levi flat hypersurfaces with real analytic boundary let x be a stein manifold of dimension at least 3 given a compact codimension 2 real analytic submanifold m of x that is the boundary of a compact levi flat hypersurface h we study the regularity of h suppose that the cr singularities of m are an o x convex set for example suppose m has only finitely many cr singularities which is a generic condition then h must in fact be a real analytic submanifold if m is real algebraic it follows that h is real algebraic and in fact extends past m even near cr singularities to prove these results we provide two variations on a theorem of malgrange that a smooth submanifold contained in a real analytic subvariety of the same dimension is itself real analytic we prove a similar theorem for submanifolds with boundary and another one for subanalytic sets

on stein neighborhood basis of real surfaces in this paper we show that a compact real surface embedded in a complex surface has a regular stein neighborhood basis provided that there are only finitely many complex points on the surface and that they are all flat and hyperbolic an application to unions of totally real planes in 2 is then given

on maps between nonminimal hypersurfaces we construct an analytic jet parametrization for nontrivial cr maps between real analytic 1 nonminimal hypersurfaces in mathbb c 2 in particular this gives a real analytic structure on the set of all such maps and shows that the automorphism groups of a 1 nonminimal hypersurface in mathbb c 2 is a lie group our results also hold for some classes of 1 nonminimal hypersurfaces in higher dimensions

occupation time large deviations of two dimensional symmetric simple exclusion process we prove a large deviations principle for the occupation time of a site in the two dimensional symmetric simple exclusion process the decay probability rate is of order t logt and the rate function is given by 2 sin 1 2 1 sin 1 2 1 2 the proof relies on a large deviations principle for the polar empirical measure which contains an interesting log scale spatial average a contraction principle permits us to deduce the occupation time large deviations from the large deviations for the polar empirical measure

on an n manifold in mathbf c n near an elliptic complex tangent in this paper we will be concerned with the local biholomorphic properties of a real n manifold m in c at a generic point such a manifold basically has the nature of the standard r in c near a complex tangent however the consideration can be much more complicated and the manifold may acquire a nontrivial local hull of holomorphy and many other biholomorphic invariants the study of such a problem was first carried out in a celebrated paper of e bishop bis where for each sufficiently non degenerate complex tangent he attached a biholomorphic invariant called the bishop invariant when the complex tangent is elliptic i e when 0 12 for a more precise definition see 2 he showed the existence of families of complex analytic disks with boundary on m that shrink down to the locus of points in m with complex tangents in particular using the well known continuity principle one sees that the image m of such families is contained in the holomorphic hull of the manifold at the time he asked whether m gives precisely the local holomorphic hull of m as well as certain uniqueness properties of the attached disks he also proposed the problem of determining the fine structure of m near such complex tangents later there appeared a sequence of papers concerning the smooth character of m in case m c here we would like to mention in particular the famous theorem proved by kenig webster in their deep work kw1 which states that the local hull of holomorphy m near an elliptic complex tangent is a smooth levi flat hypersurface with m c as part of its smooth boundary in another important paper of moser webster mw a systematic normal form theory was employed for the understanding of the local biholomorphic invariants of m in case m is real analytic when the bishop invariant 6 0 their method works in any dimension and even for some hyperbolic complex tangents but it breaks down for complex tangents with 0 among other things they showed that m can be biholomorphically mapped into the affine space

flattening of cr singular points and analyticity of the local hull of holomorphy i this is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in mathbb c n with n ge 3 near a non degenerate cr singular point the problem is motivated from the study of the complex plateau problem that seeks for the levi flat hypersurface bounded by a given real submanifold and is motivated by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space the present article is focused on the case of cr singular points with at least one elliptic direction we solve the holomorphic flattening problem and thus provide a complete description of the local hull of holomorphy in this setting the results in this paper and those in flattening of cr singular points and analyticity of the local hull of holomorphy ii p 60 2014 are taken from our arxiv post flattening of cr singular points and analyticity of the local hull of holomorphy 2012 we split flattening of cr singular points and analyticity of the local hull of holomorphy 2012 into two independent articles to avoid it being too long

nonalgebraizable real analytic tubes in ℂ n we give necessary conditions for certain real analytic tube generic submanifolds in n to be locally algebraizable as an application we exhibit families of real analytic non locally algebraizable tube generic submanifolds in n during the proof we show that the local cr automorphism group of a minimal finitely nondegenerate real algebraic generic submanifold is a real algebraic local lie group we may state one of the main results as follows let m be a real analytic hypersurface tube in n passing through the origin having a defining equation of the form v y where z w x iy u iv open image in new windown 1 open image in new window assume that m is levi nondegenerate at the origin and that the real lie algebra of local infinitesimal cr automorphisms of m is of minimal possible dimension n i e generated by the real parts of the holomorphic vector fields z1 z n 1 w then m is locally algebraizable only if every second derivative 2yky l is an algebraic function of the collection of first derivatives y1 ym

