tyz expansion for the kepler manifold the main goal of the paper is to address the issue of the existence of kempf s distortion function and the tian yau zelditch tyz asymptotic expansion for the kepler manifold an important example of non compact manifold motivated by the recent results for compact manifolds we construct kempf s distortion function and derive a precise tyz asymptotic expansion for the kepler manifold we get an exact formula finite asymptotic expansion of n 1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter m rightarrow infty hbar m 1 rightarrow 0 standing for planck s constant and x rightarrow infty x in mathbb c n moreover the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the kepler manifold we show that our estimates are sharp by analyzing the nonharmonic behaviour of tm for m rightarrow infty the arguments of the proofs combine geometrical methods quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic gevrey spaces

on rigidity of grauert tubes over riemannian manifolds of constant curvature it is well known that a real analytic manifold x admits a complexification xc a complex manifold that contains x as the fixed point set of an antiholomorphic involution this can be seen as follows the transition functions defining the manifold x are real analytic local diffeomorphisms of rn the taylor expansions of these transition functions can be considered as local biholomorphisms of cn hence they serve as transition functions of a complex manifold the germ of the complexification xc is unique every sufficiently small tubular neighborhood of x in the tangent bundle tx admits a real analytic diffeomorphism into xc that fixes x therefore a sufficiently small tubular neighborhood ofx in tx has a complex structure and can be considered as a complexification of x in general the complex structure on the tubular neighborhood is not unique since there are manyways to embed it intoxc there have been a lot of interest in finding canonical complex structures on tubular neighborhoods ofx in tx with additional datum of a real analytic riemannian metric g on x a canonical complex structure can be specified for sufficiently small tubular neighborhoods of x in tx see gs ls s1 there is a unique complex structure on such that the map f i i c is holomorphic wherever it is defined

complex symplectomorphisms and pseudo kahler islands in the quantization of toric manifolds let p be a delzant polytope we show that the quantization of the corresponding toric manifold x p in toric kahler polarizations and in the toric real polarization are related by analytic continuation of hamiltonian flows evaluated at time t sqrt 1 s we relate the quantization of x p in two different toric kahler polarizations by taking the time sqrt 1 s hamiltonian flow of strongly convex functions on the moment polytope p by taking s to infinity we obtain the quantization of x p in the singular real toric polarization recall that x p has an open dense subset which is biholomorphic to mathbb c n the quantization of x p in a toric kahler polarization can also be described by applying the complexified hamiltonian flow of the abreu guillemin symplectic potential g at time t sqrt 1 to an appropriate finite dimensional subspace of quantum states in the quantization of t mathbb t n in the vertical polarization by taking other imaginary times t k sqrt 1 k in mathbb r we describe toric kahler metrics with cone singularities along the toric divisors in x p for convex hamiltonian functions and sufficiently negative imaginary part of the complex time we obtain degenerate kahler structures which are negative definite in some regions of x p we show that the pointwise and l 2 norms of quantum states are asymptotically vanishing on negative definite regions

the complex monge ampere equation zoll metrics and algebraization let m be a real analytic riemannian manifold an adapted complex structure on tm is a complex structure on a neighborhood of the zero section such that the leaves of the riemann foliation are complex submanifolds this structure is called entire if it may be extended to the whole of tm we prove here that the only real analytic zoll metric on the n sphere with an entire adapted complex structure on tm is the round sphere using similar ideas we answer a special case of an algebraization question raised by the first author characterizing some stein manifolds as affine algebraic in terms of plurisubharmonic exhaustion functions satisfying the homogeneous complex monge ampere equation the result presented here is an extension to higher dimensions of an observation attributed to w stoll for the case of riemann surfaces

