su 2 2 invariant g_2 instantons we initiate the systematic study of g 2 instantons with su 2 2 symmetry as well as developing foundational theory we give existence non existence and classification results for these instantons we particularly focus on mathbb r 4 times s 3 with its two explicitly known distinct holonomy g 2 metrics which have different volume growths at infinity exhibiting the different behaviour of instantons in these settings we also give an explicit example of sequences of g 2 instantons where bubbling and removable singularity phenomena occur in the limit

yang mills connections over manifolds with grassmann structure let m be a manifold with grassmann structure i e with an isomorphism of the cotangent bundle t m e h with the tensor product of two vector bundles e and h we define the notion of a half flat connection w in a vector bundle w m as a connection whose curvature f s2e 2h end w 2t m end w under appropriate assumptions for example when the grassmann structure is associated with a quaternionic kahler structure on m half flatness implies the yang mills equations inspired by the harmonic space approach we develop a local construction of holomorphic half flat connections w over a complex manifold with holomorphic grassmann structure equipped with a suitable linear connection any such connection w can be obtained from a prepotential by solving a system of linear first order odes the construction can be applied for instance to the complexification of hyper kahler manifolds or more generally to hyper kahler manifolds with admissible torsion and to their higher spin analogs it yields solutions

tangent cones to positive 1 1 de rham currents we consider positive 1 1 de rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood without this assumption counterexamples to the uniqueness of tangent cones can be produced already in cn hence our result is optimal the key idea is an implementation for currents in an almost complex setting of the classical blow up of curves in algebraic or symplectic geometry unlike the classical approach in cn we cannot rely on plurisubharmonic potentials primary subject 53c38 secondary subjects 49q20 35j99

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

a weak compactness theorem of the donaldson thomas instantons on compact kahler threefolds abstract in tanaka 18 we introduced a gauge theoretic equation on symplectic 6 manifolds which is a version of the hermitian einstein equation perturbed by higgs fields and called it a donaldson thomas equation to analytically approach the donaldson thomas invariants in this article we consider the equation on compact kahler threefolds and study some of the analytic properties of solutions to them using analytic methods in higher dimensional yang mills theory developed by nakajima 1987 14 nakajima 1988 15 and tian 2000 20 with some additional arguments concerning an extra nonlinear term coming from the higgs fields we prove that a sequence of solutions to the donaldson thomas equation of a unitary vector bundle over a compact kahler threefold has a converging subsequence outside a closed subset whose real two dimensional hausdorff measure is finite provided that the l 2 norms of the higgs fields are uniformly bounded we also prove an n 2 compactness theorem of solutions to the equations on compact kahler threefolds

singularities of hermitian yang mills connections and harder narasimhan seshadri filtrations this is the first of a series of papers where we relate tangent cones of hermitian yang mills connections at a singularity to the complex algebraic geometry of the underlying reflexive sheaf in this paper we work on the case when the sheaf is locally modeled on the pullback of a holomorphic vector bundle from the projective space and we shall impose an extra assumption that the graded sheaf determined by the harder narasimhan seshadri filtrations of the vector bundle is reflexive in general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a harder narasimhan seshadri filtration of an algebraic tangent cone which is a certain torsion free sheaf on the projective space in this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable

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

complex dynamics in the 1 3 spatial resonance abstract the interaction between two steady state bifurcations with spatial wave numbers in the ratio 1 3 is considered periodic boundary conditions are assumed the resulting o 2 equivariant normal form truncated at third order exhibits a number of global bifurcations that may result in complex dynamics the origin of this behavior is elucidated with the help of careful numerical simulations and analysis of appropriate return maps the results generalize to other 1 n resonances

elliptic yang mills equation we discuss some recent progress on the regularity theory of the elliptic yang mills equation we start with some basic properties of the elliptic yang mills equation such as coulomb gauges monotonicity and curvature estimates next we discuss singularity of stationary yang mills connections and compactness theorems on yang mills connections with bounded l2 norm of curvature we also discuss in some detail self dual solutions of the yang mills equation and describe a compactification of their moduli space

