title: Negative Point Mass Singularities in General Relativity**

**Abstract: In this talk we will discuss a geometric
inequality which is in the same spirit as the Positive Mass Theorem
and the Penrose Inequality for black holes. Whereas the
cases of equality of these first two theorems are respectively
Minkowski space (which can be thought of as Schwarzschild with zero
mass) and the Schwarzschild spacetime with positive mass, the case of
equality for the inequality we will discuss is the Schwarzschild
spacetime with negative mass. **

**Physically speaking, when positive amounts of energy are
concentrated as much as possible, black holes results.**
**However, when negative amounts of energy are "concentrated"
as much as possible, it is in fact possible to form point
singularities in each spacelike slice (which form a timelike curve of
singularities in the spacetime).**

**As usual we will focus on maximal, spacelike slices of
spacetimes as a first step. The assumption of nonnegative
energy density on these slices implies that these Riemannian
3-manifolds have nonnegative scalar curvature. However, we will
allow these 3-manifolds to have singularities which contribute
negatively to the total mass. The standard example is the
negative Schwarzschild metric on R^3 minus a ball of radius m/2, (1 -
m/2r)^4 \delta_{ij}. This metric (which has total mass -m) has
zero scalar curvature everywhere but has a singularity at r = m/2.
We will propose a definition for the mass of a singularity, and prove
a sharp lower bound on the ADM mass in terms of the masses of the
singularities in the 3-manifold.**

**September 28: Damin Wu (Stanford University)**

**Title: Higher Canonical Asymptotics of K\"{a}hler-Einstein
metrics on Quasi-Projective Manifolds**

**Abstract: In this talk we will discuss a canonical
asymptotic expansion up to infinite order of the K\"{a}hler-Einstein
metric on the quasi-projective manifold, which can be compactified by
adding a divisor with simple normal crossings. Characterized by the
log-filtration of the Cheng-Yau H\"{o}lder ring, the asymptotics
is obtained by constructing an initial K\"{a}hler metric, and
deriving certain iteration formula and the isomorphism theorems of
the Monge-Amp\`{e}re operators. This work may be viewed as a parallel
to the asymptotics of Fefferman, Lee and Melrose on the strictly
pseudoconvex domain in $\mathbb{C}^n. At the end we will mention some
possible applications related to complex geometry and algebraic
geometry.**

**October 5: Giuseppe Tinaglia (Stanford University)**

**Title: Structure theorems for disks embedded in R^3**

**Abstract: In this talk we will discuss the shape of embedded
disks with bounded constant mean curvature. In particular, we will
prove that an embedded disk with bounded constant mean curvature and
Gaussian curvature large at a point contains a multi-valued graph
around that point on the scale of the norm square of the II
fundamental form. Roughly speaking, it looks like helicoids. This
generalizes Colding and Minicozzi’s result for minimal surfaces.**

**October 12: Raphael Ponge (UC Berkeley)**

**Title: New invariants for CR and contact manifolds**

**Abstract: In this talk I will explain the construction of
several new invariants for CR and contact manifolds as noncommutative
residue traces of various geometric pseudodifferential projections.
In the CR setting these operators arise from the
$\overline{\partial}_b $ complex and include the Szeg\"o
projections. In the contact setting they stem from the generalized
Szeg\"o projections at arbitrary integer levels of
Epstein-Melrose and from the contact complex of Rumin. In particular,
we recover and extend recent results of Hirachi and Boutet de Monvel
and answer a question of Fefferman.**

**October 19: Franck Pacard (Paris XII and MSRI)**

**Title: Constant scalar curvature K\"ahler metrics on the
blow up of K\"ahler constant scalar curvature manifolds**

**Abstract: I will report some recent work on the existence
of constant scalar curvature metrics on the blow up at finitely many
points of a K\"ahler manifold which already carries a constant
scalar curvature metric. I will also discuss the case of extremal
K\"ahler metrics. **

