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Akito Futaki

Publications and source records attributed to Akito Futaki.

35 records · Page 2Linked to original sources
arXiv2014

The weighted Laplacians on real and complex metric measure spaces

In this short note we compare the weighted Laplacians on real and complex (Kähler) metric measure spaces. In the compact case Kähler metric measure spaces are considered on Fano manifolds for the study of Kähler-Einstein metrics while real metric measure spaces are considered with Bakry-Émery 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.

math.DG↗
arXiv2012

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

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 $\mathbb C^n$ to the toric Calabi-Yau cones over Sasaki-Einstein manifolds.

math.DG↗
arXiv2012

Some topics on Ricci solitons and self-similar solutions to mean curvature flow

In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower diameter bound for compact manifolds with shrinking Ricci solitons. Such a bound can be obtained from an eigenvalue estimate for a twisted Laplacian, called the Witten-Laplacian. In the second part we discuss self-similar solutions to the mean curvature flow on cone manifolds. Many results have been obtained for solutions in $\bfR^n$ or $\bfC^n$. We see that many of them extend to cone manifolds, and in particular results on $\bfC^n$ for special Lagrangians and self-shrinkers can be extended to toric Calabi-Yau cones. We also see that a similar lower diameter bound can be obtained for self-shrinkers to the mean curvature flow as in the case of shrinking Ricci solitons.

math.DG↗
arXiv2012

On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking Ricci solitons

We prove a lower bound estimate for the first non-zero eigenvalue of the Witten-Laplacian on compact Riemannian manifolds. As an application, we derive a lower bound estimate for the diameter of compact gradient shrinking Ricci solitons. Our results improve some previous estimates which were obtained by the first author and Y. Sano in [12], and by B. Andrews and L. Ni in [1]. Moreover, we extend the diameter estimate to compact self-similar shrinkers of mean curvature flow.

math.DG↗
arXiv2011

Asymptotic Chow polystability in Kähler geometry

It is conjectured that the existence of constant scalar curvature Kähler metrics will be equivalent to K-stability, or K-polystability depending on terminology (Yau-Tian-Donaldson conjecture). There is another GIT stability condition, called the asymptotic Chow polystability. This condition implies the existence of balanced metrics for polarized manifolds $(M, L^k)$ for all large $k$. It is expected that the balanced metrics converge to a constant scalar curvature metric as $k$ tends to infinity under further suitable stability conditions. In this survey article I will report on recent results saying that the asymptotic Chow polystability does not hold for certain constant scalar curvature Kähler manifolds. We also compare a paper of Ono with that of Della Vedova and Zuddas.

math.DG↗
arXiv2011

An integral invariant from the view point of locally conformally Kähler geometry

In this paper we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980's. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifolds.

math.DG↗
arXiv2011

Constructing Kähler-Ricci solitons from Sasaki-Einstein manifolds

We construct gradient Kähler-Ricci solitons on Ricci-flat Kähler cone manifolds and on line bundles over toric Fano manifolds. Certain shrinking and expanding solitons are pasted together to form eternal solutions of the Ricci flow. The method we employ is the Calabi ansatz over Sasaki-Einstein manifolds, and the results generalize constructions of Cao and Feldman-Ilmanen-Knopf.

math.DG↗
arXiv2010

Lower diameter bounds for compact shrinking Ricci solitons

It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci solitons.

math.DG↗
arXiv2010

Hilbert series and obstructions to asymptotic semistability

Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature Kähler (cscK) metric. A natural question is whether or not the other obstructions are linearly dependent on the obstruction to the existence of a cscK metric. The purpose of this paper is to see that this is not the case by exhibiting toric Fano threefolds in which these obstructions span at least two dimension. To see this we show that on toric Fano manifolds these obstructions are obtained as derivatives of the Hilbert series. This last observation should be regarded as an extension of the volume minimization of Martelli, Sparks and Yau.

math.DG↗
arXiv2009

Multiplier ideal sheaves and geometric problems

In this expository article we first give an overview on multiplier ideal sheaves and geometric problems in Kählerian and Sasakian geometries. Then we review our recent results on the relationship between the support of the subschemes cut out by multiplier ideal sheaves and the invariant whose non-vanishing obstructs the existence of Kähler-Einstein metrics on Fano manifolds.

math.DG↗
arXiv2009

Momentum construction on Ricci-flat Kähler cones

We extend Calabi ansatz over Kähler-Einstein manifolds to Sasaki-Einstein manifolds. As an application we prove the existence of a complete scalar-flat Kähler metric on Kähler cone manifolds over Sasaki-Einstein manifolds. In particular there exists a complete scalar-flat Kähler metric on the toric Kähler cone manifold constructed from a toric diagram with a constant height.

math.DG↗
arXiv2008

Einstein metrics and GIT stability

In this expository article we review the problem of finding Einstein metrics on compact Kähler manifolds and Sasaki manifolds. In the former half of this article we see that, in the Kähler case, the problem fits better with the notion of stability in Geometric Invariant Theory if we extend the problem to that of finding extremal Kähler metrics or constant scalar curvature Kähler (cscK) metrics. In the latter half of this paper we see that most of ideas in Kähler geometry extend to Sasaki geometry as transverse Kähler geometry. We also summarize recent results about the existence of toric Sasaki-Einstein metrics.

math.DG↗
arXiv2008

Multiplier ideal sheaves and integral invariants on toric Fano manifolds

We extend Nadel's results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal sheaves on toric del Pezzo surfaces which do not admit Kähler-Einstein metrics. We also show that one can define multiplier ideal sheaves for Kähler-Ricci solitons and extend the result of Nadel using the holomorphic invariant defined by Tian and Zhu.

math.DG↗
arXiv2007

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

In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant $f_1$ for the first Chern class case becomes an obstruction to the existence of transverse Kähler metric of constant scalar curvature. We prove the existence of transverse Kähler-Ricci solitons (or {\it Sasaki-Ricci soliton}) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if $S$ is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a Sasaki-Einstein structure on $S$. As an application we obtain irregular toric Sasaki-Einstein metrics on the unit circle bundles of the powers of the canonical bundle of the two-point blow-up of the complex projective plane.

math.DG↗
arXiv2007

Uniqueness and examples of compact toric Sasaki-Einstein metrics

In [11] it was proved that, given a compact toric Sasaki manifold of 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 $S^5 \sharp k(S^2 \times S^3)$ for each positive integer $k$.

math.DG↗
arXiv2006

Harmonic total Chern forms and stability

In this paper we will perturb the scalar curvature of compact Kahler manifolds by incorporating it with higher Chern forms, and then show that the perturbed scalar curvature has many common properties with the unperturbed scalar curvature. In particular the perturbed scalar curvature becomes a moment map, with respect to a perturbed symplectic structure, on the space of all complex structures on a fixed symplectic manifold, which extends the results of Donaldson and Fujiki on the unperturbed case.

math.DG↗