arXiv research

This note discusses the higher K-energy functionals which were defined by Bando and Mabuchi, and integrate higher Futaki invariants. Two new formulas for the higher K-energy functionals are given, and the second K-energy is shown to be related to Donaldson's Lagrangian applied to metrics on the tangent bundle.

Extends K-energy to complexified Kähler classes for scalar curvature study.

The paper introduces $G$-Sasaki manifolds and studies K-energy criteria.

Explain convexity of K-energy leading to unique metrics.

Using Perelman's results on Kahler Ricci flow, we prove that the K energy is bounded from below if and only if the F functional is bounded from below in the canonical Kahler class.

We show that K-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kahler class long time solutions of Calabi flow minimize both K-energy and Calabi energy. Lastly, by applying convergence results from the theory of minimizing …

Establishes convexity and coercivity of K-energy functional for complex tori.

In this paper, we prove the equivalence of the existence of extremal Kahler metrics and the properness of the modified K energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K energy, the infimum of the Calabi energy and the extremal polynomials. In particular, we give an exa…

K-energy is not strictly convex on certain complex manifolds, but under specific conditions, it is.

The paper explores stability and coercivity for toric polarizations, linking them to K-energy.

Given a compact polarized Kähler manifold $X\hookrightarrow\mathbb{CP}^N$, the space of Bergman metrics on $X$, parameterized by $\mathrm{SL}(N+1,\mathbb{C})$, corresponds to a dense set in the space of Kähler potentials in the Kähler class as $N\to\infty$. Critical points of the $k$th K-energy functional, which is def…

In this paper, we discuss a Donaldson's version of the modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified $K$-energy.

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…

In this paper, we give a criterion for the properness of the K-energy in a general Kahler class of a compact Kahler manifold by using Song-Weinkove's result. As applications, we give some Kahler classes on $\mathbb{C}\mathbb{P}^2\#3\overline {\mathbb{C}\mathbb{P}^2}$ and $\mathbb{C}\mathbb{P}^2\#8\overline {\mathbb{C}\…

Constructs a path integral for quantum Mabuchi K-energy.

We develop some pluripotential theoretic techniques for the transversally holomorphic foliation of a Sasakian manifold. We prove the convexity of the K-energy along weak geodesics for Sasakian manifolds. This implies that the K-energy is bounded below if a constant scalar curvature structure exists with those metrics m…

In this paper, we give a result on the properness of the K-energy, which answers a question of Song-Weinkove in any dimensions. Moreover, we extend our previous result on the properness of K-energy to the case of modified K-energy associated to extremal Kahler metrics.

The paper studies K-energy on compactifications of Lie groups and proves the existence of Kahler-Einstein metrics.

Proves weight polytope matches with energy vectors in toric varieties.

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

Formula found for energy slope in complex geometry.

Study geodesic distances and convexity in contact sets.

The paper examines conditions for the equator map to be minimizing or unstable for higher order energy functionals.

We partially confirm an old conjecture of Donaldson that if there exists a cscK metrics in a given Kähler class, then there is no degenerated geodesic ray which is tamed by a bounded ambient geometry unless it parallels to a holomorphic line consists of cscK metrics only. We also prove that for simple test configuratio…

The Mabuchi K-energy map is exhibited as a singular metric on the refined CM polarization of any equivariant family $\mathbf{X}\overset{p}{\to} S$. Consequently we show that the generalized Futaki invariant is the leading term in the asymptotics of the reduced K-energy of the generic fiber of the map $p$. Properness of…

Stability of a new map derived from the equator map is analyzed.

We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kahler potentials on a compact Kahler manifold thus confirming a conjecture of Chen and give some applications in Kahler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally …

Paper proves various types of varieties minimize a specific energy.

In this paper, we discuss the relative $K$-stability and the modified $K$-energy associated to the Calabi's extremal metric on toric manifolds. We give a sufficient condition in the sense of convex polytopes associated to toric manifolds for both the relative $K$-stability and the properness of modified $K$-energy. In …

Based on Donaldson's method, we prove that, for an integral Kahler class, when there is a Kahler metric of constant scalar curvature, then it minimizes the K-energy. We do not assume that the automorphism group is discrete.

We prove the convergence of geodesic distance during the quantization of the space of Kähler potentials. As applications, this provides alternative proofs of certain inequalities about the K-energy functional in the projective case.

Sharp bounds for Calabi energy derived from geodesic rays and Mabuchi K-energy.

Paper proves existence of constant scalar curvature Kähler metrics under certain conditions.

Study on constant mu-scalar curvature Kähler metrics, generalizing cscK and Kähler-Ricci solitons.

Let $(X,ω)$ be a compact connected Kähler manifold and denote by $(\mathcal E^p,d_p)$ the metric completion of the space of Kähler potentials $\mathcal H_ω$ with respect to the $L^p$-type path length metric $d_p$. First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to $\mathcal E^p$ is …

In this note we give a simplified proof of a recent result of X.X. Chen, which together with work of G. Szekelyhidi implies that on a sufficiently small deformation of a polarized constant scalar curvature Kahler manifold the K-energy has a lower bound.

We provide a new proof of a result of X.X.Chen and G.Tian : for a polarized extremal Kähler manifold, an extremal metric attains the minimum of the modified K-energy. The proof uses an idea of C.Li adapted to the extremal metrics using some weighted balanced metrics.

Let (X,L) be a polarized Kähler manifold that admits an extremal Kähler metric in c1(L). We show that on a nearby polarized deformation that preserves the symmetry induced by the extremal vector field of (X,L), the modified K-energy is bounded from below. This generalizes a result of Chen, Székelyhidi and Tosatti to ex…

Let $(X,ω)$ be a compact Kähler manifold and $\mathcal H$ the space of Kähler metrics cohomologous to $ω$. If a cscK metric exists in $\mathcal H$, we show that all finite energy minimizers of the extended K-energy are smooth cscK metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As a…

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.

J.Eells and L. Lemaire introduced $k$-harmonic maps, and Wang Shaobo showed the first variation formula. In this paper, we give the second variation formula of $k$-energy, and give a notion of index, nullity and weakly stable. We also study $k$-harmonic maps into the product Riemannian manifold, and $k$-harmonic curves…

Let X be a smooth, linearly normal n dimensional complex projective variety. Assume that the projective dual of X has codimension one with defining polynomial D(X). In this paper the log of the norm of D(X) is expressed as the restriction to the Bergman metrics of an energy functional on X. We show how, for smooth plan…

In this paper, we study the limiting properties of the $K$ energy for smooth hypersurfaces in the projective spaces. Our result generalizes the result of Ding-Tian (W. Ding and G. Tian. Kähler-Einstein metrics and the generalized Futaki invariant. {\em Invent Math}, 110:315-335, 1992.) in the case of hypersurfaces. In …

Researchers introduce new energies to study constant scalar curvature metrics.

We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a very recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manif…

Paper studies K-polystability of cscK manifolds with holomorphic vector fields.

The limiting behavior of the normalized Kähler-Ricci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi K-energy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabu…

Extends pluripotential theory to Sasaki manifolds for cscs metrics.