**October 26: Larry Guth (Stanford University)**

**Title: K-Dilation and Topology of Mappings**

**The k-dilation of a mapping is a
generalization of its Lipshitz constant defined by Gromov and Lawson
in the early 80's. If the k-dilation is less than D, it means
that each k-dimensional submanifold of the domain with volume V is
mapped to a region of k-dimensional volume less than DV. The
k-dilation of a map is defined for each integer k between 1 and the
dimension of the domain.**

**We will consider the following
question. If a map f between two Riemannian manifolds has small
k-dilation, what can we conclude about its homotopy type? We
will discuss homotopically complicated maps between round spheres of
various dimensions and then we will discuss degree non-zero maps
between n-dimensional ellipsoids.**

**November 2: Jeff Viaclovsky (MIT)**

**Title: Isolated Singularities of Ricci Curvature Equations in
Conformal Geometry**

**Abstract: I will discuss solutions of a general class of fully
nonlinear equations in conformal geometry with isolated
singularities, in the case of non-negative Ricci curvature. We prove
that such solutions either extend to a H\"older continuous
function across the singularity, or else have the same singular
behavior as the fundamental solution of the conformal Laplacian. I
will then discuss the solution of the \sigma_k-Yamabe Problem, for k
> n/2**

**November 9: Pengzi Miao (UC Santa Barbara)**

**Title: An Estimate on the Electrostatic Capacity and Its
Application in General Relativity**

**Abstract: In this talk we will give a new upper estimate of the
classic electrostatic capacity of bounded domains in $R^3$. The
estimate applies to domains in certain asymptotically flat
3-manifolds with non-negative scalar curvature. As an application, we
obtain a lower estimate of the ADM mass of these manifolds in
terms of a harmonic function with prescribed geometric boundary
value. This is a joint work with H. Bray.**

**November 16: Andrejs Treibergs (University of Utah)**

**Title: A Brownian motion capture problem and an eigenvalue
estimate**

**Abstract: Consider the problem of n predators X_1,...,X_n
chasing a single prey X_0, all independent standard Brownian motions
on the real line. If the prey starts to the right of the predators,
X_k(0) < X_0(0) for all k=1,...,n, then the first capture time is
T_n = inf{ t > 0 : X_0(t) = X_k(t) for some k }. Equivalently,
this is the first exit time for a Brownian motion that starts at an
interior point of the corresponding cone in R^(n+1). Bramson and
Griffeath (1991) showed that the expected capture time E(T_n) is
infinite for n = 1, 2, 3 and, based on simulation, conjectured that
E(T_n) is finite for n = 4. This conjecture was proved by W. V. Li
and Q. M. Shao (2001) for n > 4 using a result of de Blassie
(1987), that the finiteness of expectation is equivalent to a
specific lower bound of the first Dirichlet eigenvalue of the domain
which is the intersection of cone with the unit n-dimensional sphere
at the origin. I will discuss my joint work with J. Ratzkin, in which
we prove the conjecture for n = 4 by establishing the eigenvalue
estimate.**

**November 30: Frederico J.** **Xavier (University of Notre
Dame)**

**Title: Geometric methods in the global inversion problem.**

**Abstract: We will survey some recent applications of
geometry and topology to the general problem of understanding the
basic mechanisms that make a locally invertible map admit a global
inverse. Here one such result (joint with S. Nollet): For
$n\geq 2$, a local biholomorphism $F:X\to\Bbb C^n$ is injective if
for each complex line $l\subset \Bbb C^n$, the pre-image $F^{-1}(l)$
embedds holomorphically as a connected domain into $\Bbb C\Bbb P^1$,
the embedding being unique up to M\"obius transformation (e.g.,
$F^{-1}(l)$ conformal to $\Bbb C$). The proof uses classical
estimates from complex analysis and the topology of the Hopf map. We
will also talk about other recent global inversion results that
relate to foliations, intersection numbers and other geometric
objects. **

**December 7: Lei Ni (UC San Diego) **

**Title: Local regularity theorems for Ricci flow.**

**Abstract: We discuss some $\epsilon$-regularity results for
Ricci flow under the assumption on the smallness of
`densities' formulated in terms of the {\it entropy} and the {\it
reduced distance} introduced by Perelman.**

**For questions: **

**Qian Wang
qwang@math.stanford.edu**