Joachim Krieger, Jacob Sterbenz
We prove that smallness of the critical Sobolev norm implies regularity for the Yang-Mills equations on (6+1) and higher dimensional Minkowski space.
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We study the late-time behavior of spherically symmetric solutions of the Yang-Mills equations on Minkowski and Schwarzschild backgrounds. Using nonlinear perturbation theory we show in both cases that solutions having smooth compactly supported initial data posses tails which decay as $t^{-4}$ at timelike infinity. Moreover, for small initial data on Minkowski background we derive the third-order formula for the amplitude of the tail and confirm numerically its accuracy.
The Yang-Mills and Yang-Mills-Higgs equations in temporal gauge are locally well-posed for small and rough initial data, which can be shown using the null structure of the critical bilinear terms. This carries over a similar result by Tao for the Yang-Mills equations in the (3+1)-dimensional case to the more general Yang-Mills-Higgs system and to general dimensions.
The $L_p$ dual Minkowski problem for $p>1$ and $q>0$
本文研究了$p>1$和$q>0$下的$L_p$对偶Minkowski问题的存在性，并讨论了解的正则性。
We prove a removable singularities theorem for point singularities of the coupled yang mills fermion equations, on a bundle over a two dimensional base space.
We consider the hyperbolic Yang-Mills equation on the Minkowski space $\R^{4+1}$. Our main result asserts that this problem is globally well-posed for all initial data whose energy is sufficiently small. This solves a longstanding open problem.
In this paper, we study the local well-posedness of the $(1+3)$-dimensional Yang-Mills-Higgs (YMH) and the Yang-Mills-Dirac (YMD) system in the Lorenz gauge. Since there is some bilinear term in (YMH), which is a lack of null structure, one may obtain the well-posedness at most the energy space. However, we attain the scaling critical regularity of (YMH) by imposing the extra weighted regularity in the angular variables. In (YMD), for the coupled system to persist in time, it is required to impose the angular regularity on the Dirac spinor as much as the Yang-Mills gauge potential and curvatur…
Global exterior stability of the Minkowski space: coupled Einstein-Yang-Mills perturbations
该研究在3+1维闵可夫斯基时空下，利用零比安基与零杨-米尔斯方程的特殊结构及加权分部积分法，证明了耦合杨-米尔斯微扰下闵可夫斯基空间的全局外部稳定性，是首个含非线性源的相关稳定性结果。
The boundary case for the supercritical deformed Hermitian-Yang-Mills equation
本文聚焦边界情形，研究超临界变形 Hermitian-Yang-Mills 方程的弱解问题。
We study the failure of the smoothness of flow maps for the $(1+3)$ dimensional Yang-Mills system in the Lorenz gauge by Knapp type counterexamples. This shows a gap between the scaling critical regularity exponents and the best attainable regularity via Picard's iteration in the Yang-Mills system under the Lorenz gauge condition.
Large data decay of Yang-Mills-Higgs fields on Minkowski and de Sitter spacetimes
本文通过局部化估计和共形变换，将 Yang-Mills-Higgs 方程的 $L^\infty$ 估计推广到 Einstein 柱面，从而把适定性结果从大数据推广到大数据，并进一步得到 de Sitter 空间上的指数衰减和 Minkowski 空间上的逆多项式衰减。
Exterior stability of the $(1+3)$-dimensional Minkowski space-time solution to the Einstein-Yang-Mills equations
本文证明爱因斯坦-杨-米尔斯方程闵可夫斯基时空解的外部稳定性，通过零标架分解在Lorenz和调和规范下建立适定性并得到色散估计，为已有结果提供新证明。
The global stability of the Minkowski space-time solution to the Einstein-Yang-Mills equations in higher dimensions
本文证明高维（n≥5）闵可夫斯基时空在洛伦兹规范下作为爱因斯坦-杨-米尔斯系统解的整体稳定性，并给出色散估计，为后续低维情形研究奠定基础。
Overview of the proof of the exterior stability of the $(1+3)$-Minkowski space-time governed by the Einstein-Yang-Mills system in the Lorenz gauge
本文在Lorenz规范下证明了取值于任意Lie代数的完全耦合Einstein-Yang-Mills系统支配的Minkowski时空外部稳定性，利用零标架分解方法处理不满足零条件的非线性困难，并证明小初值解收敛到零Yang-Mills场和Minkowski时空。
Retrieving Yang--Mills--Higgs fields in Minkowski space from active local measurements
本文利用源到解数据，在 Standard Model 满足的非退化条件下，通过 Lie 代数中心的 abelian 分量与波的非线性相互作用，从 Minkowski 空间中的 Yang–Mills–Higgs 方程反演 Yang–Mills 势和 Higgs 场（精确到规范）。
The Lp centro-sectional Minkowski problem
本文引入Lp中心截面闵可夫斯基问题，解决了p>1且q>0时的存在性，讨论了正则性与唯一性，并证明了p较大时的Lp Brunn-Minkowski型不等式。
The Generalized Gaussian Minkowski Problem
本文研究广义高斯概率空间中的$L_p$ Minkowski问题，利用变分方法给出归一化情形存在性的充要条件，并重点证明$p\geq1$时无需归一化的光滑解存在性。
We prove that the Yang-Mills $α$-functional satisfies the Palais-Smale condition. This guarantees the existence of critical points, which are called Yang-Mills $α$-connections. It was shown by Hong, Tian and Yin in [10] (to appear in Comm. Math. Helv.) that as $α\to 1$, a sequence of Yang-Mills $α$-connections converges to a Yang-Mills connection away from finitely many points. We prove an energy identity for such a sequence of Yang-Mills $α$-connections. As an application, we also prove an energy identity for the Yang-Mills flow at the maximal existence time.