0 2 symmetric bifurcation theory for convection rolls abstract the interaction of two steady state modes for the two dimensional benard problem with lateral periodic boundary conditions p b c and free horizontal boundary conditions is considered the p b c generate an action of o 2 on the navier stokes equations the boussinesq approximation induces an additional z 2 symmetry the o 2 z 2 equivariant bifurcation equations are determined and two fifth order terms are identified two types of mixed mode solutions and a travelling wave solution are found using macsyma the fifth order terms are calculated and the stability of the solutions to phase perturbations depending on the prandtl number is determined

matryoshka of special democratic forms special p forms are forms which have components 1 p equal to 1 1 or 0 in some orthonormal basis a p form p r d is called democratic if the set of nonzero components 1 p is symmetric under the transitive action of a subgroup of o d z on the indices 1 d knowledge of these symmetry groups allows us to define mappings of special democratic p forms in d dimensions to special democratic p forms in d dimensions for successively higher p p and d d in particular we display a remarkable nested structure of special forms including a u 3 invariant 2 form insixdimensions ag2 invariant3 forminsevendimensions aspin 7 invariant4 form in eight dimensions and a special democratic 6 formin ten dimensions the latter has the remarkable property that its contraction with one of five distinct bivectors yields in the orthogonal eight dimensions the spin 7 invariant 4 form we discuss various properties of this ten dimensional form

an introduction to potential theory in calibrated geometry in this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry these functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties moreover they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce a number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds this paper introduces and investigates questions of pseudo convexity in the context of a general calibrated manifold x phi analogues of totally real submanifolds are introduced and used to construct enormous families of strictly phi convex spaces with every topological type allowed by morse theory specific calibrations are used as examples throughout in a sequel the duality between phi pluri sub harmonic functions and phi positive currents is investigated this study involves boundaries generalized jensen measures and other geometric objects on a calibrated manifold

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

notes on a slice distance for singular lp bundles abstract a slice distance for the class of weak abelian l p bundles in 3 dimensions was introduced in 20 where it was used to prove the closure of such class of bundles for the weak l p convergence we further investigate this distance here and we prove more properties of it for example we show that it is holder continuous on the slices of an l p bundle using the same distance we give a notion of a boundary trace which allows to use the direct method for minimization problems on weak bundles with fixed trace we then state some conjectures and some open questions

the limit of the hermitian yang mills flow on reflexive sheaves abstract in this paper we study the asymptotic behavior of the hermitian yang mills flow on a reflexive sheaf we prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the harder narasimhan seshadri filtration this answers a question by bando and siu

quaternionic maps between quaternionic kahler manifolds in this paper we consider quaternionic maps between quaternionic kahler manifolds we will derive a monotonicity inequality a bochner type formula and small energy regularity for quaternionic maps

partial regularity of a minimizer of the relaxed energy for biharmonic maps in this paper we study the relaxed energy for biharmonic maps from an m dimensional domain into spheres for an integer m 5 by an approximation method we prove the existence of a minimizer of the relaxed energy of the hessian energy and that the minimizer is biharmonic and smooth outside a singular set sigma of finite m 4 dimensional hausdorff measure when m 5 we prove that the singular set sigma is i rectifiable moreover we also prove a rectifiability result for the concentration set of a sequence of stationary harmonic maps into manifolds crown copyright c 2011 published by elsevier inc all rights reserved

energy identity for stationary yang mills given a principal bundle p rightarrow m over a riemannian manifold with compact structure group g let us consider a stationary yang mills connection a with energy int m f a 2 le lambda if we consider a sequence of such connections a i then it is understood by tian ann math 151 1 193 268 2000 that up to subsequence we can converge a i rightarrow a to a singular limit connection such that the energy measures converge f a i 2 dv g rightarrow f a 2dv g nu where nu e x d lambda n 4 is the n 4 rectifiable defect measure our main result is to show without additional assumptions that for n 4 a e point the energy density e x may be computed explicitly as the sum of the bubble energies arising from blow ups at x each of these bubbles may be realized as a yang mills connection over s 4 itself this energy quantization was proved in riviere commun anal geom 10 4 683 708 2002 assuming a uniform l 1 hessian bound on the curvatures in the sequence in fact our second main theorem is to show this hessian bound holds automatically precisely given a connection a as above we have the a priori estimate int m nabla 2 f a c lambda dim g m for the curvature it is important to note this result is proved in tandem with the energy quantization and not before it indeed we will in fact prove an effective version of the energy identity and it is this effective version which will lead to both the l 1 hessian bound and the classical energy quantization results in the course of the proof we will provide a quantitative version of the bubble tree decomposition which hold in all dimensions with effective estimates for a fixed stationary connections to produce strongest estimates in the paper we introduce an epsilon gauge condition which generalizes the usual coulomb gauge and which will exist with effective control even over singular regions on these epsilon gauges we will provide a new superconvexity estimate which will be a key tool in analyzing higher dimensional annular regions

