arXiv research

The paper proves Mabuchi solitons and constants on Fano admissible manifolds.

Fano varieties get Mabuchi solitons if they have extremal Kähler metrics.

Continuity method proves existence of Mabuchi solitons on Fano manifolds.

Article establishes criteria for multiplier Hermitian-Einstein metrics on KSM-manifolds.

For Fano manifolds T. Mabuchi introduced a generalization of the Kähler-Einstein metric, which is characterized as the critical point of the Ricci-Calabi functional. We show that a Fano manifold admits Mabuchi's metric if and only if it is uniformly relatively D-stable.

As a generalization of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki invariant, Mabuchi solitons are critical points of a Calabi-type energy functional. We study their existence on toric Fano varieties and the underlying algebraic stability notion: relative Ding stability. As a toy model for a YTD…

Mabuchi solitons generalize Kähler-Einstein metrics on Fano manifolds, which constitute a Yau-Tian-Donaldson type correspondence with relative Ding stability. Comparing with Kähler-Ricci solitons, there is a distinct necessary condition for the existence. We show this condition can be implied by the uniformly relative …

This paper connects soliton-type metrics with weighted CSCK metrics on Fano manifolds.

We investigate the Kähler-Ricci flow modified by a holomorphic vector field. We find equivalent analytic criteria for the convergence of the flow to a Kähler-Ricci soliton. In addition, we relate the asymptotic behavior of the scalar curvature along the flow to the lower boundedness of the modified Mabuchi energy.

Proves existence of weighted-cscK metrics on Kähler manifolds.

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

We give necessary and sufficient conditions for existence of solutions to a general system of complex Monge-Ampère equations on Fano horosymmetric manifolds. In particular, we get necessary and sufficient conditions for existence of coupled Kähler-Ricci solitons, Mabuchi metrics and twisted Kähler-Einstein metrics in t…

The paper studies how certain solitons on Fano manifolds extend to nearby deformations.

Study proves Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.

Paper proves existence of weighted constant scalar curvature metrics.

We investigate how to obtain various flows of Kähler metrics on a fixed manifold as variations of Kähler reductions of a metric satisfying a given static equation on a higher dimensional manifold. We identify static equations that induce the geodesic equation for the Mabuchi's metric, the Calabi flow, the pseudo-Calabi…

Extremal Kahler metrics and Sasaki-Einstein metrics characterized via coercive energy.

For $φ$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $\int_X e^{-φ}.$ We prove that the logarithm of the volume is concave along bounded geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from…

In this paper, we study Mabuchi metrics on Fano manifolds. We prove that Mabuchi metrics exist if the modified Ding functional is proper modulo a reductive subgroup of its automorphism group. On the other hand, the inverse that Mabuchi metrics implies the properness is obtained by using Darvas-Rubinstein's properness p…

Study on Mabuchi functional's convexity using ε-geodesics.

The purpose of this paper is to generalize the convexity of Mabuchi's functional to the conic setting. We first established a frame to study conic cscK metrics, and then the conic Mabuchi functional was introduced in such a way that conic cscK metrics are its critical points. Finally we proved the convexity of the coni…

Study proves convergence of quantized geodesics to Mabuchi geodesics.

Study convexity of Mabuchi functional in big cohomology classes.

New findings on Mabuchi energy and stability of manifolds.

Study small eigenvalues of Toeplitz operators and their relation to Mabuchi geodesics.

Let X be a smooth, linearly normal algebraic variety. It is shown that the Mabuchi energy of X restricted to the Bergman metrics is completely determined by the X-hyperdiscriminant of format (n-1) and the Chow form of X. As a corollary it is shown that the Mabuchi energy is bounded from below for all degenerations in G…

We show that a projective manifold is stable if and only if the Mabuchi energy is proper on the space of algebraic metrics. We show that stability implies finite automorphism group.

Suppose $(X,ω)$ is a compact Kähler manifold. We introduce and explore the metric geometry of the $L^{p,q}$-Calabi Finsler structure on the space of Kähler metrics $\mathcal H$. After noticing that the $L^{p,q}$-Calabi and $L^{p'}$-Mabuchi path length topologies on $\mathcal H$ do not typically dominate each other, we …

Derives scalar curvature formula in generalized Kähler geometry.

We give a criterion for the coercivity of the Mabuchi functional for general Kähler classes on Fano manifolds in terms of Tian's alpha invariant. This generalises a result of Tian in the anti-canonical case implying the existence of a Kähler-Einstein metric. We also prove the alpha invariant is a continuous function on…

We are analysing the convexity and continuity properties of the Mabuchi functional along weak geodesics. The key technical point in our paper is the global approximation of weak geodesics obtained via a well-chosen family of Monge-Ampère equations.

An explicit seminorm $||f||_{#}$ on the vector space of Chow vectors of projective varieties is introduced, and shown to be a generalized Mabuchi energy functional for Chow varieties. The singularities of the Chow varieties give rise to currents supported on their singular loci, while the regular parts are shown to rep…

In the previous papers \cite{L1, L2} the author constructed Mabuchi and Aubin-Yau functionals over any complex surfaces and three-folds, respectively. Using the method in \cite{L2}, we construct those functionals over any complex manifolds of the complex dimension bigger than or equal to 2.

The paper proves the existence of singular cscK metrics on smoothable varieties.

In this note we construct Mabuchi $\mathcal{L}^{\rm M}_ω$ functional and Aubin-Yau functionals $\mathcal{I}^{\rm AY}_ω, \mathcal{J}^{\rm AY}_ω$ on any compact complex surfaces, and establish a number of properties. Our construction coincides with the original one in the Kähler case.

We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature Kähler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of Kähler metrics.

We prove existence, uniqueness and convergence of solutions of the degenerate J-flow on Kahler surfaces. As an application, we establish the properness of the Mabuchi energy for Kahler classes in a certain subcone of the Kahler cone on minimal surfaces of general type.

Donaldson conjectured \cite{Dona96} that the space of Kähler metrics is geodesic convex by smooth geodesic and that it is a metric space. Following Donaldson's program, we verify the second part of Donaldson's conjecture completely and verify his first part partially. We also prove that the constant scalar curvature me…

Negative curvature proven in Sasaki manifold space completion.

Let $Ω$ be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in $Ω$. We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application we study the existence of local Kähler-Ei…

Study Mabuchi rays on toric Kähler manifolds to understand quantization.

Study weighted constant scalar curvature on non-compact toric fibrations, proving K-stability conditions.

The J-flow is a parabolic flow on Kahler manifolds. It was defined by Donaldson in the setting of moment maps and by Chen as the gradient flow of the J-functional appearing in his formula for the Mabuchi energy. It is shown here that under a certain condition on the initial data, the J-flow converges to a critical metr…

Proves invariance of weighted extremal Kähler metrics under smooth blowups.

Uniqueness of weighted extremal metrics on Kähler manifolds proven.

Infinite volumes of Bergman spaces on product manifolds.

We study the convergence of the Kähler-Ricci flow on a Fano manifold under some stability conditions. More precisely we assume that the first eingenvalue of the $\bar\partial$-operator acting on vector fields is uniformly bounded along the flow, and in addition the Mabuchi energy decays at most logarithmically. We then…

Continuous metrics on ample bundles lie in infinite-dimensional cones.