self similarity of period doubling branching 3 d reversible mappings abstract we consider 3 dimensional 3 d mappings which are reversible i e possess a time reversal symmetry in the mappings studied here the reversibility is such that it guarantees the existence of one parameter families curves of symmetric periodic orbits in the phase space it is found that such curves can often intersect one another in particular a curve of n cycles can intersect a curve of 2n cycles which in turn can intersect a curve of 4n cycles etc we show that the tree of branching curves of successively doubled periods in the 3 d phase space possesses some geometric self similarity in particular we identify the scaling factors 4 018 16 363 and 8 721 previously studied 1 parameter 2 d area preserving reversible mappings in which these scalings also occur are special cases of 3 d reversible mappings in that they have an integral of motion the more general mappings studied here have no such integral we discuss the reasons for our result in terms of normal forms

real ellipsoids and double valued reflection in complex space we develop the theory of double valued reflection for a real ellipsoidal hypersurface m in complex euclidean n space this leads to a pair of meromorphic involutions on the complexification of m the main problem is to understand the dynamics of their composition which is a reversible contact trnasformation by means of the segre polar correspondence this map is transformed into a kind of generalized complex null cone billiard map relative to two nonsingular n dimensional complex quadrics in projective n 1 space next we develop a generalization of classical confocal ellipsoidal coordinates which we use together with the maurer cartan equations of the orthogonal group to demonstrate the integrable nature of the dynamics of the reversible map we show that certain complex integral curves of the characteristic systems of suitably invariant contact forms of the structure are invariant by the map the integration of these systems is in turn reduced to solving systems of abelian differential equations by means of generalized jacobi inversion finally we indicate how to choose the contact form so that the real loci of these characteristic curves will be closed curves the complex curves will then yield generalized stationary curves in the sense of lempert for the ellipsoidal domain

a normal form for a real 2 codimensional submanifold in cn 1 near a cr singularity abstract we construct a formal normal form for a class of real submanifolds m c n 1 of codimension 2 near a cr singularity approximating the sphere our result gives a generalization of huang yin s normal form in c 2 to the higher dimensional analog case

flattening of cr singular points and analyticity of the local hull of holomorphy ii abstract this is the second article of the two papers in which we investigate the holomorphic and formal flattening problem of a non degenerate cr singular point of a codimension two real submanifold in c n with n 3 the problem is motivated from the study of the complex plateau problem that looks for the levi flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space the present article is focused on non degenerate flat cr singular points with at least one non parabolic bishop invariant we will solve the formal flattening problem in this setting the results in this paper and those in 23 are taken from our earlier arxiv post 22 we split 22 into two independent articles to avoid it being too long

defining equations for real analytic real hypersurfaces in ⁿ a defining function for a real analytic real hypersurface can be utliquely written as 2 re h e where h is holomorphic and e contains no pure terms we study how h and e change when we perform a local biholomorphic change of coordinates or multiply by a unit one of the main results is necesary and sufficient conditions on the first nonvanishing homogeneous part of e expanded in terms of h beyond eoo that serve as obstructions to writing a defining equation as 2 re h e where e is independent of h we also find necessary pluriharmonic obstructions to doing this which arise from the easier case of attempting to straighten the hypersurface

orthosymmetric block reflectors we develop a general theory of reflectors and block reflectors in a class of non euclidean scalar product spaces generated by orthosymmetric scalar product matrices j these j reflectors are generalizations of ordinary householder transformations and we show that they can always be expressed in a householder like representation reflection and mapping properties of j reflectors are completely described block j reflectors can be used for block annihilation in qr like factorizations where q is j unitary and r is block upper triangular

convergent normal form and canonical connection for hypersurfaces of finite type in c2 abstract we study the holomorphic equivalence problem for finite type hypersurfaces in c 2 we discover a geometric condition which is sufficient for the existence of a natural convergent normal form for a finite type hypersurface we also provide an explicit construction of such a normal form as an application we construct a canonical connection for a large class of finite type hypersurfaces to the best of our knowledge this gives the first construction of an invariant connection for levi degenerate hypersurfaces in c 2

factoring formal maps into reversible or involutive factors abstract an element g of a group is called reversible if it is conjugate in the group to its inverse an element is an involution if it is equal to its inverse this paper is about factoring elements as products of reversibles in the group g n of formal maps of c n 0 i e formally invertible n tuples of formal power series in n variables with complex coefficients the case n 1 was already understood 25 each product f of reversibles has linear part l f of determinant 1 the main results are that for n 2 each map f with det l f 1 is the product of 2 3 ceiling log 2 n reversibles and may also be factored as the product of 9 6 ceiling log 2 n involutions where the ceiling of x is the smallest integer x