moduli spaces of contact instantons abstract we construct the moduli space of contact instantons an analogue of yang mills instantons defined for contact metric 5 manifolds and initiate the study of their structure in the k contact case we give sufficient conditions for smoothness of the moduli space away from reducible connections and show the dimension is given by the index of an operator elliptic transverse to the reeb foliation the moduli spaces are shown to be kahler when the 5 manifold m is sasakian and hyperkahler when m is transverse calabi yau we show how the transverse index can be computed in various cases in particular we compute the index for the toric sasaki einstein spaces y p q

the twistor sections on the wolf spaces let m be a compact quaternion symmetric space a wolf space and v m an irreducible homogeneous vector bundle on m with its canonical connection whose rank is less than or equal to the dimension of m we classify the zero loci of the transversal twistor sections with a reality condition there exists a bijection between such zero loci and the real representations of simple compact connected lie groups with non trivial principal isotropy subgroups which are neither tori nor discrete groups next we obtain an embedding of the wolf space into a real grassmannian manifold using twistor sections which turns out to be a minimal embedding finally we focus our attention on the norm squared s 2 of a twistor section s we identify the subset s m where this function attains the maximum value under a suitable hypothesis such sets are classified and determine totally geodesic submanifolds of the wolf spaces moreover s 2 is a morse function in the sense of bott and its critical manifolds consist of the zero locus and s m

almgren s type regularity for semicalibrated currents abstract in analogy with almgren s theorem for integral area minimizing currents of general dimension and codimension we prove that an m dimensional semicalibrated integral current in a n m dimensional c 3 e 0 manifold semicalibrated by a c 2 e 0 m form has singular set of hausdorff dimension at most m 2

uniqueness of volume minimizing submanifolds calibrated by the first pontryagin form one way to understand the geometry of the real grassmann manifold gk r k n parameterizing oriented k dimensional subspaces of rk n is to understand the volume minimizing subvarieties in each homology class some of these subvarieties can be determined by using a calibration in previous work one of the authors calculated the set of 4 planes calibrated by the first pontryagin form p1 on gk r k n for all k n 4 and identified a family of mutually congruent round 4 spheres which are consequently homologically volume minimizing in the present work we associate to the family of calibrated planes a pfaffian system on the symmetry group so k n r an analysis of which yields a uniqueness result namely that any connected submanifold of gk r k n calibrated by p1 is contained in one of these 4 spheres a similar result holds for p1 calibrated submanifolds of the quotient grassmannian g k r k n of non oriented k planes

the 1 2 spatial resonance on a hexagonal lattice in two layered rayleigh benard problems this paper investigates pattern formation resulting from the interaction between steady modes with wavenumbers in the ratio 1 2 a two layered rayleigh benard problem with infinite extent is examined two layers are separated by a non deformable and thin dividing plate this physical set up is known to provide an exact 1 2 resonance between critical modes restricting the wavevectors of interacting modes on a hexagonal lattice we derive 12 dimensional amplitude equations up to cubic order among steady solutions of the equations the stable ones are hexagons and mixed hexagons travelling waves corresponding to relative equilibria are also found to be stable depending on parameter values

nonequilibrium mode coupling theory for uniformly sheared systems we develop a nonequilibrium mode coupling theory for uniformly sheared systems starting from microscopic thermostatted sllod equations of motion our theory aims at describing stationary state properties including rheological ones of sheared systems and this is accomplished via two steps first a set of self consistent equations is formulated based on the projection operator formalism and on the mode coupling approach for the transient density correlators which measure the correlations between the density fluctuations in the initial equilibrium state and the ones at later times after the shearing force is turned on the transient time correlation function formalism is then used which combined with the mode coupling approximation expresses stationary state properties in terms of the transient density correlators a detailed comparison of our theory is also presented with the related mode coupling theory which is based on the smoluchowski equation for brownian particles under stationary shearing

