arXiv research

The paper examines Yang-Mills-Higgs pairs on vector bundles and proves stability and energy identity.

New energy functional and fields for Yang-Mills theory, proving monotonicity and vanishing theorems.

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 converge…

We generalize our previous results (Theorem 1 and Corollary 2 in arXiv:1412.4114) and Theorem 1 in arXiv:1502.00668) on the existence of an $L^2$-energy gap for Yang-Mills connections over closed four-dimensional manifolds and energies near the ground state (occupied by flat, anti-self-dual, or self-dual connections) t…

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 energ…

Study bounds the index of minimal submanifolds using energy measures and Yang-Mills-Higgs equations.

Proves energy expression on Poincaré-Einstein spaces.

Proves existence of Yang-Mills fields for specific curvature conditions.

Removes singularities for Yang-Mills-Higgs fields in higher dimensions.

In this sequel to arXiv:1510.03817, we apply our abstract Lojasiewicz-Simon gradient inequality to prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Lojasiewicz-Simon gradient …

The paper proves parabolic gap theorems for Yang-Mills energy.

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 dim…

Study on Yang-Mills equation near instanton-anti-instanton configurations with energy constraints.

Formula found for energy slope in complex geometry.

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…

We study the converse to the statement that instantons are minimizers of the Yang--Mills energy in four dimensions. We show that given an energy minimizing connection, A, the curvature of A takes values in a subbundle of the adjoint bundle which decomposes as a sum of instantons.

In this monograph, we develop results on global existence and convergence of solutions to abstract gradient flows on Banach spaces for a potential function that obeys the Lojasiewicz-Simon gradient inequality. We prove a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over closed, smooth Riem…

Study on Yang-Mills equations on conformally compact manifolds, finding obstructions and asymptotics.

In this paper we prove that over an asymptotically locally flat (ALF) Riemannian four-manifold the energy of an "admissible" SU(2) Yang--Mills is always integer. This result sharpens the previously known energy identity for such Yang--Mills instantons over ALF geometries. Furthermore we demonstrate that this statement …

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).

We prove that the pullback of the SU(n)-soliton of Chern class $c_2=1$ over $\mathbb S^4$ via the radial projection $π:\mathbb B^5\setminus\{0\}\to\mathbb S^4$ minimizes the Yang-Mills energy under the fixed boundary trace constraint. In particular this shows that stationary Yang-Mills connections in high dimension can…

Consider a Yang-Mills connection over a Riemann manifold $M=M^n$, $n\ge 3$, where $M$ may be compact or complete. Then its energy must be bounded from below by some positive constant, if $M$ satisfies certain conditions, unless the connection is flat.

Let $F: [0, \infty) \to [0, \infty)$ be a strictly increasing $C^2$ function with $F(0)=0$. We unify the concepts of $F$-harmonic maps, minimal hypersurfaces, maximal spacelike hypersurfaces, and Yang-Mills Fields, and introduce $F$-Yang-Mills fields, $F$-degree, $F$-lower degree, and generalized Yang-Mills-Born-Infeld…

Sengupta's lower bound for the Yang-Mills action on smooth connections on a bundle over a Riemann surface generalizes to the space of connections whose action is finite. In this larger space the inequality can always be saturated. The Yang-Mills critical sets correspond to critical sets of the energy action on a space …

Finite-time blow-up in Yang-Mills flow for small energy initial connections.

Study on Yang-Mills fields blow-up in 4D, proving certain configurations impossible.

We prove that monotonicity of density and energy inequality imply the rectifiability of the singular sets for Yang-Mills flow.

