On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy

Analytic torsion, vortices and positive Ricci curvature

We characterize the global maximizers of a certain non-local functional defined on the space of all positively curved metrics on an ample line bundle L over a Kahler manifold X. This functional is an adjoint version, introduced by Berndtsson, of Donaldson's L-functional and generalizes the Ding-Tian functional whose critical points are Kahler-Einstein metrics of positive Ricci curvature. Applications to (1) analytic torsions on Fano manifolds (2) Chern-Simons-Higgs vortices on tori and (3) Kahler geometry are given. In particular, proofs of conjectures of (1) Gillet-Soulé and Fang (concerning …

A new parabolic flow in Kaehler manifolds

In this paper, we introduce a new parabolic equation on Kähler manifolds. The static point of this flow is related to the existence of a lower bound of the Mabuchi energy. In this paper, we prove the flow always exists for all times for any initial smooth data. Further more, if the initial metric has non-negative bisectional curvature, we prove the flow converges to a static metric eventually.

Some boundedness properties of solutions to the complex Yang-Mills equations on closed $4$-manifolds

In this article, we study the analytical properties of the solutions of the complex Yang-Mills equations on a closed Riemannian four-manifold $X$ with a Riemannian metric $g$. The main result is that if $g$ is $good$ and the connection is an approximate ASD connection, then the extra field has a positive lower {bound}. As an application, we obtain some gap results for Yang-Mills connections and Kapustin-Witten equations.

可光滑化簇上的奇异cscK度量

Singular cscK metrics on smoothable varieties

该研究证明了退化klt Kähler簇族中Mabuchi泛函强制阈值的下半连续性，建立了ℚ-Gorenstein可光滑化klt簇上奇异cscK度量的存在性，发展了族中多复变位势理论的新强拓扑并给出一致估计。

Lower bound of modified $K$-energy on a Fano manifold with degeneration for Kähler-Ricci solitons

In this paper, we extend Tosatti's method to study the lower boundedness of modified $K$-energy on a Fano manifold and apply this result to study the relative $K$-stability of the deformation space of a Kähler Ricci soliton.

Covariant Riesz transform on differential forms for $1<p\leq2$

In this paper, we study $L^p$-boundedness ($1<p\leq 2$) of the covariant Riesz transform on differential forms for a class of non-compact weighted Riemannian manifolds without assuming conditions on derivatives of curvature. We present in particular a local version of $L^p$-boundedness of Riesz transforms under two natural conditions, namely the curvature-dimension condition, and a lower bound on the Weitzenböck curvature endomorphism. As an application, the Calderón-Zygmund inequality for $1< p\leq 2$ on weighted manifolds is derived under the curvature-dimension condition as hypothesis.

On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows

Let $X$ be a compact Kähler manifold with a given ample line bundle $L$. In \cite{Don05}, Donaldson proved that the Calabi energy of a Kähler metric in $c_1(L)$ is bounded from below by the supremum of a normalized version of the minus Donaldson--Futaki invariants of test configurations of $(X,L)$. He also conjectured that the bound is sharp. In this paper, we prove a metric analogue of Donaldson's conjecture, we show that if we enlarge the space of test configurations to the space of geodesic rays in $\mathcal{E}^2$ and replace the Donaldson--Futaki invariant by the radial Mabuchi K-energy $\…

Higher Energies in Kahler Geometry I

Let $X\hookrightarrow \cpn $ be a smooth complex projective variety of dimension $n$. Let $λ$ be an algebraic one parameter subgroup of $G:=\gc$. Let $ 0\leq l\leq n+1$. We associate to the coefficients $F_{l}(λ)$ of the normalized weight of $λ$ on the $mth$ Hilbert point of $X$ new energies $F_{\om,l}(\vp)$. The (logarithmic) asymptotics of $F_{\om,l}(\vp)$ along the potential deduced from $λ$ is the weight $F_{l}(λ)$. $F_{\om,l}(\vp)$ reduces to the Aubin energy when $l=0$ and the K-Energy map of Mabuchi when $l=1$. When $l\geq 2$ $F_{\om,l}(\vp)$ coincides (modulo lower order terms) with th…

Mixed Integral Norms for Ricci Flow

We prove that a Ricci flow cannot develop a finite time singularity assuming the boundedness of a suitable space-time integral norm of the curvature tensor. Moreover, the extensibility of the flow is proved under a Ricci lower bound and the boundedness of a space-time integral norm of the scalar curvature.

Extremal Kähler metrics and energy functionals on projective bundles

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 example where the modified K energy is bounded from below but not proper.