oxidation of self duality to 12 dimensions and beyond using partial curvature flows and the transitive action of subgroups of rm o d mathbb z on the indices 1 ldots d of the components of the yang mills curvature in an orthonormal basis we obtain a nested system of equations in successively higher dimensions d each implying the yang mills equations on d dimensional riemannian manifolds possessing special geometric structures this matryoshka of self duality equations contains the familiar self duality equations on riemannian fourfold as well as their generalisations on complex kahler threefold and on seven and eight dimensional manifolds with g2 and spin 7 holonomy the matryoshka allows enlargement oxidation to a remarkable system in 12 dimensions invariant under sp 3 there are hints that the underlying geometry is related to the sextonions a six dimensional algebra between the quaternions and octonions

removable singularities in yang mills fields we show that a field satisfying the yang mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite we obtain the result that every yang mills field overr4 with bounded functional l2 norm may be obtained from a field ons4 r4 hodge or coulomb gauges are constructed for general small fields in arbitrary dimensions including 4

ddvv type inequality for skew symmetric matrices and simons type inequality for riemannian submersions abstract in this paper we will first derive a ddvv type optimal inequality for real skew symmetric matrices then we apply it to establish a simons type integral inequality for riemannian submersions with totally geodesic fibres and yang mills horizontal distributions in this way we show phenomenons of duality between submanifold geometry and riemannian submersion particularly between second fundamental form of a submanifold and integrability tensor of a riemannian submersion

anti self duality of curvature and degeneration of metrics with special holonomy we study the structure of noncollapsed gromov hausdorff limits of sequences mni of riemannian manifolds with special holonomy we show that these spaces are smooth manifolds with special holonomy off a closed subset of codimension 4 additional results on the the detailed structure of the singular set support our main conjecture that if the mni are compact and a certain characteristic number c mni is bounded independent of i then the singularities are of orbifold type off a subset of real codimension at least 6

the singular set of 1 1 integral currents we prove that 2 dimensional integer multiplicity 2 dimensional rectifiable currents which are almost complex cycles in an almost complex manifold admitting locally a compatible positive symplectic form are smooth surfaces aside from isolated points and therefore are j holomorphic curves

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

uniqueness of tangent cones to positive p p integral cycles we prove that every positive p p integral cycle in an arbitrary almost complex manifold possesses at every point a unique tangent cone the argument relies on an algebraic blow up perturbed in order to face the analysis issues of this problem in the almost complex setting primary subjects 53c38 secondary subjects 49q05 49q20 35j99

riemannian geometry over different normed division algebras we develop a unifed theory to study geometry of manifolds with different holonomy groups they are classified by 1 real complex quaternion or octonion number in the appropriate cases and 2 being special or not specialty is an orientation with respect to the corresponding normed algebra mathbb a for example special riemannian mathbb a manifolds are oriented riemannian calabi yau hyperkahler and g 2 manifolds respectively for vector bundles over such manifolds we introduce special mathbb a connections they include holomorphic hermitian yang mills anti self dual and donaldson thomas connections similarly we introduce special frac 1 2 mathbb a lagrangian submanifolds as maximally real submanifolds they include special lagrangian complex lagrangian cayley and co associative submanifolds we also discuss geometric dualities from this viewpoint fourier transformations on mathbb a geometry for flat tori and a conjectural syz mirror transformation from special mathbb a geometry to special frac 1 2 mathbb a lagrangian geometry on mirror special mathbb a manifolds

the uniqueness of tangent cones for yang mills connections with isolated singularities 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 control over the rate of the asymptotic convergence of the connection to the tangent connection if furthermore the connection is stationary or the tangent connection is integrable with a stronger result in the latter case 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 exists for all time and has asymptotic limit if the initial value is close to a smooth local minimizer of the yang mills functional

ddvv type inequality for hermitian matrices abstract in this paper we extend ddvv type inequalities involving the frobenius norm of commutators from real symmetric and skew symmetric matrices to hermitian and skew hermitian matrices

the resolution of the yang mills plateau problem in super critical dimensions abstract we study the minimization problem for the yang mills energy under fixed boundary connection in supercritical dimension n 5 we define the natural function space a g in which to formulate this problem in analogy to the space of integral currents used for the classical plateau problem the space a g can be also interpreted as a space of weak connections on a real measure theoretic version of reflexive sheaves from complex geometry we prove the existence of weak solutions to the yang mills plateau problem in the space a g we then prove the optimal regularity result for solutions of this plateau problem on the way to prove this result we establish a coulomb gauge extraction theorem for weak curvatures with small yang mills density this generalizes to the general framework of weak l 2 curvatures previous works of meyer riviere and tao tian in which respectively a strong approximability property and an admissibility property were assumed in addition

