Akito Futaki

Uniqueness and Examples of Compact Toric Sasaki-Einstein Metrics

In [11] it was proved that, given a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on

Self-similar solutions to the mean curvature flows on Riemannian cone manifolds and special Lagrangians on Toric Calabi-Yau cones

S^5 \sharp k(S^2 \times S^3)

Eta Invariants and Automorphisms of Compact Complex Manifolds

for each positive integer k.

An Obstruction to the Existence of Einstein Kähler Metrics.

We define a family of integral invariants containing those which are closely related to asymptotic Chow semi-stability of polarized manifolds. It also contains an obstruction to the existence of Kahler–Einstein metrics and its natural extensions by the author, Calabi and Bando as Kahlerian invariants and by Morita and the author as invariant polynomials of the automorphism groups of compact complex manifolds.

Kähler-Einstein Metrics and Integral Invariants

The self-similar solutions to the mean curvature flows have been defined and studied on the Euclidean space. In this paper we initiate a general treatment of the self-similar solutions to the mean curvature flows on Riemannian cone manifolds. As a typical result we extend the well-known result of Huisken about the asymptotic behavior for the singularities of the mean curvature flows. We also extend the results on special Lagrangian submanifolds on

Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

\mathbb C^n

Bilinear forms and extremal Kähler vector fields associated with Kähler classes

to the toric Calabi-Yau cones over Sasaki-Einstein manifolds.

Invariant polynomials of the automorphism group of a compact complex manifold

In this short note we compare the weighted Laplacians on real and complex (Kahler) metric measure spaces. In the compact case Kahler metric measure spaces are considered on Fano manifolds for the study of Kahler–Einstein metrics while real metric measure spaces are considered with Bakry–Emery Ricci tensor. There are twisted Laplacians which are useful in both cases but look alike each other. We see that if we consider noncompact complete manifolds significant differences appear.