GEOMETRIC PARTIAL DIFFERENTIAL EQUATIONS
(3 May - 26 Jun 2004)
~ Abstracts ~
Entire spacelike hypersurfaces of
prescribed Gauss curvature in Minkowski space
Huaiyu Jian, Dept. of Mathematical Sciences, Tsinghua
University
This talk is based on a joint work with Bo Guan and Richard Schoen. We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) of prescribed Gauss curvarure in Minkowski space Rn,1. We will classify all entire spacelike K-hypersurfaces invariant under a subgroup of isometries of Rn,1, determine the asymptotic behaviour of entire K-hypersurfaces at infinity in terms of tangent cones and study the Minkowski type problem: to find spacelike hypersurfaces with prescribed Gauss curvarure and tangent cone .
Convex hypersurfaces of
prescribed Weingarten curvatures
Weimin Sheng, Dept. of Mathematics, Zhejiang University,
China
In this paper we study the existence of closed convex hypersurfaces in the Euclidean space Rn+1 with a Weingarten curvature prescribed as a function of their unit normal.
This is my joint work with Neil and Xujia.
Conformal invariants and partial differential equations
Alice Chang, Dept. of Mathematics, Princeton University
Conjugate functions and
semi-conformal mappings
Mike Eastwood, Dept. of Pure Mathematics, University of
Adelaide
Suppose f is a smooth function of two variables. Is there a smooth function g such that |grad f| = |grad g| and <grad f,grad g> = 0? The answer is yes if and only if f is harmonic. What about the same question for a function of three or more variables? Joint work with Paul Baird derives a differential inequality that must be satisfied by f and a differential equation in the case of a function of three variables. When f admits a conjugate, the pair (f,g) provides a semiconformal mapping into R^2. In particular, harmonic morphisms provide examples of conjugate pairs but there are more besides. The problem of finding a conjugate is conformally invariant so it not surprising that our constraints are also conformally invariant.
Conformally Einstein metrics in
dimension n
Rod Gover, Dept. of Mathematics, University of Auckland
I will discuss a range of new results concerning metrics which are conformally related to Einstein metrics. This includes a new proof that the Fefferman-Graham obstruction tensor obstructs such metrics. Also for Riemannian 4-manifolds we discuss a natural conformally invariant two tensor that vanishes if and only if a metric is conformal to an Einstein metric. Similar sharp obstructions are obtained in general dimension and signature with a slight restriction on the class of metrics.
A new Liouville-type theorem for
pseudo-concave domains and applications
Jianguo Cao, Dept. of Mathematics, University of Notre
Dame, USA
The classical Liouville Theorem states that there is no non-constant bounded holomorphic function on Euclidean complex plane. We will extend the classical Liouville Theorem to any C^2-smooth pseudo-concave domains in CP^n with n>1. Moreover, we show that there is no non-zero L2-integrable (p, 0)-forms on pseudo-concave domains above with p>0.
This new Liouville-type theorem has several applications in geometry. Among other applications, we show that there is no C^2-smooth Levi-flat real hyper-surfaces in CP^n with n>1. Such an application greatly improved and clarified the earlier results of Siu, who recently published two papers about Levi-flat hypersurfaces in the Annals of Mathematics.
This is a joint work with M. Shaw and L. Wang.
On prescribed scalar curvature
equation
Ji Min, Institute of Mathematics, Academia Sinica, China
Some existence theorems for the prescribed scalar curvature equation are given, improving some important results previous.
Q-curvature flow on S4
Andrea Malchiodi, School of Mathematics, Institute for
Advanced Study, USA
We consider the problem of prescribing the Q-curvature of S4 via a conformal transformation. Given a smooth positive function f, we define a flow on metrics in the conformal class of the standard one gS4. We prove that, either the metric converges to a solution of the problem, or it becomes "round" and concentrated near some point. Convergence is then proved, using Morse theory, under suitable assumptions on f.
The talk regards a joint work with M.Struwe.
An introduction to Kleinian
groups
William Abikoff, Department of Mathematics, University of
Connecticut
The basic ideas and recent results will be discussed. In particular the near complete solution of the Thurston program for hyperbolic 3-manifolds by Minsky et al, Agol together with some discussion of Perelman's work but not much.
