A Schwarz lemma for Kähler affine metrics and the canonical potential of a proper convex cone

Abstract

This is an account of some aspects of the geometry of Kähler affine metrics based on considering them as smooth metric measure spaces and applying the comparison geometry of Bakry-Emery Ricci tensors. Such techniques yield a version for Kähler affine metrics of Yau's Schwarz lemma for volume forms. By a theorem of Cheng and Yau there is a canonical Kähler affine Einstein metric on a proper convex domain, and the Schwarz lemma gives a direct proof of its uniqueness up to homothety. The potential for this metric is a function canonically associated to the cone, characterized by the property that its level sets are hyperbolic affine spheres foliating the cone. It is shown that for an n-dimensional cone a rescaling of the canonical potential is an n-normal barrier function in the sense of interior point methods for conic programming. It is explained also how to construct from the canonical potential Monge-Ampère metrics of both Riemannian and Lorentzian signatures, and a mean curvature zero conical Lagrangian submanifold of the flat para-Kähler space.The ∞-Ricci tensor was introduced by A. Lichnerowicz in [46] where he used it to prove a generalization of the Cheeger-Gromoll splitting theorem.The idea related to these tensors relevant in what follows is that a lower bound on the (N + n)dimensional Bakry Emery Ricci tensor of the metric measure structure (k, φ) on the n-manifold M corresponds to a lower bound on the usual Ricci curvature of some metric associated to (k, φ) on some (n + N )-dimensional manifold fibering over M . In the example relevant here, N = n and the fibering manifold is the tube domain over an n-dimensional convex cone.One aim of this paper is to explain that it is profitable to regard a Riemannian Hessian metric g ij = ∇ i dF j as the mm-space determined by g ij in conjunction with the volume form H(F )Ψ = H(F ) 1/2 vol g , which is the pullback via the differential dF of the parallel volume form dual to Ψ on the dual affine space R n+1 * . More generally, a Kähler affine structure (g, ∇) is identified with the local mm-structure formed by g and the half Koszul one-form 1 2 H i .

Remarks on curvature in the transportation metric

“…The well-posedness of (2.1) was proved by Cheng and Yau, who continued the investigations of Calabi and Nirenberg. The formulation below is taken from [12]. We assume throughout this paper that we are given the standard Euclidean coordinate system {x i }.The interior of Ω = supp(µ) is equipped with the metric defines the dual convex potential Ψ, satisfying ∇Φ•∇Ψ(y) = y and pushing forward ν onto µ.…”

“…Extension to the metric-measure space is based on the estimates for weighted Laplacians of the distance function (see, for instance, [1]). A more general statement with a proof can be found in [12], Theorem 3.3.…”

“…In particular, the Ricci and Bakry-Émery tensors coincide in this case and Theorem 1.1 immediately implies Corollary 1.2. (Calabi [5], see also Fox [12], Loftin [21] and Sasaki [23]). Consider a proper open convex cone Ω ⊂ R n .…”

The associated families of semi-homogeneous complete hyperbolic affine spheres

“…Let AutK denote the automorphism group of the cone K. The equiaffine invariance of (2.1) implies F is invariant under unimodular automorphisms g ∈ AutK (see [10] or [12]). Then the dimension of (2.1) can be effectively reduced by the generic dimension of the orbits of AutK.…”

Equiaffine geometry of level sets and ruled hypersurfaces with equiaffine mean curvature zero

“…This F is ‐logarithmically homogeneous and so satisfies . See for details. Consequently, F and Ω satisfy the conditions of Lemma .…”