Stanford University Geometry Seminar: Fall 2006 Unless otherwise noted, all seminars are on Wednesdays 4:00 - 5:00 pm in Room 383-N (Third floor of Math Building, Bldg 380).  There is tea at 3:30 on Wednesdays in the second floor lounge of the Math Building (382-T).

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 of
minimal 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 compactness
of 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