real submanifolds of maximum complex tangent space at a cr singular point i we study a germ of real analytic n dimensional submanifold of mathbf c n that has a complex tangent space of maximal dimension at a cr singularity under some assumptions we show its equivalence to a normal form under a local biholomorphism at the singularity we also show that if a real submanifold is formally equivalent to a quadric it is actually holomorphically equivalent to it if a small divisors condition is satisfied finally we investigate the existence of a complex submanifold of positive dimension in mathbf c n that intersects a real submanifold along two totally and real analytic submanifolds that intersect transversally at a possibly non isolated cr singularity

divergent birkhoff normal forms of real analytic area preserving maps in this paper we provide a real analytic area preserving map that possesses a divergent birkhoff normal form near an elliptic fixed point the method uses the small divisor theory and the work of gong in the study of the hamiltonian system

quantitative estimates for periodic points of reversible and symplectic holomorphic mappings we show that reversible holomorphic mappings of bf c 2 have periodic points accumulating at an elliptic fixed point of general type on the contrary we also show the existence of holomorphic symplectic mappings that have no periodic points of certain periods in a sequence of deleted balls about an elliptic fixed point of general type the radii of the balls are carefully chosen in terms of the periods which allows us to show the existence of holomorphic mappi ngs of bf c 2 that are not reversible with respect to any c 1 involution with a holomorphic linear part and that admit no invariant totally real and c 1 real surfaces

on normal forms of complex points of codimension 2 submanifolds abstract in this paper we present some linear algebra behind quadratic parts of quadratically flat complex points of codimension two real submanifold in a complex manifold assuming some extra nondegenericity and using the result of hong complete normal form descriptions can be given and in low dimensions we obtain a complete classification without any extra assumptions

equivalences of real submanifolds in complex space we show that for any real analytic submanifold m in cn there is a proper real analytic subvariety v m such that for any p m v any real analytic submanifold min cn and any p m the germs m p and m p of the submanifolds m and mat p and prespectively are formally equivalent if and only if they are biholomorphically equivalent as an ap plication for p m v the problem of biholomorphic equivalence of the germs m p and m p is reduced to that of solving a system of polyno mial equations more general results for k equivalences are also stated and proved

convergence of the chern moser beloshapka normal forms in this article we first describe a normal form of real analytic levi nondegenerate submanifolds of c n of codimension d 1 under the action of formal biholomorphisms that is of perturbations of levi nondegenerate hyperquadrics we give a sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic we show that our techniques can be adapted in the case d 1 in order to obtain a new and direct proof of chern moser normal form theorem

formal and finite order equivalences we show that two families of germs of real analytic subsets in mathbb c n are formally equivalent if and only if they are equivalent of any finite order we further apply the same technique to obtain analogous statements for equivalences of real analytic self maps and vector fields under conjugations on the other hand we provide an example of two sets of germs of smooth curves that are equivalent of any finite order but not formally equivalent

convergence and finite determination of formal cr mappings in this paper we study the convergence and finite determination of formal holomorphic mappings of c 0 taking one real submanifold into another by a formal holomorphic mapping h c 0 c 0 we mean an n vector h h1 hn where each hj is a formal power series in n indeterminates with no constant term if m and m are real smooth submanifolds through 0 in c defined near the origin by z z 0 and z z 0 respectively where and are vector valued smooth defining functions then we say that a formal mapping h c 0 c 0 sends m into m if the vector valued power series h z h z is a matrix multiple of z z for real analytic hypersurfaces we shall prove the following theorem 1 let m and m be real analytic hypersurfaces through the origin in c n 2 assume that neither m nor m contains a nontrivial holomorphic subvariety through 0 then any formal mapping h c 0 c 0 sending m into m is convergent the condition that m above does not contain a nontrivial holomorphic subvariety is necessary see remark 2 3 as a corollary we obtain the following characterization

on mixed polynomials of bidegree n 1 specifying the bidegrees n m of mixed polynomials p z z of the single complex variable z with complex coefficients allows to investigate interesting roots structures and counting intermediate between complex and real algebra multivariate mixed polynomials appeared in recent papers dealing with milnor fibrations but in this paper we focus on the univariate case and m 1 which is closely related to the important subject of harmonic maps here we adapt to this setting two algorithms of computer algebra vandermonde interpolation and a bissection exclusion method for root isolation implemented in maple they are used to explore some interesting classes of examples

a bishop surface with a vanishing bishop invariant we give a solution to the equivalence problem for bishop surfaces with the bishop invariant 0 as a consequence we answer in the negative a problem that moser asked in 1985 after his work with webster in 1983 and his own work in 1985 this will be done in two major steps we first derive the formal normal form for such surfaces we then show that two real analytic bishop surfaces with 0 are holomorphically equivalent if and only if they have the same formal normal form up to a trivial rotation our normal form is constructed by an induction procedure through a completely new weighting system from what is used in the literature our convergence proof is done through a new hyperbolic geometry associated with the surface