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

equivalence problems for first order lagrangians on the line abstract complete solutions to and applications of the equivalence problems for first order particle lagrangians under the pseudo groups of contact point and fiber preserving transformations both with and without the addition of divergence terms are presented

some malmquist type theorems of partial differential equations on cn abstract in this paper using the techniques and some basic results from the value distribution theory in several complex variables we investigate the existence problem of meromorphic solutions of a class of partial differential equations on c n some malmquist type theorems of partial differential equations on c n are established a rellich type theorem of partial differential equations on c n is also obtained several examples are given to complement the theory

orbits of the geodesic flow and chains on the boundary of a grauert tube a c omega riemannian manifold x g determines an integrable complex structure on a tubular neighborhood t epsilon x of the zero section in t x and a cr structure on the boundary m epsilon there are two natural families of curves on m epsilon the orbits of the geodesic flow and a cr invariant family called chains it is natural to ask whether they are related we show that if orbits of the geodesic flow are chains on m epsilon for all epsilon sufficiently small then x g is einstein as a partial converse we show that if x g is harmonic then orbits of the geodesic flow are chains to prove this we study the fefferman metric associated with m epsilon

singular monge ampere foliations this paper generalizes results of lempert and szoke on the structure of the singular set of a solution of the homogeneous monge ampere equation on a stein manifold their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution in addition their requirement that the square of the solution be c 3 everywhere is replaced by a smoothness condition on the blowup of the singular set under these conditions the singular set is shown to inherit a finsler metric which in the real analytic case uniquely determines the solution of the monge ampere equation these results are proved using techniques from contact geometry

canonical complex structures associated to connections and complexifications of lie groups the notion of adapted complex structure is extended from riemannian manifolds to general koszul connections the case of the canonical connection of a lie group and the levi civita connection of a pseudo riemannian manifold is studied

twistor cr manifolds and three dimensional conformal geometry a cr i e partially complex 5 manifold is contructed as a sphere bundle over an arbitrary 3 manifold with conformal metric this so called twistor cr manifold is show to capture completely the original geometry and necessary and sufficient conditions are given for an abstract cr manifold to arise via the construction the above correspondence is then used to prove that a twistor cr manifold is locally imbeddable as a real hypersurface in c3 only if it is real analytic with respect to a suitable atlas introduction there is a very beautiful interplay between the conformal geometry of riemannian manifolds and complex analysis examples of which are found both in the classical theorem asserting the existence of isothermal coordinates on a surface and in the theory of self dual riemannian 4 manifolds penrose 1976 atiyah et al 1978 intermediating between these two theories is a relationship between 3dimensional conformal geometry and cr partially complex manifolds of dimension 5 in this paper we will explain this relationship focusing on the imbedding problem when can these partially complex manifolds be realized as real hypersurfaces in complex 3 manifolds the idea of associating a cr manifold with a conformal riemannian 3 manifold originated with penrose 1975 1983 who starting with a real analytic space like hypersurface in a real analytic lorentzian 4 manifold constructed a real hypersurface in a complex 3 manifold if the space like hypersurface is totally geodesic or merely all umbilic penroses s cr manifold coincides in our terminology with the twistor cr manifold of the conformal structure induced on the space like hypersurface after establishing notation and conventions in 0 principal results on cr manifolds are reviewed and the notions of cr vector bundle and cr contact form are introduced in 1 a general machine for constructing involutive distributions is produced and in 2 this construction is used to associate to every conformal riemannian 3 manifold a cr 5 manifold the cr manifolds arising in this way are characterized abstractly in 4 in terms of properties presented in 3 then in 5 it is shown that these cr manifolds are isomorphic to real hypersurfaces of complex received by the editors march 15 1983 and in revised form july 8 1983 1980 mathematics subject classificcation primary 32f25 secondary 53a30 83c99 work supported in part by nsf grant mcs 812 0790 001984 american mathematical society 0002 9947 84 1 00 25 per page

phase space bounds for quantum mechanics on a compact lie group let k be a compact connected lie group and k bbb c its complexification i consider the hilbert space cal h l 2 left k bbb c nu t right of holomorphic functions introduced in h1 where the parameter t is to be interpreted as planck s constant in light of l s the complex group k bbb c may be identified canonically with the cotangent bundle of k using this identification i associate to each f in cal h l 2 left k bbb c nu t right a phase space probability density the main result of this paper is theorem 1 which provides an upper bound on this density which holds uniformly over all f and all points in phase space specifically the phase space probability density is at most a t left 2 pi t right n where n dim k and a t is a constant which tends to one exponentially fast as t tends to zero at least for small t this bound cannot be significantly improved

