SGS XII: Monday December 17, 2007
University of Alabama at Birmingham

Let (M̃ⁿ , g), n ≥ 3, be a compact smooth Riemannian manifold of dimension n with smooth boundary ∂M , M̃ = M \ ∂M be the interior of M̃ , and let Ric_g denote the Ricci tensor of g. We are interested in the question whether there exists a complete metric g̃ on M with negative Ricci tensor in the conformal class of g satisfying the equation:

det(-Ric_g̃ ) = 1.

More generally, let ƒ be a smooth symmetric function defined in a open convex symmetric cone Λ⊂ Rⁿ which contains Λ⁺ = {Λ ∈ Rⁿ : λ_i > 0} satisfying certain ellipticity structure conditions. Let Λ^-[g] denote the collection of metrics g on M in the conformal class of g such that λ(Ric_g̃) = (λ₁ , ... , λ_n ), the eigenvalues of g̃ ^-1 Ric_g̃, belongs to -Λ everywhere on M.

Problem 1. Find a complete metric g̃ ∈ Λ^-[g] on M with:

ƒ (λ(Ric_g̃)) = 1 in M.

In this talk we present some recent results from joint work with Huaiyu Jian and discuss open questions. Our result implies, in particular, that on any smooth domain in Rⁿ contained in a half-space there exists a complete conformally flat metric with negative Ricci tensor satisfying det(-Ric_g̃ ) = 1.

The first step in finding a spacetime solution to the Einstein gravitational field equations via the inital value formulation is to construct initial data which satisfy the Einstein constraint equations. There are three ways of carrying out this construction which have been found to be useful: the conformal and conformal thin sandwich methods, the gluing techniques, and the quasi-spherical approaches. We describe each of these, we discuss their advantages and disadvantages, we outline some of their recent successful applications, and we present some of the outstanding questions remaining to be solved from each of these perspectives.

A basic step in the classical black hole uniqueness theorems is Hawking's theorem on the topology of black holes, which asserts that cross sections of the event horizon in (3+1)-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. Recent interest and developments in the study of higher dimensional black holes has drawn attention to the question of what are the allowable black hole topologies in higher dimensions. This question was addressed in a recent paper with Rick Schoen in which we obtain a natural generalization of Hawking's theorem to higher dimensions. In this talk we discuss this work and further related developments. The results we describe are based on the geometry of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces.

The connection between harmonic maps and representations of discrete groups has been studied extensively in recent years. A harmonic map is classically defined between Riemannian manifolds and is a critical point of the energy functional. The seminal work of Gromov and Schoen on p-adic rigidity initiated the study of harmonic maps in the singular setting. The focus of this talk are harmonic maps from a Riemannian complex to metric spaces of non-positive curvature. We discuss conditions when one can prove that such a harmonic map is totally geodesic or even constant. In the case the harmonic map is equivariant with respect to a representation of the fundamental group of the domain space to the isometry group of a NPC space, one can in this way deduce the rigidity of the representation. This research is motivated by the study of representations of lattices in non-Archimedian groups in connection with Margulis superrigidity.

Complete Riemannian manifolds with nonnegative Ricci curvature have been intensively studied and well understood. Riemannian manifolds with a negative lower bound for Ricci curvature are considerably more complicated and less understood. It turns out the bottom of spectrum of the Laplace operator plays an important role. I will first survey some recent results on such manifolds with positive bottom of spectrum. Then I will discuss a new rigidity theorem which characterizes hyperbolic manifolds. The proof uses idea from potential theory and Brownian motion on Riemannian manifolds

The Einstein-Maxwell equations under axial symmetry and stationarity reduce to a problem of singular harmonic map into the complex hyperbolic plane. Using this model, G. Weinstein showed that there exists a family of spacetimes having multiple black holes which are parameterized uniquely by the number of black holes, masses, angular momenta, charges and the distances between them. He also pointed out that such spacetimes might possibly possess a conical singularity along the axis connecting the black holes, which can be interpreted as the forces. For a better understanding of this singularity, it is necessary to study the regularity of the above harmonic map problem. This will be the purpose of the talk. The idea is to recast the harmonic map system as a singular homogeneous quasilinear elliptic system. We show that, to a certain extent, the newly formed system shares a common feature with classical elliptic systems: smallness in normalized energy implies regularity. The regularity for the original harmonic map problem is then established by exploiting the richness in symmetries of the hyperbolic plane. We show that any solution can always be expressed as the sum of an explicit solution corresponding to a static spacetime and a smooth nonlinear contribution.