I'll also present some recent results, joint with Bill Harvey, on classes of Kleinian groups, which are extremal with respect to geometric inequalities bounded by the number of generators.
Local pointwise and Harnack estimate for solutions of
the σk equation
Zheng-Chao Han, Rutgers University
Some recent results on the mean
curvature flow
Xu-Jia Wang, Centre for Mathematics and its Applications,
The Australian National University
In this talk I will briefly review recent progress on the singularity behavior of the Ricci and mean curvature flows. Then I will discuss the classification of ancient convex solutions to the mean curvature flow.
Weak solutions of fully
nonlinear PDE
Neil Trudinger, Centre for Mathematics and Its
Applications, The Australian National University
In these talks, we explore generalized notions of solution for nonlinear elliptic partial differential equations and associated boundary value problems, with the main focus on equations of Monge-Ampere type.
Topics will include:
- Alexandrov-Bakelman, Monge-Kantorovich and Perron solutions;
- Hessian measures in Euclidean space and Heisenberg groups.
These two parts are essentially independent. In 1, we build upon the fundamental geometric approach of Alexandrov, while in 2, we follow the recent analytic approach of Trudinger and Wang.
Bubbling accumulations for nonlinear elliptic equations
with critical nonlinearity
Juncheng Wei, Department of Mathematics, Chinese University
of Hong Kong
Blow-up solutions of nonlinear elliptic equations in
Rn with the critical Sobolev exponent
Man-Chun Leung, Department of Mathematics, National
University of Singapore
Special periodic solutions of
schrodinger flow
Weiyue Ding, School of Mathematical Sciences, Peking
University
We discuss a class of special solutions of Schrodinger flows when the target manifolds are Kahler-Einstein with non-trivial holomorphic vector fields. The solution has the form u(t, x) = S(t) o f(x), where f(x) is a mapping from the domain into the target K-E manifold, and S(t) is the one-parameter family of isometries generated by a Killing field. As an example, we show such periodic solutions exist for the case where both domain and target manifolds are the 2-sphere.
The recent study of two curvature
flows
Zheng Yu, East China Normal University
In this talk, we mainly report the recent study of the following two curvature flows: Riemannian metrics along Ricci curvature direction and hyper-surface along discriminated curvature in normal direction respectively. The first flow is about the negative gradient flow of the L2 norm of Ricci curvature on a compact closed Riemannian manifold. We should provide the proof of the local existence and give some new results on the blowing up of this forth order degenerate flow. For the second flow, we will study one mixed discriminated curvature flow of hyper-surface in Rn, which will include the local existence and its limit behavior of the flow with conditions on initial surfaces, such as strictly convex and even just the convex surfaces.
Jet Isomorphism Theorems in CR
and conformal geometries beyond the obstruction
Kengo Harichi, Graduate School of Mathematical Sciences,
University of Tokyo
For odd dimensional conformal manifolds, the ambient metric of Fefferman-Graham gives an isomorphism between the (jets of) curvature of the conformal structure and the (jets of) curvature of the ambient metric --- via this isomorphism one can easily write down all (scalar) local conformal invariants. The ambient metric construction is obstructed for even dimensional conformal manifolds and for CR manifolds, and in these cases one need to consider ambient metrics with logarithmic singularities. In this talk, I will explain how to formulate isomorphism theorems for such ambient metrics. (This is an interim report on a project with Robin Graham.)
Hyperbolic cone-surfaces, classical
Schottky groups, generalized Markoff maps and McShane’s
identity
Zhang Ying, Department of Mathematics, National University
of Singapore
In this talk I will report joint work with S. P. Tan and Y. L. Wong on generalizations of McShane’s identities.
McShane discovered in 1990 a remarkable identity concerning the lengths of all simple closed geodesics on a once-punctured torus with complete finite-area hyperbolic structure (and later extended it to more general hyperbolic surfaces with cusps). Recently, Mirzakhani has generalized the identity for bordered hyperbolic surfaces and found beautiful applications of the identity by using it to obtain a recursive formula for the Weil-Petersson volume of moduli spaces of such Riemann surfaces, and to obtain some counting estimates for simple closed geodesics on hyperbolic surfaces.