characterization of strictly convex domains biholomorphic to a circular domain recently parabolic exhaustions have been used successfully to classify complex spaces stoll 6 and 7 burns 2 p wong 9 here we define parabolic exhaustions for strictly convex domains in c and give necessary and sufficient conditions for such a domain to be biholomorphically equivalent to a circular domain or even to the ball the results presented in this note are part of the ph d thesis the author is completing under the direction of professor w stoll details and more implications will appear at a later date a strictly convex domain d c c is a domain for which there exists a defining function whose real hessian is strictly positive on tx dd for all x g dd let a be the open unit disk in c and let s be the unit sphere in c let d c c be a strictly convex domain and p e d be any point for b g s lempert 4 constructs an extremal map f d b a d which is holomorphic with f 0 6 v and f 0 6 x where 1 x 0 is the length of b in the infinitesimal kobayashi metric of d at p these conditions determine f d b uniquely and the map extends smoothly to an embedding f u b s d also f a x s d is of class c and surjective one and only one function r d r exists such that r f z b z for z b g a x s then r is a continuous exhaustion of d positive and of class c on d p also r 1 on 3d we refer to r as the lempert exhaustion of d at p

on meromorphic solutions of first order briot bouquet type pdes abstract utilizing a theorem of coman and poletsky see 4 which characterizes the factorization of algebraically dependent meromorphic functions and the parametrization of the polynomial functional equation they satisfy along with results from nevanlinna theory we obtain results on the meromorphic solutions of the first order briot bouquet type pdes p u u x 0

on nonparametric multivariate binary discrimination aitchison aitken 1976 introduced a novel and ingenious nonparametric method for estimating probabilities in a multidimensional binary space the technique is designed for use in multivariate binary discrimination their estimator depends crucially on an unknown smoothing parameter a and aitchison aitken proposed a maximum likelihood method for determining a from the sample unfortunately this leads to an adaptive estimator which can behave very erratically when there are a number of empty or near empty cells present we demonstrate this both theoretically and by example to overcome these difficulties we introduce another method of estimating a which is designed to minimize a global function of the mean squared error

on meromorphic solutions of some algebraic partial differential equations oncn abstract applying the techniques from nevanlinna theory of value distribution theory of meromorphic functions oncn we investigate the existence problem of some meromorphic solutions and obtain a satisfactory malmquist type theorem for a class of algebraic partial differential equations oncnwhich improves some earlier main results we give several examples to complement our results

maximal complexifications of certain riemannian homogeneous manifolds let m g k be a homogeneous riemannian manifold with dim c g c dim r g where g c denotes the universal complexification of g under certain extensibility assumptions on the geodesic flow of m we give a characterization of the maximal domain of definition in tm for the adapted complex structure and show that it is unique for instance this can be done for generalized heisenberg groups and naturally reductive homogeneous riemannian spaces as an application it is shown that the case of generalized heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex

the complex time segal bargmann transform abstract we introduce a new form of the segal bargmann transform for a connected lie group k of compact type we show that the heat kernel t x t 0 x k has a space time analytic continuation to a holomorphic function c z re 0 z k c where k c is the complexification of k the new transform is defined by the integral b f z k c z k 1 f k d k z k c if s 0 and d s s the disk of radius s centered at s this integral defines a holomorphic function on k c for each f l 2 k s we construct a heat kernel density s on k c such that for all s as above b s b l 2 k s is an isometric isomorphism from l 2 k s onto the space of holomorphic functions in l 2 k c s when t s the transform b t t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type when t 0 2 s the transform b s t coincides with the one introduced by the first two authors

on nonparametric discrimination using density differences summary we propose a technique for nonparametric discrimination in which smoothing parameters are chosen jointly according to a criterion based on the difference between two densities the approach is suitable for categorical continuous and mixed data and uses information from both populations to determine the smoothing parameter for any one population in the case of categorical data optimal performance is sometimes achieved using negative smoothing parameters a property which does not emerge if the smoothing parameters are chosen individually

