Atlas: On Geodesics in a Function Space of Kahler metrics by Eugenio Calabi

Atlas home || Conferences | Abstracts | about Atlas

International Congress on Differential Geometry in memory of Alfred Gray (1939-1998)
September 18-23, 2000
Universidad del País Vasco
Bilbao, Spain

Organizers
M. Fernández (chairman), R. Ibáñez, M. Macho-Stadler

View Abstracts
Conference Homepage

On Geodesics in a Function Space of Kähler metrics
by
Eugenio Calabi
University of Pennsylvania, USA
Coauthors: Xiuxiong Chen

Oral Communication

Consider an n-dimensional, compact, complex manifold M admitting Kähler metrics, among which we pick a ``background" metric g₀, to represent the function space \Omega consisting of all Kähler metrics in the same deRham cohomology class. In terms of a local, holomorphic coordinate system (z), any Kähler metric g in \Omega is expressed by an hermitian matrix (g_{\alpha[`(\beta)]}) = (g_{\alpha[`(\beta)]}(z, [`z])), related to the (g_{(0)\alpha[`(\beta)]}) corresponding to g₀ by the equation

g_{\alpha[`(\beta)]}=g_{(0)\alpha[`(\beta)]} +\partial²\Phi/\partialz^\alpha \partialz^[`(\beta)]; g_{\alpha[`(\beta)]}dz^\alphadz^[`(\beta)] > 0 (1)

The function \Phi(z, [`z]), called the distorsion potertial of g with respect to g₀, is of class C^{1, 1}, globally defined on M, and is uniquely determined by the pair (g_o, g) up to an additive constant; otherwise, in terms of g₀, it is arbitrary, except for the restriction implied by the inequality (1).

The purpose of this report is to communicate some recent contributions to the study of the function space of all metrics g in \Omega, with emphasis on the rectifiable, parametrized paths in \Omega. Such paths are denoted by g(t) or, in terms of the distorsion potentials, by \Phi(t)=\Phi(t;z, [`z]), for t varying in the unit interval I. The ``velocity" of such a path at the ponit g(t) is represented by the function \Phi': M --> R, (essentially, the ``time" derivative of \Phi(t)), \Phi'(t)=\partial\Phi/\partialt-\int_M\partial\Phi/\partialt dVol(g(t))/\int_MdVol(g(t)). Thus, the tangent space of \Omega at each g is isomorphic to the function space of all functions j'(z, [`z]):M --> R of class C^{1, 1}, that are orthogonal to constants in terms of the L₂-norm with respect to the volume element dVol(g).

In addition to the ``obvious" C^{1, 1}-topology, the space \Omega has also a ``natural" weaker topology induced by path lenght metric defined by the pre-Hilbert space metric on the tangent space,

dS²= ó
õ

M
(j'(z,

z

))² dVol(g); (2)

this metric was defined originally (in succession, but independently) by T. Mabuchi, by S. Semmes and S. Donaldson, and hence will be called the MSD-metric.

One can calculate directly the Levi-Civita connection in \Omega corresponding to the metric (2): the resulting equation for a geodesic path in \Omega, with the parameter t proportional to the arc lenght, becomes

\Phi^''(t) = \partial\Phi'/\partialt = g_(t)^{\alpha[`(\beta)]} (\partial\Phi'/\partialz^\alpha) (\partial\Phi'/\partialz^[`(\beta)]). (3)

Semmes and Donaldson proved, independently, that, for any two distinct metrics in \Omega, the lenght of any path connecting them, in terms of (2), has a positive lower bound. Donaldson also showed that the lower bound is achieved by a weak solution of (3) for the corresponding boundary value problem in (M×I, M×{0, 1}). Locally, the p.d.e. implicit in (3) is a sort of degenerate-elliptic Monge-Ampère equation in Cⁿ×R; it is generally believed that, at least if the boundary data are sufficiently smooth, then the corresponding geodesic paths are likewise smooth, unique and depend continuously on the boundary data. Intuitively this is expected, especially since the MSD metric (2) and the connection (3) are known to be, formally, those of an infinite dimensional symmetric space of non-positive curvature; however the usual geometric tools needed to verify these properties fail, because the space is nowhere locally complete with respect to the MSD metric.

