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Chong Song

Publications and source records attributed to Chong Song.

At least 19 recordsLinked to original sources
arXiv2026

Yang-Mills-Higgs-Schr\"odinger flow

In this paper, we initiate the study of the Yang-Mills-Higgs-Schr\"odinger(YMHS) flow, i.e. the Hamiltonian flow of the Yang-Mills-Higgs functional defined on a symplectic fiber bundle. The YMHS flow provides a natural gauge-theoretic extension of the classical Schr\"odinger flow within the framework of symplectic reduction theory, and generalizes the Chern-Simons-Schr\"odinger equations to a non-Abelian gauge and non-linear fiber setting. We study its geometric structures and establish the local well-posedness of the corresponding Cauchy problem on compact Riemann surfaces.

math.DG↗
arXiv2026

Gradient estimate and Liouville theorem for a semilinear parabolic equation with variable coefficient

In this paper, we investigate the semilinear parabolic equation $\partial_t f-\Delta f=a(x,t)f^p$ on a complete Riemannian manifold with Ricci curvature bounded from below. By means of Nash-Moser iteration, we establish Li-Yau type gradient estimates for positive solutions to this equation, where the coefficient function $a$ can be either strictly sign-definite or sign-changing. As an application, we derive Liouville theorems for ancient and eternal solutions on manifolds with nonnegative Ricci curvature, generalizing a number of classical results. Our proof features the incorporation and tuning of two parameters in the auxiliary quantities to accommodate the variable coefficient $a$ and to extend the admissible range of $p$.

math.AP↗
arXiv2026

Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow

In this paper, we observe that the (spacelike) mean curvature flow of a submanifold in a (pseudo-)Euclidean space is equivalent to a harmonic-Ricci flow with coupling constant $\alpha=-1$ (or $+1$), for the corresponding Gauss map and the induced metric. The solitons of these two flows are also equivalent. As an application, we get a monotonicity formula for the spacelike mean curvature flow.

math.DG↗
arXiv2024

Isolated singularities of 3-dimensional Yang-Mills-Higgs fields

In this paper, we derive decay estimates near isolated singularities of 3-dimensional (3d) Yang-Mills-Higgs fields defined on a fiber bundle, where the fiber space is a compact Riemannian manifold and the structure group is a connected compact Lie group. As an application, we obtain removable singularity theorems for 3d Yang-Mills-Higgs fields under different types of energy conditions, which generalizes classical removable singularity theorems for 3d Yang-Mills fields~\cite{S84,SS84} and 3d harmonic maps~\cite{L85}.

math.DG↗
arXiv2020

Harmonic map flow for almost-holomorphic maps

Let $\Sigma$ be a compact oriented surface and $N$ a compact K\"ahler manifold with nonnegative holomorphic bisectional curvature. For a solution of harmonic map flow starting from an almost-holomorphic map $\Sigma \to N$ (in the energy sense), the limit at each singular time extends continuously over the bubble points and no necks appear.

math.DG↗
arXiv2019

Isolated Singularities of Yang-Mills-Higgs fields on surfaces

We study isolated singularities of two dimensional Yang-Mills-Higgs fields defined on a fiber bundle, where the fiber space is a compact Riemannian manifold and the structure group is a compact connected Lie group. In general the singularity can not be removed due to possibly non-vanishing limit holonomy around the singular points. We establish a sharp asymptotic decay estimate of the Yang-Mills-Higgs field near a singular point, where the decay rate is precisely determined by the limit holonomy. Our result can be viewed as a generalization of the classical removable singularity theorem of two dimensional harmonic maps.

math.DG↗
arXiv2019

Local existence and uniqueness of Skew Mean Curvature Flow

The Skew Mean Curvature Flow(SMCF) is a Schr\"odinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general dimensional SMCF in Euclidean spaces.

math.DG↗
arXiv2019

Harmonic maps with free boundary from degenerating bordered Riemann surfaces

We study the blow-up analysis and qualitative behavior for a sequence of harmonic maps with free boundary from degenerating bordered Riemann surfaces with uniformly bounded energy. With the help of Pohozaev type constants associated to harmonic maps defined on degenerating collars, including vertical boundary collars and horizontal boundary collars, we establish a generalized energy identity.