maximal complexifications of certain homogeneous riemannian manifolds let m g k be a homogeneous riemannian manifold with dim c g c dim r g where g c denotes the universal complexification of g under certain extensibility assumptions on the geodesic flow of m we give a characterization of the maximal domain of definition in tm for the adapted complex structure and show that it is unique for instance this can be done for generalized heisenberg groups and naturally reductive homogeneous riemannian spaces as an application it is shown that the case of generalized heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex

monge ampere equations and moduli spaces of manifolds of circular type abstract a bounded manifold of circular type is a complex manifold m of dimension n admitting a bounded exhaustive real function u defined on m minus a point x o so that a it is a smooth solution on m x o to the monge ampere equation d d c u n 0 b x o is a singular point for u of logarithmic type and e u extends smoothly on the blow up of m at x o c d d c e u 0 at any point of m x o this class of manifolds naturally includes all smoothly bounded strictly linearly convex domains and all smoothly bounded strongly pseudoconvex circular domains of c n a set of modular parameters for bounded manifolds of circular type is considered in particular for each biholomorphic equivalence class of them it is proved the existence of an essentially unique manifold in normal form it is also shown that the class of normalizing maps for an n dimensional manifold m is a new holomorphic invariant with the following property it is parameterized by the points of a finite dimensional real manifold of dimension n 2 when m is a non convex circular domain while it is of dimension n 2 2 n when m is a strictly linearly convex domain new characterizations of the circular domains and of the unit ball are also obtained

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

the structure of legendre foliations the local and global structure of legendre foliations of contact manifolds is analysed the main invariant of a legendre foliation is shown to be a quadratic form on the tangent bundle to the foliation the fundamental quadratic form the equivalence problem is solved in the case when the fundamental quadratic form is nondegenerate and a generalization of chern s solution to the equivalence problem for finsler manifolds is obtained a normal form for legendre foliations is given which is closely related to weinstein s structure theorem for lagrangian foliations it is shown that every compact simply connected leaf of a legendre foliation is diffeomorphic to a sphere

the asymptotic expansion of a cr invariant and grauert tubes cr manifolds the abstract models of real hypersurfaces in complex manifolds are 2n 1 dimensional manifolds m with a codimension one subbundle h of the tangent bundle which carries a complex structure the cr refers to cauchy riemann because for m c c n l the subbundle h consists of induced cauchy riemann operators there is a wealth of geometry and analysis associated with these structures especially when the cr manifolds are strictly pseudoconvex for example two strictly pseudoconvex domains are biholomorphically equivalent if and only if their boundaries are cr equivalent a fundamental problem in cr geometry is to find computable invariants associated with the cr structures the global cr invariant we will consider in this paper is the chem simons type invariant discovered by bums and epstein b e 1 it is a real valued global cr invariant of a compact 3dimensional strictly pseudo convex cr manifold whose holomorphic tangent bundle is trivial cheng and lee independently found this invariant and extend the definition of b e invariant to a relative invariant on an arbitrary compact 3 dimensional cr manifold cf c l we will evaluate this g asymptotically on the boundary of small grauert tubes before posing the question in a more precise form we will first say a few workds about grauert tubes let x be a real analytic manifold then every coordinate patch u c r n can be thickened to obtain an open set r c c n since the coordinate changes of x are real analytic maps by taking power series expansions and by shrinking c u to get convergence they can be extended holomorphically to such enlarged domains and thus they can be used as holomorphic transition

geometry of nonlinear connections abstract we show that locally diffeomorphic exponential maps can be defined for any second order differential equation and give a possibly nonlinear covariant derivative for any possibly nonlinear connection we introduce vertically homogeneous connections as the natural correspondents of homogeneous second order differential equations we provide significant support for the prospect of studying nonlinear connections via certain closely associated second order differential equations one of the most important is our generalized ambrose palais singer correspondence

complex monge ampere equations and totally real submanifolds abstract we study the dirichlet problem for complex monge ampere equations in hermitian manifolds with general non pseudoconvex boundary our main result theorem 1 1 extends the classical theorem of caffarelli kohn nirenberg and spruck in c n we also consider the equation on compact manifolds without boundary attempting to generalize yau s theorems in the kahler case as applications of the main result we study some connections between the homogeneous complex monge ampere hcma equation and totally real submanifolds and a special dirichlet problem for the hcma equation related to donaldson s conjecture on geodesics in the space of kahler metrics