We generalize McShane’s identity to hyperbolic surfaces with cone-type singularities (with cone angles up to p) as well as with geodesic boundary and cusps. We obtain a unified identity using complex translation lengths — a cone point with angle θ will be assumed to have complex length θi. This generalization covers all other related identities previously obtained by McShane. We extend the generalized identity to classical Schottky groups to give some unexpected identities. We also show that it extends to generalized Markoff maps satisfying Bowditch’s Q-conditions, which include quasi-fuchsian representations of torus group into PSL(2;C), thus extending Bowditch’s work on Markoff maps. 1
Circle packings on projective
surfaces
Ser-Peow Tan, Department of Mathematics, National
University of Singapore
The study of circle packings date back to antiquity, but have seen a surge of activity in the last thirty years. For a closed orientable surface, a complex projective structure on the surface (called a projective surface here) is a structure modeled on the Riemann sphere with transition functions which are restrictions of elements of the group of complex projective transformations PSL(2,C) acting on the Riemann sphere. In this talk, we are interested in the interplay between these two fields, specifically, we are interested in circle packings on projective surfaces, and the related deformation spaces. The first crucial observation is that circles/disks are fundamental geometric objects in projective geometry and so circle packings make sense on projective surfaces. The second is that the Koebe-Andreev-Thurston circle packing theorem guarantees that the deformation space of projective structures admitting a circle packing of fixed combinatorial type is non-empty. This talk surveys some on-going studies (joint work with Sadayoshi Kojima and Shigeru Mizushima from Tokyo Inst. Of Tech) by describing a foundational basis for this study, a conjectural picture of the various deformation spaces and relations between them, and some of the results we have obtained so far.
On the behavior of solutions of
equations involving the p-Laplacian
Reynaldo Rey, University of the Philippines
In this talk, I will discuss the existence and behavior of solutions of the equation Δpυ + λ|υ|g-2 υ = 0 under varying assumptions on p and g.
Blowup solutions of inhomogeneous
nonlinear Schrodinger equations
Hongyan Tang, Institute of Mathematics, Academia Sinica,
China
This talk focuses on blowup solutions to the Cauchy problem of an inhomogeneous nonlinear Schrodinger equation on torus T2. The L2- concentration property for general initial data, the location of singularities and blowup rate of the L2-minimal blowup solutions will be presented. This is a joint work with Peter Pang and Youde Wang.
Q-curvature and spectral
invariants
Thomas Branson, Department of Mathematics, The University
of Iowa
The Q-curvature on an even-dimensional pseudo-Riemannian manifold is a local invariant whose properties generalize those of the 2-dimensional Gauss curvature. Q was introduced in talks and private communications by [Branson, 1987]; in 4 dimensions, Q was used to analyze the functional determinant in [Branson-{\O}rsted, Proc. AMS 1991] and [Branson-Chang-Yang, Commun. Math. Phys. 1992]. The general even dimensional version was worked out in detail in [Branson, Seoul National University Lecture Lecture Notes #4, 1993]. More recent work by several major contributors has solidified the importance of the Q-curvature, as well as the foundations of its construction.
The Q-curvature
- has a 1-homogeneous (in the conformal factor) conformal deformation law;
- naturally appears in functional determinant quotient formulas for pairs of metrics in a conformal class, and in the related exponential class inequality that estimates these determinants;
- has as its total metric variation the Fefferman-Graham obstruction tensor (and thus is an action principle for a generalization of Weyl relativity from four to arbitrary even dimensions).
In fact, the conformal deformation law from (1) above is by the critical Graham-Jenne-Mason-Sparling operator P, an important conformally invariant operator of order the dimension. (2) indicates that Q and P are implicit in the spectral theory of more elementary conformal objects, for example the conformal Laplacian.
We start with first principles to explore the above aspects of Q, and some generalizations of the functional determinant quotient to torsion quantities naturally associated with conformal structure.
