The paper studies stability of F-Yang-Mills connections on complex projective spaces.
problem Stability of F-Yang-Mills connections on complex projective spaces.
method Inspired by Lawson-Simons, the paper proves stability conditions and structures for F-Yang-Mills connections.
result Conditions for weakly stable F-Yang-Mills connections on complex projective spaces.
The study refines stability results for Yang-Mills fields and harmonic maps.
problem Stability of Yang-Mills fields and harmonic maps.
method Refinement of stability results using Jacobi operator over S^m.
result Refined stability results and Morse index estimates.
In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of F-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabil…
Stability of Yang-Mills connections' Morse indices and nullity in 4D.
problem Stability of Yang-Mills connections' Morse indices and nullity in 4D under weak convergence.
method Proves stability results of the Morse index plus nullity of Yang-Mills connections in dimension 4 under weak convergence.
result Stability of the sum of Morse indices and nullity of a sequence of Yang-Mills connections.
Surveying progress on deformed Hermitian-Yang-Mills equation and its connections to stability.
problem Solvability of the deformed Hermitian-Yang-Mills equation and its relation to geometric stability.
method Utilizing geometric invariant theory (GIT) and Bridgeland stability theory to analyze the equation.
result On the blow-up of \(\mathbb{P}^2\), line bundles admitting a solution of the deformed Hermitian-Yang-Mills equation are Bridgeland stable, but not conversely.
Stability of Minkowski space-time in higher dimensions proven for arbitrary small perturbations.
problem Stability of Minkowski space-time solution to Einstein-Yang-Mills equations in higher dimensions.
method Global stability proof for arbitrary small perturbations using wave coordinates and gauge invariant norms.
result Global stability of Minkowski space-time in higher dimensions n≥5 for arbitrary small perturbations. The paper examines Yang-Mills-Higgs pairs on vector bundles and proves stability and energy identity.
problem Stability and energy identity of Yang-Mills-Higgs pairs on vector bundles.
method Bubble-neck decomposition and analysis of weakly stable pairs.
result A sequence of Yang-Mills-Higgs pairs converges to a Yang-Mills-Higgs pair with uniformly bounded energy.
Flat Yang-Mills connections on pinched manifolds.
problem Stability of Yang-Mills connections on compact manifolds.
method Pinching conditions and weak stability criteria.
result No non-flat weakly stable Yang-Mills connections on δ(n)-pinched compact simply-connected Riemannian manifolds.
Paper proves no stable Yang-Mills fields on spheres.
problem Existence of stable Yang-Mills fields on spheres.
method Analyzes C2 neighborhoods of Euclidean sphere metrics and warped product manifolds. result No nontrivial weakly stable Yang-Mills connections in specified neighborhoods.
New criterion found for Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.
problem Existence of Hermitian-Yang-Mills metrics on non-compact Kähler manifolds.
method Algebraic criterion and stability condition introduced.
result New stability condition is both sufficient and necessary for the existence of Hermitian-Yang-Mills metrics.
Stability of Minkowski space-time in Einstein-Yang-Mills system proven.
problem Stability of Minkowski space-time in Einstein-Yang-Mills system.
method Null frame decomposition, wave coordinates, dispersive estimates.
result Solutions converge to zero Yang-Mills curvature and Minkowski space-time.
Stability of singularity formation in Yang-Mills fields in higher dimensions.
problem Stability of self-similar blowup profiles for Yang-Mills equations in (1+d)-dimensions. method Analysis of explicitly known equivariant self-similar blowup solution and small equivariant perturbations.
result Global-in-space asymptotic stability of the self-similar blowup solution for Yang-Mills equations in (1+d)-dimensions for d≥5. Proves stability of Minkowski space-time in Einstein-Yang-Mills system.
problem Stability of Minkowski space-time governed by Einstein-Yang-Mills system.
method Null frame decomposition, well-posedness of Cauchy development, convergence to Minkowski space-time.
result Exterior stability of Minkowski space-time in Lorenz gauge without spherical symmetry.
Stability of Morse index for Yang-Mills connections in 4D.
problem Stability of critical points in Yang-Mills energy relaxation.
method Establishing lower semi-continuity of Morse index and upper continuity of Morse index plus nullity.
result Yang-Mills fields are more stable than harmonic maps in 4D.
