Recent Results in Nonlinear Elliptic Equations and their Interactions with Geometry Organized by: Frank Pacard, Neil Trudinger and Paul Yang October 31, 2005 to November 4, 2005 Lecture Schedule & Talk Abstracts Monday, October 31 9:30 – 10:30 Gang Tian (Princeton Univ.), "Curvature estimate for 4-dimensional Einstein equation" 10:30 – 11:00 Morning Tea – Sixth Floor 11:00 – 12:00 Xiuxiong Chen (Univ. of Wisconsin), "On the Calabi functional in Kahler manifold" 12:00 – 1:00 Jeff Viaclovsky (MIT), “Volume Growth, Curvature Decay, and Critical Metrics” Abstract: I will discuss an upper volume growth estimate for spaces with quadratic curvature decay, and some local regularity theorems for critical metrics (such as anti-self-dual metrics or constant scalar curvature Kahler metrics). Finally, I will give some applications to orbifold and multi-fold compactness theorems for manifolds with critical metrics. 1:00 – 2:30 Lunch 2:30 – 3:30 Xavier Cabre (ICREA-Univ. Politecnica Catalunya), "Regularity of radial minimizers and extremal radial solutions of semilinear elliptic equations" 3:30 – 4:00 Travel to 60 Evans Hall, UCB 4:10 – 5:10 (At 60 Evans Hall, UCB) Cristian Gutierrez (Temple University), “The Monge-Ampere equation: Overview and Recent Results” Abstract: This is essentially a self contained talk describing basic facts about the Monge-Ampere equation, such us weak solutions and solvability of the Dirichlet problem. We will also discuss some regularity issues for solutions and finally consider a Monge-Ampere type equation appearing in the constructions of reflectors. 5:15 – 7:00 Reception at Craig Evan’s home, 1517 Hawthorn Terrace, Berkeley

Tuesday, November 1 9:30 – 10:30 Richard Schoen (Stanford University), "Blowup issues for the Yamabe equation on high dimensional manifolds" Abstract: We will describe recent joint work with Marcus Khuri in which we prove appropriate vanishing of the Weyl tensor at points of blowup for (local) Yamabe solutions in high dimensions. We show that the vanishing obtained is that which is required to have the mass defined, and we use this to extend low dimensional arguments to show there is no blowup for global solutions in high dimensions. 10:30 – 11:00 Morning Tea – Sixth Floor 11:00 – 12:00 Simon Brendle (Stanford University), "Global convergence of the Yamabe flow in dimension 6 and higher" Abstract: Let M be a compact manifold of dimension n \geq 3. Along the Yamabe flow, a Riemannian metric on M is deformed according to the equation \frac{\partial g}{\partial t} = -(R_g - r_g) \, g, where R_g is the scalar curvature associated with the metric g and r_g denotes the mean value of R_g. It is known that the Yamabe flow exists for all time. Moreover, if 3 \leq n \leq 5 or M is locally conformally flat, then the solution approaches a metric of constant scalar curvature as t \to \infty. I will describe how this result can be generalized to dimensions 6 and higher under a technical condition on the Weyl tensor. The proof is based on the construction of a family of test functions. Each of these test functions has Yamabe energy less than the Yamabe constant of the standard sphere S^n. 12:00 – 1:00 Olivier Druet (ENS Lyons), “The prescribed scalar curvature problem” Abstract : We will present a proof of the Kazdan-Warner conjecture concerning the prescribed scalar curvature problem in the null case. 1:00 – 2:30 Lunch 2:30 – 3:30 Bernd Amman (Univ. Nancy), "The first Dirac eigenvalue in a conformal class" Abstract: Let (M,g_0) be a compact n-diemnsional Riemannian manifold equiped with a fixed spin structure. Let [g_0] be the set of all metrics conformal to g_0 having volume 1. We study the first positive eigenvalue of the Dirac operator as a function on [g_0]. At first, we sketch the proof that the first positive Dirac eigenvalue is not bounded from above. Then we turn our attention to the infimum, denoted by \mu(M,[g_0]). We will show that \mu(M,[g_0]) is always positive. In order to discuss whether this infimum is attained, we reformulate the problem as a variational problem. The infimum is attained if \mu(M,[g_0])<\mu(\mathbb{S}^n) where \mathbb{S}^n denotes the round sphere. Roughly speaking, this inequality avoids concentration of minimizing sequences for our functional. We discuss the Euler- Lagrange equation of the variational problem. In dimension 2 the spinorial Weierstrass representation can be used to transform the Euler-Lagrange equation into an evenly branched conformal immersion into R^3 such that the image has constant mean curvature. The existence of certain periodic constant mean curvature surfaces is obtained as a corollary. In the remaining part we discuss several conditions implying the inequality \mu(M,[g_0])<\mu(\mathbb{S}^n). With an Aubin type construction of a test spinor, one sees that this inequality holds when M is not conformally flat and {\rm dim} M>6. Other conditions are known if M is conformally flat and for lower dimension. 3:30 – 4:00 Afternoon Tea – Sixth Floor 4:00 – 5:00 Xiaohua Zhu (Beijing University), “Kahler-Ricci flow and certain complex Monge-Ampere equation" Abstract: In this talk, I will discuss a problem about convergence of the Kahler-Ricci flow on a complex compact manifold which admits a Kahler-Ricci soliton. We will use the capacity method to discuss some estimates on certain complex Monge-Ampere equations associated to the Kahler-Ricci flow. This is a joint work with Prof. Gang Tian..