Analytical aspects of the singular
Yamabe problem
Frank Pacard, Département de Mathématiques, Universitéde de
Paris XII
Abstract : The singular Yamabe problem can be stated as follows : Given a compact set K Ì Sn, does there exist a metric defined on Sn-K which is complete, conformal to the standard metric on the sphere and has constant scalar curvature R?
It is known that there are obstructions to the existence of such a metric. In particular, the dimension of K is very sensitive to the sign of R. In the negative and zero scalar curvature cases R ≤ R, recent progress has been done by D. Labutin and a clear picture of necessary and sufficient conditions for the solvability of the singular Yamabe problem is now available in this setting.
The case where R > 0 seems more rigid and only partial results are available so far. In this lectures, I will explain how to solve the singular Yamabe problem in the positive case R > 0, when K is a disjoint union of closed submanifolds.
The decomposition of global
conformal invariants
Spyridon Alexakis, Department of Mathematics, Princeton
University
It had been stipulated in the Physics literature that a scalar Riemannian invariant of weight -n (n even) whose integral over any n-dimensional manifold (Mn,gn) remains invariant under conformal transformations of the metric can be written as a linear combination of a conformally invariant scalar, the divergence of a vector field and a multiple of the Pfaffian of the curvature tensor. I intend to outline my proof of this conjecture and to discuss some of its applications to the structure of the Q-curvature of an even-dimensional manifold.
The ambient obstruction tensor and
Q-curvature
Robin Graham, Department of Mathematics, University of
Washington
This talk will focus on the ambient obstruction tensor, a generalization of the Bach tensor in four dimensions to higher even-dimensional conformal geometry. A proof will be outlined of the result, joint with Kengo Hirachi, that the obstruction tensor arises as the variational derivative of the integral of Branson's Q-curvature. The connection between Q-curvature and the obstruction tensor is made via the renormalized volume of a Poincare metric associated to the conformal structure.
Discrete Painleve property
Ramajayam Sahadevan, University of Madras
It is well known that the Painleve analysis originally advocated by Paul Painleve plays an important role, in general, in the analysis of nonlinear ordinary and partial differential equations and integrability in particular. In this seminar a discrete version of the Painleve property applicable to partial differential - difference and pure difference equations will be presented. Also, we show how this analytical technique provides an effective tool to predict integrable situations of discrete nonlinear systems including mapping.
Self-improving properties of Poincar´e inequalities
Seng Kee Chua, Department of Mathematics, National
University of Singapore
Nucleation of superconductivity and
smectics
Xingbin Pan, Department of Mathematics, National University
of Singapore
DeGennes's discovery of analogy between superconductivity and smectics proves useful in the study of phase transitions of liquid crystals. In this talk, we review recent progress on nucleation of superconductivity, and discuss some possible analogy in liquid crystals.
Compactness and non-compactness
issues in Yamabe type equations
Olivier Druet, UMPA, Ecole Normale Supérieure de Lyon,
France
We consider sequences of solutions of elliptic PDE's with critical Sobolev growth on compact Riemannian manifolds. And we ask the question of the asymptotic behaviour of these sequences. We will see that the dimension and the geometry of the manifold have an influence on the blow up phenomena that may occur. In particular, we will see that blow up can occur only with some special perturbations of the Yamabe equation.
Conformally invariant equations
Paul Yang, Department of Mathematics, Princeton University
I will briefly survey the analytic aspects of several conformally invariant equations such as the Yamabe equation, the Paneitz equation and the fully nonlinear second order equations. I will also talk about certain geometric applications to the topology of 4-manifolds, and the structure of certain Kleinian groups.
Asymptotic symmetries for conformal scalar curvature
equations with singularity
Lei Zhang, Department of Mathematics, Texas A & M
University
Bubbling location of f-harmonic
maps and inhomogeneous Landau-Lifshitz systems
Wang Youde, Institute of Mathematics, Academy of
Mathematics and System Sciences, Chinese Academy of Sciences
In this talk I will discuss the removal singularity of f-harmonic map from a Riemann surface, the bubble location of a Palais-smale sequence of f-harmonic map and finally we get the bubble location of inhomogeneous Landau-Lifshitz equation when it blows up at infinity.