supermembrane limit of yang mills theory we consider yang mills theory with n 1 super translation group in eleven auxiliary dimensions as the structure group the gauge theory is defined on a direct product manifold 3 s1 where 3 is a three dimensional lorentzian manifold and s1 is a circle we show that in the infrared limit when the metric on s1 is scaled down the yang mills action supplemented by a wess zumino type term reduces to the action of an m2 brane

a removal singularity theorem of the donaldson thomas instanton on compact kahler threefolds abstract we consider a perturbed hermitian einstein equation which we call the donaldson thomas equation on compact kahler threefolds in 12 we analysed some analytic properties of solutions to the equation in particular we proved that a sequence of solutions to the donaldson thomas equation has a subsequence which smoothly converges to a solution to the donaldson thomas equation outside a closed subset of the hausdorff dimension two in this article we prove that some of these singularities can be removed

compactness theorems for coupled yang mills fields in this paper we consider coupled yang mills fields on vector bundle e over compact riemannian manifold m under appropriate conditions on the curvature and the higgs field two compactness theorems are proved

principal bundles on projective varieties and the donaldson uhlenbeck compactification let h be a semisimple algebraic group we prove the semistable reduction theorem for semistable principal h bundles over a smooth projective variety x defined over the field c when x is a smooth projective surface and h is simple we construct the algebro geometric donaldson uhlenbeck compactification of the moduli space of semistable principal h bundles with fixed characteristic classes and describe its points for large characteristic classes we show that the moduli space of stable principal h bundles is non empty

asymptotic laws and preasymptotic correction formulas for the relaxation near glass transition singularities within the mode coupling theory mct for the dynamics of simple liquids the leading corrections to the asymptotic solutions for the relaxation in the vicinity of an ideal glass transition are derived the formulas are used to determine the range of validity of the scaling law description of the mct results for the a and b processes in glass forming systems solutions of the mct equations of motion are calculated for a hard sphere colloidal suspension model and compared with the derived analytical results the leading order formulas are shown to describe the major qualitative features of the bifurcation scenario near the transition and the leadingplus next to leading order formulas are demonstrated to give a quantitative description of the evolution of structural relaxation for the model s1063 651x 97 06005 4

g2 manifolds and associative submanifolds via semi fano 3 folds we construct many new topological types of compact g 2 manifolds i e riemannian 7 manifolds with holonomy group g 2 to achieve this we extend the twisted connected sum construction first developed by kovalev and apply it to the large class of asymptotically cylindrical calabi yau 3 folds built from semi fano 3 folds constructed previously by the authors in many cases we determine the diffeomorphism type of the underlying smooth 7 manifolds completely we find that many 2 connected 7 manifolds can be realised as twisted connected sums in a variety of ways raising questions about the global structure of the moduli space of g 2 metrics many of the g 2 manifolds we construct contain compact rigid associative 3 folds which play an important role in the the higher dimensional enumerative geometry gauge theory calibrated submanifolds approach to defining deformation invariants of g 2 metrics by varying the semi fanos used to build different g 2 metrics on the same 7 manifold we can change the number of rigid associative 3 folds we produce

regularity theory for 2 dimensional almost minimal currents i lipschitz approximation we construct lipschitz q valued functions which carefully approximate integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense this result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of 2 dimensional integral currents area minimizing in riemannian manifolds semicalibrated and spherical cross sections of 3 dimensional area minimizing cones

special quantum field theories in eight and other dimensions we build nearly topological quantum field theories in various dimensions we give special attention to the case of eight dimensions for which we first consider theories depending only on yang mills fields two classes of gauge functions exist which correspond to the choices of two different holonomy groups in so 8 namely su 4 and spin 7 the choice of su 4 gives a quantum field theory for a calabi yau fourfold the expectation values for the observables are formally holomorphic donaldson invariants the choice of spin 7 defines another eight dimensional theory for a joyce manifold which could be of relevance in m and f theories relations to the eight dimensional supersymmetric yang mills theory are presented then by dimensional reduction we obtain other theories in particular a four dimensional one whose gauge conditions are identical to the non abelian seiberg witten equations the latter are thus related to pure yang mills self duality equations in 8 dimensions as well as to the n 1 d 10 super yang mills theory we also exhibit a theory that couples 3 form gauge fields to the second chern class in eight dimensions and interesting theories in other dimensions