Kahler Ricci Solitons Research Articles

Deformation of Curved BPS Domain Walls and Supersymmetric Flows on 2d Kähler-Ricci Soliton

We consider some aspects of the curved BPS domain walls and their supersymmetric Lorentz invariant vacua of the four dimensional N = 1 supergravity coupled to a chiral multiplet. In particular, the scalar manifold can be viewed as a two dimensional Kahler-Ricci soliton generating a one-parameter family of Kahler manifolds evolved with respect to a real parameter, τ. This implies that all quantities describing the walls and their vacua indeed evolve with respect to τ. Then, the analysis on the eigenvalues of the first order expansion of BPS equations shows that in general the vacua related to the field theory on a curved background do not always exist. In order to verify their existence in the ultraviolet or infrared regions one has to perform the renormalization group analysis. Finally, we discuss in detail a simple model with a linear superpotential and the Kahler-Ricci soliton considered as the Rosenau solution.

Kähler-Ricci flow, Morse theory, and vacuum structure deformation of <i>N</i> = 1 supersymmetry in four dimensions

We address some aspects of four-dimensional chiral N = 1 supersym- metric theories on which the scalar manifold is described by Kahler geometry and can further be viewed as Kahler-Ricci soliton generating a one-parameter family of Kahler geometries. All couplings and solu- tions, namely the BPS domain walls and their supersymmetric Lorentz invariant vacua turn out to be evolved with respect to the flow parameter related to the soliton. Two models are discussed, namely N = 1 theory on Kahler-Einstein manifold and U(n) symmetric Kahler-Ricci soliton with positive definite metric. In the first case, we find that the evolution of the soliton causes topological change and correspondingly, modifies the

The Kähler-Ricci flow on Kähler manifolds with 2-non-negative traceless bisectional curvature operator

The authors show that the 2-non-negative traceless bisectional curvature is preserved along the Kahler-Ricci flow. The positivity of Ricci curvature is also preserved along the Kahler-Ricci flow with 2-non-negative traceless bisectional curvature. As a corollary, the Kahler-Ricci flow with 2-non-negative traceless bisectional curvature will converge to a Kahler-Ricci soliton in the sense of Cheeger-Gromov-Hausdorff topology if complex dimension n ≥ 3.

Convergence of Kähler-Ricci flow

In this paper, we prove a theorem on convergence of Kähler-Ricci flow on a compact Kähler manifold which admits a Kähler-Ricci soliton.

Kähler–Ricci solitons on compact complex manifolds with C1(M) > 0

In this paper, we discuss the relation between the existence of Kahler–Ricci solitons and a certain functional associated to some complex Monge–Ampere equation on compact complex manifolds with positive first Chern class. In particular, we obtain a strong inequality of Moser–Trudinger type on a compact complex manifold admitting a Kahler–Ricci soliton.

An Equivariant Version of the K–energy

In this note, we present a connection between equivariant Bott–Chern classes and Kahler–Ricci solitons. We also propose a generalized version the of the K–energy.

A new holomorphic invariant and uniqueness of Kähler-Ricci solitons

In this paper, a new holomorphic invariant is defined on a compact Kahler manifold with positive first Chern class and nontrivial holomorphic vector fields. This invariant generalizes the Futaki invariant. We prove that this invariant is an obstruction to the existence of Kahler-Ricci solitons. In particular, using this invariant together with the main result in [TZ1], we solve completely the uniqueness problem of Kahler-Ricci solitons. Two functionals associated to the new holomorphic invariant are also discussed. The main result here was announced in [TZ2].

Kähler-Einstein metrics for manifolds with nonvanishing Futaki character

In this paper, we generalize the concept of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki character. Similar to Kahler-Einstein metrics, these new metrics have various nice properties. In addition, the equations for the metrics are in general neither those of extremal Kahler metrics nor those of Kahler-Ricci solitons.

Kähler-Ricci soliton typed equations on compact complex manifolds withC 1(M) > 0

As a generalization of Calabi’s conjecture for Kahler-Ricci forms, which was solved by Yau in 1977, we discuss the existence of Kahler-Ricci soliton typed equation on a compact Kahler manifold (M, g) with positive first Chem C1 (M) > 0 as well as the uniqueness. For a given positively definite (1,1)-form Ω ∈ C1 (M) of M and a holomorphic vector field X on M, we prove that there is a Kahler form ω in the Kahler class [ωg] solving the Kahler-Ricci soliton typed equation if and only if, i) X is belonged to a reductive subalgebra of holomorphic vector fields and the imaginary part of X generates a compact one-parameter transformations subgroup of M; and ii) LX Ω is a real-valued (1,1)-form. Moreover, the solution ω is unique in the class [ωg].

Uniqueness of Kahler—Ricci solitons on compact Kahler manifolds

We introduce a new holomorphic invariant on any compact Kähler manifolds with positive first Chern class and nontrivial holomorphic vector fields, which contains the Futaki invariant as a special case. This invariant is shown to be an obstruction to the existence of Kähler-Ricci solitons. By solving a complex Monge-Ampère equation, we prove the uniqueness of Kähler-Ricci solitons.

Fano–Ricci limit spaces and spectral convergence

We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact Kahler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kahler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kahler-Ricci limit soliton and the space of all $L^2$ holomorphic vector fields with smooth potentials is a Lie algebra with respect to the Lie bracket, then the Lie algebra has the same structure as smooth Kahler-Ricci solitons. In particular if a $\Q$-Fano variety admits a Kahler-Ricci limit soliton and all holomorphic vector fields are $L^2$ with smooth potentials then the Lie algebra has the same structure as smooth Kahler-Ricci solitons. If the sequence consists of Kahler-Ricci solitons then the Ricci limit space is a weak Kahler-Ricci soliton on a $\mathbb{Q}$-Fano variety and the space of limits of $1$ eigenfunctions for the weighted $\barpartial$-Laplacian forms a Lie algebra with respect to the Poisson bracket and admits a similar decomposition as smooth Kahler-Ricci solitons.

Non-Kähler Ricci flow singularities modeled on Kähler–Ricci solitons

We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are shrinking Kahler-Ricci solitons. Specifically, the singularity model for these solutions is expected to be the discovered in [FIK03]. Our partial results support the conjecture that the blowdown soliton is stable under Ricci flow, as well as the conjectured stability of the subspace of Kahler metrics under Ricci flow.