arXiv research

Study Kähler-Ricci solitons on bounded domains, proving they are Kähler-Einstein.

We introduce a flow of Kähler structures over Fano manifolds with formal limit at infinite time a Kähler-Ricci soliton. This flow correspond to a Perelman's modified backward Kähler-Ricci type flow that we call Soliton-Kähler-Ricci flow. It can be generated by the Soliton-Ricci flow. We assume that the Soliton-Ricci fl…

Simply-connected shrinking Kähler-Ricci solitons are proven.

We prove that any shrinking Kahler Ricci soliton has only one end, and that any expanding Kahler Ricci soliton with proper potential has only one end.

Smooth approximations of Kähler-Ricci solitons found using quantized metrics and Futaki invariants.

Unique soliton found on resolved cones.

In this paper, we consider the twisted Kähler-Ricci soliton, and show that the existence of twisted Kähler-Ricci soliton with semi-positive twisting form is closely related to the properness of some energy functionals. We also consider the conical Kähler-Ricci soliton, and obtain some existence results. In particular, …

The paper analyzes Kähler-Ricci flow near Kähler-Ricci solitons under complex structure deformation.

In this paper, we prove that a sequence of weak almost Kähler-Ricci solitons under further suitable conditions converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded belo…

Unique Kähler-Ricci solitons found for S^1-invariant metrics.

The paper shows how to stabilize perturbed Kähler-Ricci solitons.

Local equivalence found between certain solitons and generalized Kähler-Ricci solitons.

Study on Kähler-Ricci solitons on toric manifolds, proving rigidity.

We investigate the linear stability of Kähler-Ricci solitons for perturbations induced by varying the complex structure within a fixed Kähler class. We calculate stability for the known examples of Kähler-Ricci solitons.

Study lower bounds on modified K-energy on Fano manifolds with Kähler-Ricci solitons.

Complete shrinking soliton found on a specific complex surface.

In this paper, we prove that any complete shrinking gradient Kähler-Ricci solitons with positive orthogonal bisectional curvature must be compact. We also obtain a classification of the complete shrinking gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature.

Derives scalar reduction for generalized Kähler-Ricci solitons, proving uniqueness.

We first show that a Kähler cone appears as the tangent cone of a complete expanding gradient Kähler-Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model (on which the soliton lives). This allows us to classify two-dimensional complete expanding gradient Kähler-Ri…

In this paper we prove the compactness result for compact Kähler Ricci gradient shrinking solitons. If $(M_i,g_i)$ is a sequence of Kähler Ricci solitons of real dimension $n \ge 4$, whose curvatures have uniformly bounded $L^{n/2}$ norms, whose Ricci curvatures are uniformly bounded from below and $μ(g_i,1/2) \ge A$ (…

Let $(Y, d)$ be a Gromov-Hausdorff limit of $n$-dimensional closed shrinking Kähler-Ricci solitons with uniformly bounded volumes and Futaki invariants. We prove that off a closed subset of codimension at least 4, Y is a smooth manifold satisfying a shrinking Kähler-Ricci soliton equation. A similar convergence result …

We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact Kähler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kähler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kähler…

This paper proves uniqueness of Kähler-Ricci flow on non-compact manifolds.

New insights into Kähler Ricci solitons and Calabi-Yau cones.

In this paper, we prove the existence of a Kahler Ricci soliton on any smooth Fano horospherical manifold by a study of the Kahler-Ricci flow. Indeed, we prove that the renormalized Kahler Ricci flow converges in the sense of Cheeger Gromov and that this limit is a Kahler-Ricci soliton.

Study of generalized almost-Kähler-Ricci solitons and their implications.

The study shows rigidity of Kähler-Ricci solitons on a specific 4D manifold.

Proof of flow convergence on Fano manifolds.

In this short paper, we improve the result of Phong-Song-Sturm on degeneration of Fano Kähler-Ricci solitons by removing the assumption on the uniform bound of the Futaki invariant. Let $\mathcal{KR}(n)$ be the space of Kähler-Ricci solitons on $n$-dimensional Fano manifolds. We show that after passing to a subsequence…

In this paper, we generalize Chen-Tian energy functionals to Kähler-Ricci solitons and prove that the properness of these functionals is equivalent to the existence of Kähler-Ricci solitons. We also discuss the equivalence of the lower boundedness of these functionals and their relation with Tian-Zhu's holomorphic inva…

Unique shrinking gradient Kähler-Ricci solitons found on non-compact toric manifolds.

Paper studies Kähler-Ricci flow convergence on Fano manifolds.

In this paper, we prove that complete gradient steady Kähler-Ricci solitons with harmonic Bochner tensor are necessarily Kähler-Ricci flat, i.e., Calabi-Yau, and that complete gradient shrinking (or expanding) Kähler-Ricci solitons with harmonic Bochner tensor must be isometric to a quotient of $N^k\times \mathbb{C}^{n…

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 so…

New Kähler solitons found that are not $U(n)$-invariant.

Kähler-Ricci solitons can be immersed into complex space forms if and only if the manifold is Einstein.

Study identifies Kähler-Einstein, Kähler-Ricci soliton, and Sasaki-Einstein metrics on log del Pezzo surfaces.

This paper studies Kähler-Ricci solitons on Heisenberg groups and related metrics.

We study a Boltzmann's type entropy functional (which appeared in existing literature) defined on Kähler metrics of a fixed Kähler class. The critical points of this functional are gradient Kähler-Ricci solitons, and the functional was known to be monotonically increasing along the Kähler-Ricci flow in the canonical cl…

Kähler-Ricci flow on spherical Fano manifolds converges to a soliton.

Study shows instability of Kähler Ricci solitons and stability of orbifold singularities.

Solves complex Monge-Ampère equation for Kähler-Ricci solitons.

We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kähler-Ricci solitons. This naturally enlarges the moduli space of Fano manifolds with Kähler-Einstein metrics, which was constructed by Odaka and Li-Wang-Xu. We discover a moment map picture for Kähler-Ricci solitons, and give comp…

We show that expanding Kähler-Ricci solitons which have positive holomorphic bisectional curvature and are asymptotic to Kähler cones at infinity must be the U(n)-rotationally symmetric expanding solitons constructed by Cao.

The paper explores Kähler-Ricci solitons with maximal symmetry in complex dimension two.

Study on shrinking solitons of generalized Ricci flow.

The paper studies a degenerate equation related to Kähler-Ricci flow on symplectic quotients.

We show that the generalized Kähler-Ricci soliton equation on 4-dimensional toric Kähler orbifolds reduces to ODEs assuming there is a Hamiltonian 2-form. This leads to an explicit resolution of this equation on labeled triangles and convex labeled quadrilaterals. In particular, we give the explicit expression of the K…