Study higher rank deformed Hermitian-Yang-Mills equations for stable vector bundles.
problem Stability conditions for higher rank vector bundles over complex manifolds.
method Establish equivalence between dHYM equations and Z-stability. result Equivalence between dHYM solutions and Z-stability for vortex type bundles. 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 asymptoti…
Study on HYM connections on Kähler manifolds, calculating moduli space virtual dimension.
problem Calculating the moduli space of Hermitian-Yang-Mills connections.
method Analytic proof of stability of Higgs bundles on compact Kähler manifolds.
result Virtual dimension of the moduli space of HYM connections calculated.
Study deformed Hermitian-Yang-Mills equation on complex projective space blowup.
problem Solving the deformed Hermitian-Yang-Mills equation on complex projective space blowup.
method Expressed the equation as an ODE and solved it using combinatorial methods under an algebraic stability condition.
result Evidence supporting a conjecture on general compact Kahler manifolds.
Proves local existence and extension principle for Einstein Yang--Mills system with spherical symmetry.
problem Local existence and stability of the spherically symmetric Einstein Yang--Mills system.
method Employed an L2-based method to prove local existence and establish an extension principle. result Established local existence and extension principle for the SSEYM with H1 data. New examples of deformed Hermitian-Yang-Mills connections found.
problem Constructing deformed Hermitian-Yang-Mills connections on manifolds.
method Constructed first higher rank, irreducible deformed Hermitian-Yang-Mills connections in both small and large radius regimes.
result Existence of solutions with any possible angle and ruling out some stability conditions.
We introduce Z-critical connections for holomorphic vector bundles and prove their existence under stability conditions.
problem Existence of Z-critical connections for holomorphic vector bundles. method Associated geometric PDEs to Bridgeland stability conditions and used infinite dimensional moment maps.
result In the large volume limit, a sufficiently smooth holomorphic vector bundle admits a Z-critical connection if and only if it is asymptotically Z-stable. Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.
problem Stability of Minkowski space-time for perturbations governed by Einstein-Yang-Mills equations.
method Proves exterior energy estimates for tensorial non-linear wave equations in Minkowski space-time.
result Proves exterior stability of Minkowski space-time for Einstein-Yang-Mills equations.
Following work of Colding-Minicozzi, we define a notion of entropy for connections over Rn which has shrinking Yang-Mills solitons as critical points. As in Colding-Minicozzi, this entropy is defined implicitly, making it difficult to work with analytically. We prove a theorem characterizing entropy stabilit…
The note proves a metric equivalence for stable bundles on surfaces.
problem Understanding stability conditions and metrics on complex projective surfaces.
method Analyzing stability in the large scaling limit and proving equivalence with deformed Hermitian-Yang-Mills metrics.
result Equivalence of stability and deformed Hermitian-Yang-Mills metrics for smooth projective surfaces.
Paper confirms conjecture for projective manifolds in supercritical phase.
problem Stability condition for deformed Hermitian-Yang-Mills equation.
method Establishes stability result not involving uniform constants.
result Confirms conjecture for projective manifolds in supercritical phase.
Stable solutions found for a specific physics model.
problem Stability of solutions to the U(1)-Yang-Mills-Higgs model. method Gluing method and detailed analysis of linearized operators.
result Found a family of stable critical points in higher dimensions.
We consider the minimum Yang-Mills energy on the complete G2-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 G2-instanton on G2-manifold and the bundle is holomorphic on Calabi-Yau 3-fold with holonomy $…
Develops harmonic metrics for Hull-Strominger system stability.
problem Existence of solutions to the Hull-Strominger system with balanced class.
method Uses non-Hermitian Yang-Mills connections and holomorphic Courant algebroids, introduces harmonic metrics.
result Expected existence of a numerical stability condition for generic families of solutions.
Proves stability of certain vector bundles on Kähler surfaces.
problem Stability of rank 2 holomorphic vector bundles on Kähler surfaces.
method Proves existence of Z-positive and Z-critical metrics leading to bundle stability. result Proves stability results for deformed Hermitian Yang-Mills and almost Hermite-Einstein equations for rank 2 bundles.
The paper studies HYM connections on stable vector bundles over Kähler manifolds.
problem Analyzing stability and convergence of Hermitian Yang-Mills connections.
method Semialgebraic decomposition of the Kähler cone into stability chambers.
result HYM connections converge to a stable HYM connection as polarisation converges.
We study Yang-Mills connections on holomorphic bundles over complex Kähler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isom…
Investigates admissible metrics on compact Kähler varieties and their stability.
problem Existence of admissible metrics on compact Kähler varieties and their stability.
method Analyzes admissible Hermitian metrics and Hermitian-Yang-Mills metrics on slope stable coherent sheaves.
result Existence of admissible metrics and Hermitian-Yang-Mills metrics under certain conditions.
