Related papers: On sharp lower bounds for Calabi type functionals …

In this short note we prove that a Kahler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results of…

We prove that constant scalar curvature K\"ahler metric "adjacent" to a fixed K\"ahler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T…

Let $(X, \omega)$ be a compact symplectic manifold and $L$ be a Lagrangian submanifold. Suppose $(X, L)$ has a Hamiltonian $S^1$ action with moment map $\mu$. Take an invariant $\omega$-compatible almost complex structure, we consider…

We consider canonical metrics on Fano manifolds. First we introduce a norm-type functional on Fano manifolds, which has Kahler-Einstein or Kahler-Ricci soliton as its critical point and the Kahler-Ricci flow can be viewed as its (reduced)…

In this paper, we study Mabuchi's K-energy on a compactification M of a reductive Lie group G, which is a complexification of its maximal compact subgroup K. We give a criterion for the properness of K-energy on the space of K \times…

We first give a precise statement on the short time existence of the Calabi flow and prove a stability result: any metric near a constant scalar curvature metric will flow to this cscK metric exponentially fast. Secondly, we prove that a…

We prove that on a K\"ahler manifold admitting an extremal metric $\omega$ and for any K\"ahler potential $\varphi_0$ close to $\omega$, the Calabi flow starting at $\varphi_0$ exists for all time and the modified Calabi flow starting at…

We present an analytic proof of the relationship between the Calabi-Futaki invariant for a K\"ahler manifold relative to a holomorphic vector field with a nondegenerate zero and the corresponding invariant of its blowup at that zero,…

In this note, we consider a sequence of test configurations compatible with a Kaehler metric in $c_1(L)$ on a polarized algebraic manifold $(X,L)$. Then an explicit formula for the Donaldson-Futaki invariant $F_1$ for the sequence will be…

We prove that for any smooth polarized complex $n$-dimensional manifold $(X, L_X)$ which admits an extremal K\"ahler metric in $c_1(L_X)$, and for any integer $k$ large enough (in terms of a bound depending on $(X, L_X)$), the…

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 a gradient estimate for Donaldson's equation \[\omega\wedge(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^{n-1}=e^F(\chi+\sqrt{-1}\partial\overline{\partial}\varphi)^n\] (and its parabolic analog) on an $n$-dimensional compact…

Over the space of K\"ahler metrics associated to a fixed K\"ahler class, we first prove the lower bound of the energy functional $\tilde E^\beta$, then we provide the criterions of the geodesics rays to detect the lower bound of $\tilde…

Let $(X, P)$ be a toric variety. In this note, we show that the $C^0$-norm of the Calabi flow $\varphi(t)$ on $X$ is uniformly bounded in $[0, T)$ if the Sobolev constant of $\varphi(t)$ is uniformly bounded in $[0, T)$. We also show that…

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…

We prove a compactness theorem for K\"ahler metrics with various bounds on Ricci curvature and the $\mathcal I$ functional. We explore applications of our result to the continuity method and the Calabi flow.

Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to…

In arXiv:0805.2192, we set up a gauge-theoretic equation on symplectic 6-manifolds, which is a version of the Hermitian-Einstein equation perturbed by Higgs fields, and call Donaldson-Thomas equation, to analytically approach the…

We argue that in type IIB LVS string models, after including the leading order moduli stabilisation effects, the moduli space for the remaining flat directions is compact due the Calabi-Yau K\"ahler cone conditions. In cosmological…

We formulate a notion of K-stability for K\"ahler manifolds, and prove one direction of the Yau-Tian-Donaldson conjecture in this setting. More precisely, we prove that the Mabuchi functional being bounded below (resp. coercive) implies…