Wednesday, November 2 9:30 – 10:30 Gerhard Huisken (Max Planck Inst. f. Gravitationsphysik), "Rigidity inequalities via inverse mean curvature flow" 10:30 – 11:00 Morning Tea – Sixth Floor 11:00 – 12:00 J. Cheng (Academica Sinica, Taipei), "The mean curvature equation in Pseudohermitian geometry" Abstract: I will make a brief report on the recent study about the mean curvature equation in pseudohermitian geometry. As a differential equation, this (p-)minimal surface equation is degenerate (hyperbolic and elliptic) in dimension 2 while subelliptic in the nonsingular domain for higher dimensions. We analyze the singular set and formulate an extension theorem. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem. As a geometric application, we prove the nonexistence of C^{2} smooth hyperbolic surfaces having bounded p-mean curvature, immersed in a pseudohermitian 3-manifold. From the variational formulation of the equation, we study the Dirichlet problem by proving the existence and the uniqueness of the (p-)minimizers. 12:00 – 1:00 Sophie Chen (Princeton University), “Boundary Estimates for Some Fully Nonlinear PDEs and Applications to Conformal Geometry” Abstract: We consider some boundary value problems motivated by four dimentional Chern-Gauss- Bonnet formula for manifolds with boundary. We present existence results for some fully nonlinear PDEs with Neumann boundary condition. As an application, we prove that given a conformally compact Einstein 4-manifold with some conditions on the conformal infinity and the renormalized volume, there exists a conformal compactification such that the \sigma_2 curvature is constant. 1:00 – 2:30 Lunch 2:30 – 3:30 Qing Han (Univ. of Notre Dame), “Smooth local isometric embedding of surfaces with Gauss curvature changing sign monotonically across a curve” Abstract: In 1986, C.-S. Lin proved the existence of sufficiently smooth local isometric embedding of surfaces with Gauss curvature changing sign cleanly. It is not clear whether such an isometric embedding is smooth if surfaces are smooth. In this talk, we give an affirmative answer to such a question. In general, we prove the existence of smooth local isometric embedding of surfaces with Gauss curvature changing sign monotonically across a curve, where the Gauss curvature may vanish up to finite or infinite order. The proof relies on a careful analysis of linearizations of the Darboux equation and a construction of smooth solutions to a class of degenerate elliptic equations. 3:30 – 4:00 Afternoon Tea – Sixth Floor 4:00 – 5:00 Nassif Ghoussoub (Univ. of British Columbia), "Selfdual variational principles for PDEs and evolution equations"

Thursday, November 3 9:30 – 10:30 Bruno Franchi (Univ. of Bologna), "Intrinsic submanifolds of Heisenberg group" 10:30 – 11:00 Morning Tea – Sixth Floor 11:00 – 12:00 Joel Spruck (Johns Hopkins Univ.), “Complete hypersurfaces of constant curvature in Hyperbolic space with prescribed asymptotic boundary" Abstract: In this talk we will consider the problem of constructing complete hypersurfaces in H^{n+1} with constant higher order curvature f(\kappa)=H_r(\kappa)=\frac{S_r(\kappa)}{S_r(1,\ldots,1)}=\sigma^r\in(0,1) and with prescribed boundary \Gamma at infinity. Using the half-space model for H^{n+1}, we transform this problem into a degenerate fully nonlinear elliptic Dirichlet problem: \begin{eqnarray*} f(u\kappa +\frac1{W} \vec{1})&=&\sigma^r ~~\mbox{in}~~\Omega\\ u&=&0 ~~\mbox{on}~~\partial \Omega. \end{eqnarray*} Here \Gamma=\partial \Omega\subset \{x_{n+1}=0\} represents the asymptotic boundary, the \kappa[u] are the Euclidean principal curvatures of S, the graph of u, and \frac1{W}=\nu^{n+1} is the last component of the upward unit normal \nu to S. The cases r=1 (mean curvature) and r=n (Gauss curvature) have received all the attention and we shall survey the known results. We will then present new existence results for the other curvatures. 