Research Description (for non-experts)

The concept of constrained optimization is one of the most basic in mathematics. My research concerns an optimization problem in geometry which, while difficult to convey completely in a brief summary, can be understood by way of analogy. The presentation here begins with intuitive geometric motivation and some definitions, and proceeds to a more precise description of my past research.

Geometric Motivation

A soap film in equilibrium that spans a closed, planar loop of wire is a portion of a plane. For a non-planar loop, the film pulls itself into a stable configuration of smallest area, but the boundary constraints prevent even small pieces of the film from being planar. This geometric optimization problem is ``extrinsic,'' in the sense that it involves the way the surface ``bends'' in space. By contrast, my work involves an ``intrinsic'' energy functional, but in many respects is similar to the minimal surface problem. The spaces with which I work are called ``Kähler manifolds,'' and are of interest for geometric reasons as well as physical ones: They arise naturally in complex algebraic geometry, several complex variables, and differential geometry, and have many attributes that make them candidates for descriptions of spacetime physics at very small--and to a lesser extent, very large--scales. The ``optimization'' problem is to find the ``shape'' of a space when a certain ``energy'' is minimized, or more specifically, to find a Kähler metric on a manifold whose curvature is distributed as evenly as possible. Roughly, the constraint arises from the topology of the space, and prevents the curvature from being constant.

Intuitive Definitions

The basic objects of differential geometry and global analysis are manifolds, spaces that locally look like Euclidean space. Complex manifolds are modelled on complex Euclidean space Cn, and admit a (local) notion of holomorphic function. The shape of a manifold is determined by a (Riemannian) metric, which assigns lengths to tangent vectors and thereby endows the manifold with intrinsic notions of distance and angle. In an arbitrary manifold, a metric determines paths of (locally) shortest length, called geodesics; these generalize great circles on the surface of a sphere. Three (sufficiently close) points determine a geodesic triangle, but the sum of the interior angles may not be equal to pi; on the surface of a doughnut, the sum is smaller than pi on the saddle-shaped ``inside'' of the hole, and is greater than pi on the convex ``outside.'' The curvature of a metric is a precise measure of this local angular defect, and a single number, the scalar curvature, quantifies the average angular defect at a point. A metric determines a scalar curvature function on the manifold.

The optimization problem in which I am interested is to minimize the integral of the square of the scalar curvature; this functional is called the Calabi energy of the metric. (The total scalar curvature turns out to be essentially topological in the Kähler situation, and is therefore not suitable for minimizing.) Intuitively, the Calabi energy gives the standard deviation of the scalar curvature, and is therefore minimized exactly when the scalar curvature is ``as nearly constant as possible.''

The manifolds with which I work satisfy a technical property called the ``Kähler condition'' that relates their differential geometry and complex analytic structure. Rather than give the definition, I will explain how the Kähler condition is used to constrain the optimization problem. In complex geometry, a ``curve'' is a (holomorphically) embedded Riemann surface, and a metric determines the area of every curve. The Kähler condition implies that

  1. A compact curve in a Kähler manifold is always homologically non-trivial (that is, it never bounds a 3-manifold in the ambient space),

  2. Its area is determined by its homology class.

The constraint on the optimization problem is to fix the areas of homology classes, i.e., to minimize the energy among metrics for which the area of an arbitrary curve is the same as with respect to some ``backgound'' metric. The set of metrics satifying such a normalization constitutes a Kähler class. The need for some constraint is seen by considering what happens if the metric is scaled by a constant factor k: the energy scales by kn-2, with n the complex dimension. Unless n=2, the energy can be made arbitrarily small by suitable scaling, so with no constraint there are no critical points. Since areas of curves change by a factor k2 under scaling, fixing the Kähler class prevents this degeneracy.

Critical Metrics, and Past Research

The variational problem described above was introduced by Calabi. In a pair of fundamental papers, Calabi established the basic properties of critical metrics, and gave examples that do not have constant scalar curvature. One striking feature is the Euler equation: A metric is critical iff the gradient vector field of the scalar curvature function is a (global) holomorphic vector field. This demonstrates the close relationship between complex analysis and differential geometry in this situation.

I used results of Calabi, together with some work of Futaki and Mabuchi, to show that every critical metric is actually an absolute minimum of the energy (in its Kähler class), and that the critical value of the energy can be computed from a priori information. This shows that the common name, ``extremal'' Kähler metric, is justified, and (as observed independently by Simanca) that the critical value of the energy varies smoothly as the constraints are varied. In a separate paper, Simanca and I characterized Kähler metrics on Hirzebruch surfaces that are locally conformal to Einstein metrics. This characterization is closely related to extremal Kähler metrics.

An observation, made jointly with Mabuchi, gave a new criterion that a manifold must satisfy in order to have an extremal Kähler metric. This criterion generalized a known condition, due to Levine, but was genuinely stronger, in the sense of excluding more manifolds.

Much of my work has revolved around the construction of extremal metrics--and in particular of metrics of constant scalar curvature--that are invariant under a circle action. My thesis described a large class of examples, and gave complex-analytic criteria for existence of metrics of constant scalar curvature on the same class of manifolds. This provided a partial converse to a theorem of Lichnerowicz, to the effect that the group of holomorphic transformations of a Kähler manifold of constant scalar curvature is constrained (precisely, is a reductive group). It is known that the complete converse is false (examples of Burns and de Bartolomeis), even for Fano manifolds (as shown recently by Tian).

Singer and I have given what we hope is a definitive treatment of an ansatz, due to Calabi, for reducing existence of Kähler metrics with specified scalar curvature to solution of an ODE. We use this ansatz to find many interesting metrics that appear to be new, including complete, scalar-flat Kähler metrics on C2 (which are consequently anti-self-dual), and many examples of complete Einstein-Kähler metrics of negative curvature. We also show in a precise sense that except for possible ``sporadic'' metrics (which we believe do not exist), all the metrics of constant scalar curvature arising in the Calabi ansatz fall under the scope of our existence theorems.

Last Change: 13 October, 2002