Simon Brendle
We describe a glueing construction for the Yang-Mills equations in dimension $n > 4$. Our method is based on a construction of approximate solutions, and a detailed analysis of the linearized operator near an approximate solution.
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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…
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 paper, we consider the deformed Hermitian-Yang-Mills equation on closed almost Hermitian manifolds. In the case of hypercritical phase, we derive a priori estimates under the existence of an admissible $\mathcal{C}$-subsolution. As an application, we prove the existence of solutions for the deformed Hermitian-Yang-Mills equation under the condition of existence of a supersolution.
Decay estimates for Yang-Mills connections and Yang-Mills-Higgs pairs in higher dimensions
本文通过抽象衰减定理和Kato不等式，证明了高维流形上Yang-Mills联络与Yang-Mills-Higgs对在二次衰减假设下的改进衰减速率。
Let $P$ be a principal U(1)-bundle over a closed manifold $M$. On $P$, one can define a modified version of the Ricci flow called the Ricci Yang-Mills flow, due to these equations being a coupling of Ricci flow and the Yang-Mills heat flow. We use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to study the stability of the volume-normalized Ricci Yang-Mills flow at Einstein Yang-Mills metrics in dimension two. In certain cases, we show the presence of a center manifold of fixed points, while in…
$F$-Yang-Mills connections are critical points of $F$-Yang Mills functional on the space of connections of a principal fiber bundle, which is a generalization of Yang-Mills connections, $p$-Yang-Mills connections and exponential Yang-Mills connections and so on. Here, $F$ is a strictly increasing $C^{2}$-function. In this paper, we extend Simons theorem for an instability of Yang-Mills connections to $F$-Yang-Mills connections. We derive a sufficient condition that any non-flat, $F$-Yang-Mills connection over convex hypersurfaces in a Euclidean space is instable. In the sphere case, this condi…
Normal-Yang-Mills and Tangent-Yang-Mills submanifolds
本文研究子流形浸入单位球体时正法和切法曲率张量的L²范数变分问题，定义了正法-杨-米尔斯和切法-杨-米尔斯子流形，并推导了相应的欧拉-拉格朗日方程，构造了非平凡例子。
We use the Yang-Mills gradient flow on the space of connections over a closed Riemann surface to construct a Morse-Bott chain complex. The chain groups are generated by Yang-Mills connections. The boundary operator is defined by counting the elements of appropriately defined moduli spaces of Yang-Mills gradient flow lines that converge asymptotically to Yang-Mills connections.
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.
In this paper, we study the hypercritical deformed Hermitian-Yang-Mills equation on compact Kähler manifolds and resolve two conjectures of Collins-Yau.
We construct Yang-Mills connections on SO(n)-bundles over spheres equipped with the Euclidean metric. We use a cohomogeneity one group action on the bundle to reduce the Yang-Mills-equation to a system of ordinary differential equations. The system is shown to have solutions by variational methods, using ideas from harmonic map theory. Examples include Yang-Mills connections on each of the countably many principal SO(6)-bundles over $S^6$, and countably many Yang-Mills connections on $TS^n$ for $n\in\{5,...,9\}$.
In the spirit of recent work of Lamm, Malchiodi and Micallef in the setting of harmonic maps, we identify Yang-Mills connections obtained by approximations with respect to the Yang-Mills α-energy. More specifically, we show that for the SU(2) Hopf fibration over the four sphere, for sufficiently small α values the SO(4) invariant ADHM instanton is the unique α-critical point which has Yang-Mills α-energy lower than a specific threshold.
We construct local examples of singular Hermitian Yang-Mills connections over $B_1\subset \mathbb{C}^3$ with uniformly bounded $L^2$-energy, but the number of essential singular points can be arbitrarily large.
In this note we introduce a Yang-Mills bar equation on complex vector bundles over compact Hermitian manifolds as the Euler-Lagrange equation for a Yang-Mills bar functional. We show the existence of a non-trivial solution of this equation over compact Kähler manifolds as well as a short time existence of the negative Yang-Mills bar gradient flow. We also show a rigidity of holomorphic connections among a class of Yang-Mills bar connections over compact Kähler manifolds of positive Ricci curvature.
In this work, we study the deformed Hermitian-Yang-Mills equation on compact Kähler manifold. We introduce the notions of coerciveness and properness of the $\mathcal{J}$-functional on the space of almost calibrated $(1,1)$-forms and show that they are both equivalent to the existence of solutions to the hypercritical deformed Hermitian-Yang-Mills equation.
On a Riemannian manifold of dimension $n$ we extend the known analytic results on Yang-Mills connections to the class of connections called $Ω$-Yang-Mills connections, where $Ω$ is a smooth, not necessarily closed, $(n-4)$-form. Special cases include $Ω$-anti-self-dual connections and Hermitian-Yang-Mills connections over general complex manifolds. By a key observation, a weak compactness result is obtained for moduli space of smooth $Ω$-Yang-Mills connections with uniformly $L^2$ bounded curvature, and it can be improved in the case of Hermitian-Yang-Mills connections over general complex man…
We establish various existence and uniqueness results for the Yang-Mills flow on cylindrical end 4-manifolds. We also show long-time existence and infinite-time convergence under certain hypotheses on the underlying data.
