the yang mills flow and the atiyah bott formula on compact kahler manifolds we study the yang mills flow on a holomorphic vector bundle e over a compact k ahler manifold x along a solution of the flow we show that the curvature endomorphism i lambda f a t approaches in l 2 an endomorphism with constant eigenvalues given by the slopes of the quotients from the harder narasimhan filtration of e this proves a sharp lower bound for the hermitian yang mills functional and thus the yang mills functional generalizing to arbitrary dimension a formula of atiyah and bott first proven on riemann surfaces furthermore we show any reflexive extension to all of x of the limiting bundle e infty is isomorphic to rm gr rm hns e verifying a conjecture of bando and siu

a vector bundle version of the monge ampere equation abstract we introduce a vector bundle version of the complex monge ampere equation motivated by a desire to study stability conditions involving higher chern forms we then restrict ourselves to complex surfaces provide a moment map interpretation of it and define a positivity condition ma positivity which is necessary for the infinite dimensional symplectic form to be kahler on rank 2 bundles on compact complex surfaces we prove two consequences of the existence of a positively curved solution to this equation stability involving the second chern character and a kobayashi lubke bogomolov miyaoka yau type inequality finally we prove a kobayashi hitchin correspondence for a dimensional reduction of the aforementioned equation

compactification of the moduli spaces of vortices and coupled vortices vortices and coupled vortices arise from yang mills higgs theories and can be viewed as generalizations or analogues to yang mills connections and in particular hermitianyang mills connections we proved an analytic compactification of the moduli spaces

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

analytic cycles bott chern forms and singular sets for the yang mills flow on kahler manifolds abstract it is shown that the singular set for the yang mills flow on unstable holomorphic vector bundles over compact kahler manifolds is completely determined by the harder narasimhan seshadri filtration of the initial holomorphic bundle we assign a multiplicity to irreducible top dimensional components of the singular set of a holomorphic bundle with a filtration by saturated subsheaves we derive a singular bott chern formula relating the second chern form of a smooth metric on the bundle to the chern current of an admissible metric on the associated graded sheaf this is used to show that the multiplicities of the top dimensional bubbling locus defined via the yang mills density agree with the corresponding multiplicities for the harder narasimhan seshadri filtration the set theoretic equality of singular sets is a consequence

bott chern currents and complex immersions in the whole paper we will say that i m m is an immersion of smooth manifolds if m is a submanifold of m and if i is the corresponding injection map in particular the topology of m is the topology induced by the topology of m in differential geometry such maps i m m are also called embeddings

a singularity removal theorem for yang mills fields in higher dimensions the purpose of this paper is to investigate the small energy behavior of weakly yang mills fields in rn for n 4 and in particular to extend the singularity removal theorem of uhlenbeck 7 to higher dimensions fix n 4 and let ft be some bounded domain in rn typically we shall restrict our attention to the cubes ft 1 2 n or ft 0 l n let g be a fixed finite dimensional compact lie group it will be convenient to consider g as embedded in some large unitary group u n let g be the lie algebra of g we define a connection on ft to be a section a of t ft q i e a g valued 1 form which is locally l2 for any connection a let 1 f a da aaa

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

long time existence for yang mills flow we establish that finite time singularities do not occur in four dimensional yang mills flow confirming the conjecture of schlatter et al am j math 120 117 128 1998 the proof relies on a weighted energy identity and sharp decay estimates in the neck region

applications of the synchrosqueezing transform in seismic time frequency analysis abstracttime frequency representation of seismic signals provides a source of information that is usually hidden in the fourier spectrum the short time fourier transform and the wavelet transform are the principal approaches to simultaneously decompose a signal into time and frequency components known limitations such as trade offs between time and frequency resolution may be overcome by alternative techniques that extract instantaneous modal components empirical mode decomposition aims to decompose a signal into components that are well separated in the time frequency plane allowing the reconstruction of these components on the other hand a recently proposed method called the synchrosqueezing transform sst is an extension of the wavelet transform incorporating elements of empirical mode decomposition and frequency reassignment techniques this new tool produces a well defined time frequency representation allowing the identification of instantaneous frequencies in seismic signals to highlight

on the blow up set of the yang mills flow on kahler surfaces the yang mills flow on a kahler surface with holomorphic initial data converges smoothly away from a singular set determined by the harder narasimhan seshadri filtration of the initial holomorphic bundle

on a real analog of bezout inequality and the number of connected components of sign conditions let r be a real closed field and q1 q r x1 xk such that for each i 1 i deg qi di for 1 i denote by qi q1 qi vi the real variety defined by qi and ki an upper bound on the real dimension of vi by convention v0 r k and k0 k suppose also that 2 d1 d2 1 k 1 d3 1 k 1 2 d4 1 k 1 3 d 1 1 k 1 2 d and that k we prove that the number of semi algebraically connected components of v is bounded by o k 1 j d kj 1 kj j d 1 this bound can be seen as a weak extension of the classical bezout inequality which holds only over algebraically closed fields and is false over real closed fields to varieties defined over real closed fields additionally if p r x1 xk is a finite family of polynomials with deg p d for all p p cardp s and d 1 k 1d we prove that the number of semi algebraically connected components of the realizations of all realizable sign conditions of the family p restricted to v is bounded by o k sd 1 j d kj 1 kj j these results have found applications in discrete geometry for proving incidence bounds 11 as well as in effcient range searching 20