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Chengjian Yao

Publications and source records attributed to Chengjian Yao.

14 recordsLinked to original sources
arXiv2026

Uniqueness for the Kähler-Yang-Mills equations

A formula for the $α$-K-energy functional for the Kähler-Yang-Mills (KYM) equations is provided in this paper. Using this formula and Chen's $ε$-geodesic equation on the space of Kähler potentials, we prove that if a solution exists to the KYM equations for a simple vector bundle on a Kähler manifold with discrete automorphism group, then it is unique and the $α$-K-energy is bounded from below. Inspired by the study of constant scalar curvature Kähler metrics, we introduce a coupled J-equation to study the $α$-K energy functional.

math.DG↗
arXiv2026

Closed $\mathrm{G}_2$-structures with $\mathbb{T}^3$-symmetry and hypersymplectic structures

We decompose linear $\mathrm{G}_2$-structure in canonical ways adapted to 3-dimensional subspaces, in terms of certain natural 1-forms and definite triple of 2-forms, and apply the decompositions to the study of $\mathrm{G}_2$-structure with $\mathbb{T}^3$-symmetry. Closed $\mathrm{G}_2$-structures $φ$ with an effective $\mathbb{T}^3$-symmetry on connected manifolds are roughly classified into two types according the orbits being non-isotropic or isotropic. Type I: if some orbit is non-isotropic, then the action is almost-free and $φ$ reduces to a good hypersymplectic orbifold with cyclic isotropic groups. Type II: if some orbit is isotropic, then the action is locally multi-Hamiltonian for $φ$. Moreover, the open and dense subset of principal orbits is foliated by $\mathbb{T}^3$-invariant hypersymplectic manifolds. If $φ$ is torsion-free, then for Type I, there arises another natural hypersymplectic structure, and a generalized Gibbons-Hawking Ansatz extending Madsen-Swann Ansatz is derived. For Type II, $φ$ is locally toric. Assuming moreover completeness and constant orbit volume, exactly three possibilities occur. Type Ia: orbits are purely non-isotropic non-associative, then the hypersymplectic 4-orbifold becomes a flat manifold. Type Ib: orbits are purely associative, then the $\mathbb{T}^3$-action is flat, and the hypersymplectic 4-orbifold becomes a hyperkähler 4-orbifold. Type II: orbits are isotropic, then all orbits are principal, and $φ$ is flat.

math.DG↗
arXiv2025

Hypersymplectic Structures Invariant Under an Effective Circle Action

A hypersymplectic structure on a 4-manifold is a triple of symplectic forms for which any non-zero linear combination is again symplectic. In 2006, Donaldson conjectured that on a compact 4-manifold any hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperkähler triple. We prove this under the assumption that the initial structure is invariant under an effective $S^1$-action. In particular we show that the underlying 4-manifold is diffeomorphic to $\mathbb{T}^4$.

math.SG↗
arXiv2024

Convergence of the hypersymplectic flow on $T^4$ with $T^3$-symmetry

A hypersymplectic structure on a 4-manifold is a triple $ω_1, ω_2, ω_3$ of 2-forms for which every non-trivial linear combination $a^1ω_1 + a^2 ω_2 + a^3 ω_3$ is a symplectic form. Donaldson has conjectured that when the underlying manifold is compact, any such structure is isotopic in its cohomolgy class to a hyperkähler triple. We prove this conjecture for a hypersymplectic structure on $T^4$ which is invariant under the standard $T^3$ action. The proof uses the hypersymplectic flow, a geometric flow which attempts to deform a given hypersymplectic structure to a hyperkähler triple. We prove that on $T^4$, when starting from a $T^3$-invariant hypersymplectic structure, the flow exists for all time and converges modulo diffeomorphisms to the unique cohomologous hyperkähler structure.

math.DG↗
arXiv2023

The dissolving limit and large volume limit of Einstein-Bogomol'nyi metrics

We study the limits of Einstein-Bogomol'nyi metrics on $\mathbf{P}^1$, which is the solution to a dimensional reduction of Einstein-Maxwell-Higgs system in dimension four, in two regimes. In one regime called the "dissolving limit" where the volume of the metrics is approaching the admissible lower bound, it exhibits a pattern that all the vortices are dissolving similar to the Bradlow limit in the study of vortices on Riemann surfaces. In another regime called the "large volume limit" where the volume of of the metrics is approaching infinity, the magnetic field is concentrating around the zeros of the Higgs field. In the meantime, the volume-normalized underlying metric is approaching the Euclidean cone metric determined by the Higgs field in the case of stable Higgs field. Moreover, by studying the large volume limit of Yang's solution for a strictly polystable Higgs field, for each natural number $N'$ we recover the Einstein-Bogomol'nyi metrics on $\mathbf{C}$ which is asymptotically cylindrical at exponential rate and with total string number $N'$ firstly discovered by Linet and Yang.

