arXiv research

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

Study proves convergence of quantized geodesics to Mabuchi geodesics.

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.

Negative curvature proven in Sasaki manifold space completion.

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 …

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…

Infinite volumes of Bergman spaces on product manifolds.

New findings on Mabuchi energy and stability of manifolds.

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.

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…

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…

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…

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

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

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

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…

Proves invariance of weighted extremal Kähler metrics under smooth blowups.

Given a compact Kähler manifold, we prove that all global isometries of the space of Kähler metrics are induced by biholomorphisms and anti-biholomorphisms of the manifold. In particular, there exist no global symmetries for Mabuchi's metric. Moreover, we show that the Mabuchi completion does not even admit local symme…

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.

Study convexity of Mabuchi functional in big cohomology classes.

Over the space of Kähler metrics associated to a fixed Kähler class, we first prove the lower bound of the energy functional $\tilde E^β$, then we provide the criterions of the geodesics rays to detect the lower bound of $\tilde {\mathfrak J}^β$-functional. They are used to obtain the properness of Mabuchi's $K$-energy…

Continuity method proves existence of Mabuchi solitons on Fano manifolds.

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

Suppose $(X,ω)$ is a compact Kähler manifold. Following Mabuchi, the space of smooth Kähler potentials $\mathcal H$ can be endowed with a Riemannian structure, which induces an infinite dimensional path length metric space $(\mathcal H,d)$. We prove that the metric completion of $(\mathcal H,d)$ can be identified with …

The space of Kähler potentials in a compact Kähler manifold, endowed with Mabuchi's metric, is an infinite dimensional Riemannian manifold. We characterize local isometries between spaces of Kähler potentials, and prove existence and uniqueness for such isometries.

Proves results on K-stability using arcs and Mabuchi functional.

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

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…

Sasaki manifolds have isometric spaces of potentials implying similar geometric properties.

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…

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 continuous geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes f…

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.

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 …

We develop the connection between equivariant completions of algebraic homogeneous spaces of reductive groups and lower bounds for the Mabuchi energy of a polarized manifold over the space of Bergman metrics. We provide a new definition of Tian's CM Polarization and discuss its properties.

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.

Study of generalized Kähler structures on complex manifolds using symplectic and Riemannian geometry.

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.

Study Mabuchi rays on toric Kähler manifolds to understand quantization.

Let (X,L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1,1)-forms in the first Chern class of L, endowed with the Mabuchi metric, is not strictly convex modulo automorphisms. How…

We study the Futaki invariant and the Mabuchi K-energy of a Kähler manifold $M$ using the Deligne pairing technique developed in earlier papers. We first prove a rather simple characterization of the Futaki character: The Futaki character on a Q-Fano variety is the eigenvalue of the action of $Aut(M)$ on $Chow(M)$, the…

The paper introduces a new system of equations for Hessian-cscK metrics.

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…

Let X be a projective manifold. We prove that the Mabuchi Energy of X is bounded below on all degenerations in B (the space of Bergman metrics) if and only if it is bounded below uniformly on B.

Uniqueness of weighted extremal metrics on Kähler manifolds proven.

In this paper, we prove that the transverse Mabuchi K-energy functional is convex along the weak geodesic in the space of Sasakian metrics. As an application, we obtain the uniqueness of constant scalar curvature Sasakian metrics modulo automorphisms for the transverse holomorphic structure.

We introduce complex generalizations of the classical Legendre transform, operating on Kähler metrics on a compact complex manifold. These Legendre transforms give explicit local isometric symmetries for the Mabuchi metric on the space of Kähler metrics around any real analytic Kähler metric, answering a question origi…