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Xiuxiong Chen

Publications and source records attributed to Xiuxiong Chen.

At least 19 recordsLinked to original sources
arXiv2026

Poincaré type J-equation

We introduce a two-parameter continuity path for the J-equation and use it to characterize the solvability of the J-equation for Kähler metrics with Poincaré type singularities along a divisor $D$, allowing simple normal crossings and self-intersections. On Kähler surfaces, we show that the classical subsolution condition in the smooth setting implies solvability in the Poincaré type setting for any smooth divisor $D$. As a consequence, if $X$ contains no curves of negative self-intersections and $K_X[D]$ is ample, then the K-energy is bounded from below on any Poincaré type Kähler class. In the smooth divisor case, we further analyze the asymptotic behavior of solutions near $D$, and show that existence of a Poincaré type solution implies existence of a solution to the J-equation on $D$.

math.DG↗
arXiv2020

On the geometry of asymptotically flat manifolds

In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin-Thorpe inequality for oriented Ricci-flat $4$-manifolds with curvature decay and controlled holonomy. As an application, we show that any complete asymptotically flat Ricci-flat metric on a $4$-manifold which is homeomorphic to $\mathbb R^4$ must be isometric to the Euclidean or the Taub-NUT metric, provided that the tangent cone at infinity is not $\mathbb R \times \mathbb R_+$.

math.DG↗
arXiv2018

Gravitational instantons with faster than quadratic curvature decay (I)

In this paper, we study gravitational instantons (i.e., complete hyperkäler 4-manifolds with faster than quadratic curvature decay). We prove three main theorems: 1.Any gravitational instanton must have known end----ALE, ALF, ALG or ALH. 2.In ALG and ALH-non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in ALG and ALH cases. 3.In ALF-D_k case, it must have an O(4)-multiplet.

math.DG↗
arXiv2018

A class of fully nonlinear equations

In this paper we consider a class of fully nonlinear equations which cover the equation introduced by S. Donaldson a decade ago and the equation introduced by Gursky-Streets recently. We solve the equation with uniform weak $C^2$ estimates, which hold for degenerate case.

math.AP↗
arXiv2018

On the constant scalar curvature Kähler metrics, general automorphism group

In this paper, we derive estimates for scalar curvature type equations with more singular right hand side. As an application, we prove Donaldson's conjecture on the equivalence between geodesic stability and existence of cscK when $Aut_0(M,J)\neq0$. Moreover, we also show that when $Aut_0(M,J)\neq0$, the properness of $K$-energy with respect to a suitably defined distance implies the existence of cscK.

math.DG↗
arXiv2018

On the constant scalar curvature Kähler metrics, existence results

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the $K$-energy is non-increasing. Moreover, we prove that the properness of $K$-energy in terms of $L^1$ geodesic distance $d_1$ in the space of Kähler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the $K$-energy in $(\mathcal{E}^1, d_1)$ are smooth.

math.DG↗
arXiv2017

On the constant scalar curvature Kähler metrics, apriori estimates

In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of a $C^0$ bound for the Kähler potential. We also discuss some local versions of these estimates which can be of independent interest.

math.DG↗
arXiv2016

Gravitational instantons with faster than quadratic curvature decay (III)

This is our third paper in a series on the gravitational instantons. In this paper, we classify ALG and ALH gravitational instantons. In ALG case, we extend Hein's construction slightly and show that it's the only ALG gravitational instanton. In ALH case, we prove a Torelli-type theorem.

math.DG↗
arXiv2016

Gravitational instantons with faster than quadratic curvature decay (II)

This is our second paper in a series to study gravitational instantons, i.e. complete hyperkäler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: 1.The asymptotic rate of gravitational instantons to the standard models can be improved automatically. 2.Any ALF-D_k gravitational instanton must be the Cherkis-Hitchin-Ivanov-Kapustin-Lindström-Roček metric.

math.DG↗
arXiv2016

Space of Ricci flows (II)

Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized Kähler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed Kähler Einstein manifolds. As applications, we prove the Hamilton-Tian conjecture and the partial-$C^0$-conjecture of Tian.

math.DG↗
arXiv2015

Kähler-Ricci flow, Kähler-Einstein metric, and K-stability

We prove the existence of Kahler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kahler-Ricci flows. The key ingredient is an algebro-geometric description of the asymptotic behavior of Kahler-Ricci flow on Fano manifolds. This is in turn based on a general finite dimensional discussion, which is interesting in its own and could potentially apply to other problems. As one application, we relate the asymptotics of the Calabi flow on a polarized Kahler manifold to K-stability assuming bounds on geometry.

math.DG↗