These is, however, an interesting class of Kählerian manifolds in which a certain (infinite dimensional) subspace of \Omega is geodesically convex, with all the regularity properties that one might expect: the manifolds in question are the toric complex manifolds and the metrics considered are the toric-invariant metrics.

An n-dimensional, compact, complex manifold M is called a toric manifold, if it admits an effectively acting group of holomorphic transformations that is isomorphic to the multiplicative group C^{* n}, and hence splits M into a finite family of orbits. Among these orbits there is a unique one, denoted by M₀, that is open and everywhere dense in M. Fixing one such group action (in case there are more choices), M may be imbedded holomorphically in an N-dimensional projective space P_C^N with a projective action of C^{* n} that is equivariant with its given action on the imbedded M. The natural coordinates (z¹, z², ..., zⁿ: z^\alpha in C^*) of C^{* n} then may used as holomorphic coordinate functions in M₀ and extend analytically as meromorphic functions in M.

The group C^{* n} has a unique maximal compact subgroup, the torus group Tⁿ; any Kähler metric in M may be averaged over the pullback action of Tⁿ, yielding a Kähler metric that is invariant under the action on Tⁿ. Such an invariant metric is called a toric (invariant) Kähler metric.

We introduce now the logarithmic coordinates (w¹, w², ..., wⁿ;z^\alpha=exp(w^\alpha)) for the universal covering space of M₀, and decompose each w^\alpha into its real and imaginary parts, w^\alpha = u^\alpha +2\pii v^\alpha. One sees then that M₀ is holomorphically equivalent to the quotient of the numerical space Cⁿ by the lattice \Lambda = {(v) in Zⁿ}. Hence any toric Kähler metric g in M₀, expressed in terms of (w) is of the form ds²=2g_{\alpha[`(\beta)]} dw^\alphad[`z]^[`(\beta)], where g_{\alpha[`(\beta)]}=\partial²\Psi(u)/\partialu^\alpha \partialu^\beta, where the local potential function \Psi is an entire, strictly convex function of (u) in Rⁿ, uniquely determined by g up to the addition of a linear polynomial. Furthermore, the regularity of the metric in the complement of M₀ in M implies that the image of Rⁿ under the gradient map {(u) --> grad\Psi(u)=(\xi)} is the interior of a polyhedral, convex body \Pi in the space Rⁿ(\xi) dual to Rⁿ(u), as well as certain asymptotic properties of \Psi(u) at infinity. The polyhedron \Pi is uniquely determined, up to a parallel translation, by the Kähler cohomology class represented by \Omega_g, while a toric Kähler metric in M in the function space \Omega' corresponding to any other deRham class produces, similarly, a polyhedron \Pi' subset Rⁿ(\xi) combinatorially equivalent to \Pi, obtained by suitable, independent parallel translations of the hyperplanes containing each face of \Pi.

We now adjust the choice of al local Kähler potential \Psi_g: Rⁿ(u) --> R of each toric Kähler metric g in \Omega so that it differs from the one corresponding to a background metric g₀ by a (necessarily bounded) distorsion potential; consequently the polyhedra \Pi subset Rⁿ(\xi) associated with any two such metrics coincide. Consider the Legendre transform of \Psi_g, ^*\Psi_g(\xi)=sup{<\xi, u>-\Psi_g(u): (u) in Rⁿ}, \Psi_g(u)=sup{<\xi, u>-^*\Psi_g(\xi): (\xi) in Int(\Pi)}, for any toric Kähler metric g in \Omega. One easily verifies that, given any two toric Kähler metrics g, g' in \Omega and the corresponding Kähler potentials \Psi(u) and \Psi'(u)=\Psi(u)+\Phi(u) (with \Phi(u) bounded), then the geodesic path in \Omega joining g and g' consists necessarily of toric metrics and is generated simply by a linear interpolation between the corresponding Legendre transforms, ^*\Psi(\xi, t)=(1-t)^*\Psi(\xi)+t ^*\Psi'(\xi).

The explicit construction of such geodesics in a few simple cases has been valuable in settling several questions, dealing, for instance, with their asymptotic behaviour when one of the endpoints tends asymptotically to a degenerated metric.

Date received: June 20, 2000