We introduce \emph{normalized exponential Yang-Mills energy functional} $\mathcal{YM}_e^0$, stress-energy tensor $S_{e,\mathcal{YM}^0 }$ associated with the normalized \emph{exponential Yang-Mills energy functional} $\mathcal{YM}_e ^0 $, $e$-conservation law. We also introduce the notion of the {\it $e$-degree} $d_e$ which connects two separate parts in the associated normalize exponential stress-energy tensor $S_{e,\mathcal{YM}^0 }$ (cf. (3.10) and (4.15)), derive monotonicity formula for exponential Yang-Mills fields, and prove a vanishing theorem for exponential Yang-Mills fields. These mon…
In this paper, we study the properties of the critical points of Yang-Mills-Higgs functional, which are called Yang-Mills-Higgs pairs. We first consider the properties of weakly stable Yang-Mills-Higgs pairs on a vector bundle over S^n (n > 3). When n > 3, we prove that the norm of its Higgs field is 1 and the connection is actually Yang-Mills. More precisely, its curvature vanishes when n > 4. We also use the bubble-neck decomposition to prove the energy identity of a sequence of Yang-Mills-Higgs pairs over a 4-dimensional compact manifold with uniformly bounded energy. We show there is a sub…
We prove that monotonicity of density and energy inequality imply the rectifiability of the singular sets for Yang-Mills flow.
We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.
We proved a uniqueness theorem of tangent connections for a Yang-Mills connection with an isolated singularity with a quadratic growth of the curvature at the singularity. We also obtained controls over the rate of the asymptotic convergence of the connection to the tangent connection under assumptions that the connection is stationary or the tangent connection is integrable. There are parallel results for the cones at infinity of a Yang-Mills connection on an asymptotically flat manifold. We also gave an application of our methods to the Yang-Mills flow and proved that the Yang-Mills flow exi…
Energy estimates for the Einstein-Yang-Mills fields and applications
本文为张量非线性波动方程建立外部能量估计，并应用于证明闵可夫斯基时空在爱因斯坦-杨-米尔斯系统下的外部稳定性，为后续$(1+3)$维情形奠定基础。
In this paper, we introduce an α-flow for the Yang-Mills functional in vector bundles over four dimensional Riemannian manifolds, and establish global existence of a unique smooth solution to the α-flow with smooth initial value. We prove that the limit of solutions of the α-flow as α\to 1 is a weak solution to the Yang-Mills flow. By an application of the α-flow, we then follow the idea of Sacks and Uhlenbeck to prove some existence results for Yang-Mills connections and improve the minimizing result of the Yang-Mills functional of Sedlacek.
We show under natural assumptions that stable solutions to the abelian Yang--Mills--Higgs equations on Hermitian line bundles over the round $2$-sphere actually satisfy the vortex equations, which are a first-order reduction of the (second-order) abelian Yang--Mills--Higgs equations. We also obtain a similar result for stable solutions on a flat $2$-torus. Our method of proof comes from the work of Bourguignon--Lawson concerning stable $SU(2)$ Yang--Mills connections on compact homogeneous $4$-manifolds.
In this work, we introduce a novel approach to the problem of gauge choice for the Yang-Mills equation on the Minkowski space $\mathbb{R}^{1+3}$, which uses the Yang-Mills heat flow in a crucial way. As this approach does not possess the drawbacks of the previous approaches, it is expected to be more robust and easily adaptable to other settings. As the first demonstration of the `structure' offered by this new approach, we will give an alternative proof of the local well-posedness of the Yang-Mills equations for initial data in $(\dot{H}^{1}_{x} \cap L^{3}_{x}) \times L^{2}_{x}$, which is a c…
A local monotonicity formula for the Yang-Mills-Higgs flow on $G$-bundles over $\mathbb{R}^{n}$ ($n>4$) is proved. It is shown that the monotone quantity coïncides on certain self-similar solutions with that appearing in existing non-local monotonicity formulæ for the Yang-Mills and Yang-Mills-Higgs flows.
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