Developed a new symmetric hyperbolic formulation for Einstein-Yang-Mills system.
problem Future stability of solutions of the Einstein-Yang-Mills system with arbitrary dimension.
method Tensorial symmetric hyperbolic formulation and local well-posedness for Cauchy problem.
result Established local well-posedness for the Cauchy problem of EYM equations in the temporal gauge.
Constructs a moduli space for PDEs, linking stability to geometric metrics.
problem Moduli space construction for involutive ideal sheaves from PDEs.
method Introduces D-Hilbert and D-Quot functors, defines Spencer stability. result Spencer poly-stability of PDE ideal implies Hermitian-Yang-Mills metric existence.
Stable blowup solutions found for supercritical Yang-Mills equations.
problem Understanding blowup solutions for supercritical Yang-Mills equations.
method Investigated equivariant self-similar blowup solutions and their stability.
result Stability of blowup mechanism for odd dimensions greater than or equal to 5.
The paper proves estimates for vortex-type equations on compact Riemann surfaces.
problem Estimating vortex-type equations on compact Riemann surfaces.
method Proves \emph{a priori} estimates for vortex-type equations.
result Recover existing estimates for vortex bundle Monge-Ampère equation, prove existence and uniqueness for Calabi-Yang-Mills equations, and get estimates for J−vortex equation. We introduce the notion of T-stability for torsion-free Higgs sheaves as a natural generalization of the notion of T-stability for torsion-free coherent sheaves over compact complex manifolds. We prove similar properties to the classical ones for Higgs sheaves. In particular, we show that only saturated flags of to…
Study shows stability of tangent bundle through conifold transitions.
problem Stability of tangent bundle through conifold transitions.
method Hermitian-Yang-Mills metric and conformally balanced metrics.
result Tangent bundle T1,0Xt admits a Hermitian-Yang-Mills metric Ht near vanishing cycles of Xt. Investigates J-equation on holomorphic vector bundles over Kähler manifolds.
problem Analyzes properties and solutions of J-equation on holomorphic vector bundles. method Introduces and studies J-equation, provides algebraic and numerical criteria. result Provides an algebraic condition (asymptotic J-stability) and a numerical criterion for vortex bundles. We study the heat flow for Yang-Mills connections on Rd×SO(d). It is well-known that in dimensions 5≤d≤9 this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an explicit example given by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stab…
Let (X,ω) be a compact Kähler manifold of complex dimension n and (L,h) be a holomorphic line bundle over X. The line bundle mean curvature flow was introduced in \cite{JY} in order to find deformed Hermitian-Yang-Mills metrics on L. In this paper, we consider the stability of the line bundle mean curvature f…
We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of the equation, and some recent progress towards its solution. In dimension 3 we …
Introduces relative stability conditions on triangulated categories.
problem Stability conditions in triangulated categories.
method Definition and deformation of relative stability conditions.
result Deformation of relative stability conditions via gluing stability conditions.
Study special Lagrangian sections in Calabi-Yau threefolds, showing stability conditions imply isomorphism to special Lagrangians.
problem Understanding stability conditions on Fukaya-Seidel categories of Calabi-Yau threefolds.
method Analyzing sections of special Lagrangian fibrations, constructing Bridgeland stability conditions, and relating to deformed Hermitian Yang-Mills connections.
result Semistability of L[2] implies isomorphism to special Lagrangian sections. The paper establishes lower bounds on Yang-Mills functionals for fibrations.
problem Analyzing the stability and nefness of direct image sheaves in fibrations.
method Generalizing mean curvature and Harder-Narasimhan filtrations to arbitrary polarized fibrations.
result Optimal lower bounds on fibered Yang-Mills functionals in terms of direct image sheaves.
In this paper, we consider the heat flow for Yang-Mills connections on R5×SO(5). In the SO(5)−equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove …
We study equations on a principal bundle over a compact complex manifold coupling a connection on the bundle with a Kahler structure on the base. These equations generalize the conditions of constant scalar curvature for a Kahler metric and Hermite-Yang-Mills for a connection. We provide a moment map interpretation of …
Yang-Mills theory is growing at the interface between high energy physics and mathematics. It is well known that Yang-Mills theory and Gauge theory in general had a profound impact on the development of modern differential and algebraic geometry. One could quote Donaldson invariants in four dimensional differential top…