We derive a generalization of the flat space Yang's and Newman's equations for self-dual Yang-Mills fields to (locally) conformally Kahler Riemannian 4-manifolds. The results also apply to Einstein metrics (whose full curvature is not necessarily self-dual). We analyse the possibility of hidden symmetries in the form of Backlund transformations, and we find a continuous group of hidden symmetries only for the case in which the geometry is conformally half-flat. No isometries are assumed.
$F$-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…
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.
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 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 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 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.
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…
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.
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.
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.
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 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 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.
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.
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).