Removable singularities of Yang-Mills-Higgs fields in higher dimensions
本文研究高维Yang-Mills-Higgs场在孤立奇点附近的衰减估计，并在conformally不变能量界下证明了其可去奇点定理。
In this paper, we investigate the Chern number inequalities on $4$-dimensional Kähler manifolds admitting the deformed Hermitian-Yang-Mills metrics under the assumption $\hatθ\in (π,2π)$.
We study a functional that derives from the classical Yang-Mills functional and Born-Infeld theory. We establish its first variation formula and prove the existence of critical points. We also obtain the second variation formula.
In this paper, we analyze the asymptotic behaviour of the Hermitian-Yang-Mills flow over a compact non-Kähler manifold $(X,g)$ with the Hermitian metric $g$ satisfying the Gauduchon and Astheno-Kähler condition.
Kähler-Yang-Mills Equations and Vortices
本文介绍Kähler-Yang-Mills方程的几何背景，重点研究与涡旋相关的该方程的维数约化问题。
In 1982, Uhlenbeck \cite {U2} established the well-known gauge fixing theorem, which has played a fundamental role for Yang-Mills theory. In this paper, we apply the idea of Uhlenbeck to establish a parabolic type of gauge fixing theorems for the Yang-Mills flow and prove existence of a weak solution of the Yang-Mills flow on a compact $n$-dimensional manifold with initial value $A_0$ in $W^{1,n/2}(M)$. When $n=4$, we improve a key lemma of Uhlenbeck (Lemma 2.7 of \cite {U2}) to prove uniqueness of weak solutions of the Yang-Mills flow on a four dimensional manifold.
Nonclosure of stationary Yang-Mills connections under weak convergence
本文构造了一列具有孤立奇点的平稳弱Yang-Mills联络，其弱极限非平稳，从而证明了平稳弱Yang-Mills联络在弱收敛下不封闭，并给出了调和映射中类似现象的Yang-Mills类比。
This is an expository article on the 1978 Atiyah-Drinfeld-Hitchin-Manin construction of Yang-Mills instanton connections over the 4-sphere.
Here we provide refinements of the stability results of Simons and Xin, concerning the stability of Yang-Mills fields and harmonic maps respectively. The result also implies the earlier Morse index estimates for both cases.
We use the energy gap result of pure Yang-Mills equation [Feehan P.M.N., Adv. Math. 312 (2017), 547-587, arXiv:1502.00668] to prove another energy gap result of complex Yang-Mills equations [Gagliardo M., Uhlenbeck K., J. Fixed Point Theory Appl. 11 (2012), 185-198, arXiv:1401.7366], when Riemannian manifold $X$ of dimension $n\geq 2$ satisfies certain conditions.
In this paper, we introduce some notions on the pair consisting of a Chern connection and a Higgs field closely related to the first and second variation of Yang-Mills- Higgs functional, such as strong Yang-Mills-Higgs pair, degenerate Yang-Mills-Higgs pair, stable Yang-Mills-Higgs pair. We investigate some properties of such pairs.
We study pseudo Yang-Mills fields on a compact strictly pseudoconvex CR manifold.
Einstein and Yang-Mills implies conformal Yang-Mills
该研究证明了在偶维共形流形上，轨连接的共形杨-米尔斯条件等价于其费弗曼-格拉姆障碍张量的消失，并展示了共形杨-米尔斯条件是爱因斯坦度量下杨-米尔斯条件的弱化。
We prove the first mathematical result relating the Yang-Mills measure on a compact surface and the Yang-Mills energy. We show that, at the small volume limit, the Yang-Mills measures satisfy a large deviation principle with a rate function which is expressed in a simple and natural way in terms of the Yang-Mills energy.
We present an elementary self-contained account of semisimple Frobenius manifolds in three dimensions, and exhibit a new family of explicit examples. These examples are constructed from Yang-Mills instantons with a certain symmetry.
In this paper we give a complete algebro-geometric characterization of analytic tangent cones of admissible Hermitian-Yang-Mills connections over any reflexive sheaves.
In this paper, we study the Yang-Mills functional on quantum Heisenberg manifolds using the appratuses developed by A. Connes and M. Rieffel. It is discovered that a connection on a projective module over a quantum Heisenberg manifold is a minimum of Yang-Mills functional whicih is a critical point that is different with critical points found by S. Kang.
By exploiting standard facts about $N=1$ and $N=2$ supersymmetric Yang-Mills theory, the Donaldson invariants of four-manifolds that admit a Kahler metric can be computed. The results are in agreement with available mathematical computations, and provide a powerful check on the standard claims about supersymmetric Yang-Mills theory.
This paper has been withdrawn by the author.
In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of $\mathcal{F}$-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabilities have connections with the study of Type-I singularities of the Yang-Mills flow. Two byproducts are also included: We show that the Yang-Mills flow in dimension four cannot develop a Type-I singularity; and we obtain a gap theorem for homothetically shrinking solitons.
In this note, we prove an ${L^{\frac{n}{2}}}$-energy gap result for Yang-Mills connections on a principal $G$-bundle over a compact manifold without using Lojasiewicz-Simon gradient inequality (arXiv:1502.00668).
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