math.DG↗
arXiv2021

Gravitating vortices with positive curvature

We give a complete solution to the existence problem for gravitating vortices with non-negative topological constant $c \geqslant 0$. Our first main result builds on previous results by Yang and establishes the existence of solutions to the Einstein-Bogomol'nyi equations, corresponding to $c=0$, in all admissible Kähler classes. Our second main result completely solves the existence problem for $c>0$. Both results are proved by the continuity method and require that a GIT stability condition for an effective divisor on the Riemann sphere is satisfied. For the former, the continuity path starts from a given solution with $c = 0$ and deforms the Kähler class. For the latter result we start from the established solution in any fixed admissible Kähler class and deform the coupling constant $α$ towards $0$. A salient feature of our argument is a new bound $S_g \geqslant c$ for the curvature of gravitating vortices, which we apply to construct a limiting solution along the path via Cheeger-Gromov theory.

math.DG↗
arXiv2020

Twisted and Singular gravitating vortices

We introduce the notion of twisted gravitating vortex on a compact Riemann surface. If the genus of the Riemann surface is greater than 1 and the twisting forms have suitable signs, we prove an existence and uniqueness result for suitable range of the coupling constant generalizing the result of arXiv:1510.03810v2 in the non twisted setting. It is proved via solving a continuity path deforming the coupling constant from 0 for which the system decouples as twisted Kähler-Einstein metric and twisted vortices. Moreover, specializing to a family of twisting forms smoothing delta distribution terms, we prove the existence of singular gravitating vortices whose Kähler metric has conical singularities and Hermitian metric has parabolic singularities. In the Bogomol'nyi phase, we establish an existence result for singular Einstein-Bogomol'nyi equations, which represents cosmic strings with singularities.

math.DG↗
arXiv2020

A report on the hypersymplectic flow

This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and 7-dimensional $G_2$-geometry. We also survey recent progress on the flow, most notably an extension theorem assuming a bound on scalar curvature. The second half contains new results. We prove that a complete torsion-free hypersymplectic structure must be hyperkähler. We show that a certain integral bound involving scalar curvature rules out a finite time singularity in the hypersymplectic flow. We show that if the initial hypersymplectic structure is sufficiently close to being point-wise orthogonal then the flow exists for all time. Finally, we prove convergence of the flow under some strong assumptions including, amongst other things, long time existence.

math.DG↗
arXiv2020

Cohomogeneity-one $G_2$-Laplacian flow on 7-torus

We prove the hypersymplectic flow of simple type on standard torus $\mathbb{T}^4$ exists for all time and converges to the standard flat structure modulo diffeomorphisms. This result in particular gives the first example of a cohomogeneity-one $G_2$-Laplacian flow on a compact $7$-manifold which exists for all time and converges to a torsion-free $G_2$ structure modulo diffeomorphisms.

math.DG↗
arXiv2017

Hypersymplectic 4-manifolds, the $G_2$-Laplacian flow and extension assuming bounded scalar curvature

A hypersymplectic structure on a 4-manifold $X$ is a triple $\underlineω$ of symplectic forms which at every point span a maximal positive-definite subspace of $Λ^2$ for the wedge product. This article is motivated by a conjecture of Donaldson: when $X$ is compact $\underlineω$ can be deformed through cohomologous hypersymplectic structures to a hyperkähler triple. We approach this via a link with $G_2$-geometry. A hypersymplectic structure $\underlineω$ on a compact manifold $X$ defines a natural $G_2$-structure $ϕ$ on $X \times \mathbb{T}^3$ which has vanishing torsion precisely when $\underlineω$ is a hyperkähler triple. We study the $G_2$-Laplacian flow starting from $ϕ$, which we interpret as a flow of hypersymplectic structures. Our main result is that the flow extends as long as the scalar curvature of the corresponding $G_2$-structure remains bounded. An application of our result is a lower bound for the maximal existence time of the flow, in terms of weak bounds on the initial data (and with no assumption that scalar curvature is bounded along the flow).

math.DG↗
arXiv2014

The Continuity Method to Deform Cone Angle

The continuity method is used to deform the cone angle of a weak conical Kähler-Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson's Openness Theorem on deforming cone angle \cite{Don} by combining it with the regularity result of Guenancia-P$\breve{\text{a}}$un \cite{GP} and Chen-Wang \cite{CW}. This continuity method uses relatively less regularity of the metric (only weak conical Kähler-Einstein) and bypasses the difficult Banach space set up; it is also generalized to deform the cone angles of a \emph{weak conical Kähler-Einstein metric} along a simple normal crossing divisor (pluri-anticanonical) on a smooth Fano manifold (assuming no tangential holomorphic vector fields).

math.DG↗
arXiv2013

Existence of Weak Conical Kähler-Einstein Metrics Along Smooth Hypersurfaces

The existence of \emph{weak conical Kähler-Einstein} metrics along smooth hypersurfaces with angle between $0$ and $2π$ is obtained by studying a smooth continuity method and a \emph{local Moser's iteration} technique. In the case of negative and zero Ricci curvature, the $C^0$ estimate is unobstructed; while in the case of positive Ricci curvature, the $C^0$ estimate obstructed by the properness of the \emph{twisted K-Energy}. As soon as the $C^0$ estimate is achieved, the local Moser iteration could improve the \emph{rough bound} on the approximations to a \emph{uniform $C^2$ bound}, thus produce a \emph{weak conical Kähler-Einstein} metric. The method used here do not depend on the bound of any background conical Kähler metrics.

math.DG↗