We consider a vector bundle $E$ over a compact Riemannian manifold $M$=$M^{n}$,$n\geq 4$,and $A$ is a Yang-Mills connection with $L^{\frac{n}{2}}$ curvature $F_{A}$ on $E$.Then we prove a mean value inequality for the density $|F_{A}|^{\frac{n}{2}}$.This inequality give rise to an energy concentrate principle for seque…

A result (Corollary 4.3) in an article by Uhlenbeck (1985) asserts that the $W^{1,p}$-distance between the gauge-equivalence class of a connection $A$ and the moduli subspace of flat connections $M(P)$ on a principal $G$-bundle $P$ over a closed Riemannian manifold $X$ of dimension $d\geq 2$ is bounded by a constant ti…

We prove that energy minimizing Yang-Mills connections on a compact $G_{2}$-manifold has holonomy equal to $G_{2}$ are $G_{2}$-instantons, subject to an extra condition on the curvature. Furthermore, we show that energy minimizing connections on a compact Calabi-Yau $3$-fold has holonomy equal to $SU(3)$ subject to a s…

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.

Study proves uniqueness of Yang-Mills field tangent cones in arbitrary dimensions.

We develop a new method for proving regularity for small energy stationary solutions of coupled gauge field equations. Our results duplicate those of Tian--Tao [7] for the pure Yang Mills equations, but our proof is simpler, and obtains bounded curvature without the use of Coulomb gauges. It relies instead on the Weitz…

Direct method finds Yang-Mills connections for SO(3) bundles.

We prove a conformally invariant estimate for the index of Schrödinger operators acting on vector bundles over four-manifolds, related to the classical Cwikel-Lieb-Rozenblum estimate. Applied to Yang-Mills connections we obtain a bound for the index in terms of its energy which is conformally invariant, and captures th…

Study on finite energy SU(2) monopoles on AC 3-manifolds, proving integrality of charge and curvature decay.

We establish that finite-time singularities do not occur in four-dimensional Yang-Mills flow, confirming the conjecture of Schlatter, Struwe, and Tahvildar-Zadeh. The proof relies on a weighted energy identity and sharp decay estimates in the neck region.

The paper studies decay near singularities of 3d Yang-Mills-Higgs fields.

We prove an $L^{2}$ energy gap result for Yang-Mills connections on principal $G$-bundles over compact Kähler surfaces with positive scalar curvature. We prove related results for compact simply-connected Calabi-Yau $2$-folds.

Study investigates singularity formation in $\alpha$-Yang-Mills-Higgs fields on spheres.

We consider the minimum Yang-Mills energy on the complete $G_{2}$-manifolds and Calabi-Yau 3-folds,the connection $A$ is a stability Yang-Mills connection on the $G$-bundle $E$.We prove that the connection must be a $G_{2}$-instanton on $G_{2}$-manifold and the bundle is holomorphic on Calabi-Yau 3-fold with holonomy $…

For any compact Lie group $G$ and closed, smooth Riemannian manifold $(X,g)$ of dimension $d\geq 2$, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal $G$-bundle over $X$ supporting a connection with $L^p$-small curvature, when $p>d/2$, to the case of a connection with …

The paper defines and analyzes higher-order Yang-Mills-Higgs functionals and their gradient flows.

In this sequel to [arXiv:1412.4114], we prove an $L^{d/2}$ energy gap result for Yang-Mills connections on principal $G$-bundles, $P$, over arbitrary, closed, Riemannian, smooth manifolds of dimension $d\geq 2$. We apply our version of the Lojasiewicz-Simon gradient inequality [arXiv:1409.1525, arXiv:1510.03815] to rem…

Solves non-Abelian Rainich problem for SU(2) gauge fields.

Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.

We investigate monotonicity properties of $p$-harmonic vector bundle-valued $k$-forms by studying the energy-momentum tensor associated with such a form. As a consequence, we obtain a unified proof of the monotonicity formulæ for $p$-harmonic maps and Yang-Mills connections, proving a monotonicity formula for $p$-Yang-…

We extend an $L^2$ energy gap result due independently to Min-Oo and Parker (1982) for Yang-Mills connections on principal $G$-bundles, $P$, over closed, connected, four-dimensional, oriented, smooth manifolds, $X$, from the case of positive Riemannian metrics to the more general case of good Riemannian metrics, includ…