math.DG↗
arXiv2017

The boundary value problem for Yang--Mills--Higgs fields

We show the existence of Yang--Mills--Higgs (YMH) fields over a Riemann surface with boundary where a free boundary condition is imposed on the section and a Neumann boundary condition on the connection. In technical terms, we study the convergence and blow-up behavior of a sequence of Sacks-Uhlenbeck type $\alpha$-YMH fields as $\alpha\to 1$. For $\alpha>1$, each $\alpha$-YMH field is shown to be smooth up to the boundary under some gauge transformation. This is achieved by showing a regularity theorem for more general coupled systems, which extends the classical results of Ladyzhenskaya-Ural'ceva and Morrey.

math.DG↗
arXiv2017

Gauss map of the skew mean curvature flow

The skew mean curvature flow (SMCF) is a natural generalization of the famous vortex filament equation. In this note, we show that the Gauss map of the SMCF satisfies a Schrödinger flow equation. In this regard, we explore the geometry of the oriented Grassmannian manifold explicitly by embedding it into the exterior product space.

math.DG↗
arXiv2017

Uniqueness of Schrödinger flow on manifolds

In this paper, we show the uniqueness of Schrödinger flow from a general complete Riemannian manifold to a complete Kähler manifold with bounded geometry. While following the ideas of McGahagan[16], we present a more intrinsic proof by using the distance functions and gauge language.

math.DG↗
arXiv2017

Skew Mean Curvature Flow

The skew mean curvature flow(SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space $\mathbb{R}^4$. A Sobolev-type embedding theorem for the second fundamental forms of two dimensional surfaces is also proved, which might be of independent interest.

math.DG↗
arXiv2016

Heat flow of Yang-Mills-Higgs functionals in dimension two

We consider the heat flow of Yang-Mills-Higgs functional where the base manifold is a Riemannian surface and the fiber is a compact symplectic manifold. We show that the corresponding Cauchy problem admits a global weak solution for any $H^1$-initial data. Moreover, the solution is smooth except finitely many singularities. We prove an energy identity at finite time singularities and give a description of the asymptotic behavior at time infinity.

math.AP↗
arXiv2016

Fast generation of three-dimensional entanglement between two spatially separated atoms via invariant-based shortcut

A scheme is proposed for the fast generation of three-dimensional entanglement between two atoms trapped in two cavities connected by a fiber via invariant-based shortcut to adiabatic passage. With the help of quantum Zeno dynamics, the technique of invariant-based shortcut to adiabatic passage is applied for the generation of two-atom three-dimensional entanglement. The numerical simulation results show that, within a short time, the scheme has a high fidelity and is robust against the decoherence caused by the atomic spontaneous emission, photon leakage, and the variations in the parameters selected. Moreover, the scheme may be possible to be implemented with the current experimental technology.

quant-ph↗
arXiv2014

Convergence of Yang-Mills-Higgs fields

In this paper, we study the convergence of Yang-Mills-Higgs fields defined on fiber bundles over Riemann surfaces where the fiber is a compact symplectic manifold and the conformal structure of the Riemann surface is allowed to vary. We show that away from the nodes, the YMH fields converges, up to gauge, to a smooth YMH field modulo finitely many harmonic spheres, while near the nodes where the conformal structure degenerates, the YMH fields converges to a pair consisting of a flat connection and a twisted geodesic (with potential) after finitely many times of blowing-up's. In particular, we prove a generalized energy identity and give a refined analysis of the neck.

math.DG↗
arXiv2012

Geometric solitons of Hamiltonian flows on manifolds

It is well-known that the LIE(Locally Induction Equation) admit soliton-type solutions and same soliton solutions arise from different and apparently irrelevant physical models. By comparing the solitons of LIE and Killing magnetic geodesics, we observe that these solitons are essentially decided by two families of isometries of the domain and the target space respectively. With this insight, we propose the new concept of geometric solitons of Hamiltonian flows on manifolds, such as geometric Schrödinger flows and KdV flows for maps. Moreover, we give several examples of geometric solitons of the Schrödinger flow and geometric KdV flow, including magnetic curves as geometric Schrödinger solitons and explicit geometric KdV solitons on surfaces of revolution.

math.DG↗
arXiv2010

Generalized Landau-Lifshitz Equation into $S^n$

In this paper, a type of integrable evolution equation--the generalized Landau-Lifshitz equation into $S^n$ is considered. We deal with this equation from a geometric point of view by rewriting it in a geometric form. Through the geometric energy method, we show the global well-posedness of the corresponding Cauchy problem.

math.DG↗