12:00 – 1:00 Refe Mazzeo (Stanford University), “Einstein metrics and higher rank geometry” Abstract: I will discuss some new methods, developed jointly with Andras Vasy, to study the Laplacian on noncompact symmetric spaces of rank greater than one and their perturbations, and shall describe some applications of this work, with Olivier Biquard, concerning the construction of Einstein metrics of this asymptotic type. 1:00 – 2:30 Lunch 2:30 – 3:30 Brian White (Stanford University), “Getting a handle on the helicord” 3:30 – 4:00 Afternoon Tea – Sixth Floor 4:00 – 5:00 Xujia Wang (Australian National Univ.), “Fully nonlinear partial differential equations and applications in geometry” Abstract: In this talk I will report two recent results on the conformal k-Hessian equation on manifolds of dimension n>2. The first one, with W.M. Sheng and N.S. Trudinger, is the existence of solutions to the so- called k-Yamabe problem for k less than or equal to n/2, assuming that the metric is admissible and the equation is variational. The second one, with N.S. Trudinger, is the compactness of admissible metrics to the conformal k-Hessian equation when k>n/2, which also yields the existence of solutions to the k- Yamabe problem if there is an admissible metric, first proved by Gursky and Viaclovsky. 5:00-6:00 MSRI Wine and Cheese Reception

Friday, November 4 9:30 – 10:30 Luis Caffarelli (Univ. of Texas, Austin) 10:30 – 11:00 Morning Tea – Sixth Floor 11:00 – 12:00 Alice Chang (Princeton Univ.), "Conformal invariants associated with a smooth measure" Abstract: In this talk, I will report some recent joint study with Gursky and Yang on the topic of conformal invariants on a Riemannanian manifold (M^n,g) equipped with a smooth measure m. In particular, we show that there is a natural definition of the Ricci and scalar curvatures associated to such a space, both of which are conformally invariant. We will compare this notion with that of Baker-Emery and G. Perelman. I will indicate methods to construct families of conformally covariant operators defined on these spaces. Certain variational problems in this setting are considered, including a generalization of the Einstein-Hibert action. 12:00 – 1:00 Rod Gover (Univ. of Auckland) “Conformal Laplacian operators and Q-curvature on Einstein manifolds” Abstract: The Q curvature is a curvature quantity on even dimensional manifolds that yields a prescription problem which, in terms of elementary geometric analysis, is a higher-dimensional generalisation of the 2-dimensional Gauss curvature prescription problem. I will give a proof that Q is constant on Einstein manifolds. Closely related to Q curvature is a family of higher order conformal Laplacians due to Graham-Jenne-Mason-Sparling (GJMS). I will show that on Einstein manifolds these factor into a composition of second order Laplacian type operators and show that the GJMS family of operators may be extended in this setting. 1:00 – 2:30 Lunch 2:30 – 3:30 Jie Qing (UC Santa Cruz), “On semi-linear conformally invariant PDE” Abstract: In this talk we will discuss the existence and compactness of positive solutions to a family of semi-linear conformally invariant equations. The family of conformally covariant operators were introduced via the scattering theory for Poincare metrics associated with a conformal manifold. We will show that, on a closed and locally conformally flat manifold with Poincare exponent small, the set of positive smooth solutions to the family of equations is compact in the C^infty topology. Therefore the existence of positive solutions follows from the existence of Yamabe metrics and a degree theory. 3:30 – 4:00 Afternoon Tea – Sixth Floor 4:00 – 5:00 Fengbo Hang (Princeton & Michigan State Univ.), “On the higher order Q curvature on the Sphere” Abstract: We will show that in the conformal class of standard metrics on S^n (n odd), the scaling invariant integral of 2mth order Q curvature maximizes at the standard metric when 2m=n+1 or n+3. When 2m>n+3, the functional has no local maximum.