kernel estimates for categorical data a new method of estimation of the smoothing parameters of the aitchison and aitken kernel method for categorical data is proposed this is based on minimizing an unbiased estimator of the mean squared error and it provides a natural approach to multivariate estimation thus allowing for the interaction structure of data in a simple way the kernel smoothing technique is shown theoretically to offer a potential improvement over the direct use of cell frequencies by means of an existence theorem akin to that used by hoerl and kennard for ridge regression the proposed smoothing parameter estimates are compared with competing methods by means of a simulation in the simple binomial case and are illustrated on a set of multivariate data used in titterington 1980 a property of invariance to the reversal of estimate formation and marginalization is proved for any multivariate kernel and estimation procedure

isometry theorem for the segal bargmann transform on a noncompact symmetric space of the complex type we consider the segal bargmann transform on a noncompact symmetric space of the complex type we establish isometry and surjectivity theorems for the transform in a form as parallel as possible to the results in the dual compact case the isometry theorem involves integration over a tube of radius r in the complexification followed by analytic continuation with respect to r a cancellation of singularities allows the relevant integral to have a nonsingular extension to large r even though the function being integrated has singularities

modular data and regularity of monge ampere exhaustions and of kobayashi distance regularity properties of intrinsic objects for a large class of stein manifolds namely of monge ampere exhaustions and kobayashi distance is interpreted in terms of modular data the results lead to a construction of an infinite dimensional family of convex domains with squared kobayashi distance of prescribed regularity properties a new sharp refinement of stoll s characterization of mathbb c n is also given

cartan decomposition of the moment map we investigate a class of actions of real lie groups on complex spaces using moment map techniques we establish the existence of a quotient and a version of luna s slice theorem as well as a version of the hilbert mumford criterion a global slice theorem is proved for proper actions we give new proofs of results of mostow on decompositions of groups and homogeneous spaces

tangentially positive isometric actions and conjugate points let m g be a complete riemannian manifold with no conjugate points and f m g b g b a principal g bundle where g is a lie group acting by isometries and b the smooth quotient with g b the riemannian submersion metric we obtain a characterization of conjugate point free quotients b g b in terms of symplectic reduction and a canonical pseudo riemannian metric on the tangent bundle tm from which we then derive necessary conditions involving g and m for the quotient metric to be conjugate point free particularly for m a reducible riemannian manifold let g tm with the lie algebra of g be the moment map of the tangential g action on tm and let gp be the canonical pseudo riemannian metric on tm defined by the symplectic form d and the map f tm m x m f z exp z exp z first we prove a theorem stating that if gp is not positive definite on the action vector fields for the tangential action along g 1 0 then b g b acquires conjugate points we proved the converse result in 2005 then we characterize self parallel vector fields on m in terms of the positivity of the gp length of their tangential lifts along certain canonical subsets of tm we use this to derive some necessary conditions on g and m for actions to be tangentially positive on relevant subsets of tm which we then apply to isometric actions on complete conjugate point free reducible riemannian manifolds when one of the irreducible factors satisfies certain curvature conditions

differential geometry of real monge ampere foliations on etudie les feuilletages qui proviennent de la condition de monge ampere recquerant le hessien d une fonction definie sur une variete affine d avoir un rang inferieur au maximal

complex zeros of real ergodic eigenfunctions we determine the limit distribution as of complex zeros for holomorphic continuations to grauert tubes of real eigenfunctions of the laplacian on a real analytic compact riemannian manifold m g with ergodic geodesic flow if phi j k is an ergodic sequence of eigenfunctions we prove the weak limit formula frac 1 lambda j z phi j k mathbb c to frac i pi partial bar partial xi g where z phi j k mathbb c is the current of integration over the complex zeros and where overline partial is with respect to the adapted complex structure of lempert szoke and guillemin stenzel

harmonic analysis with respect to heat kernel measure this paper surveys developments over the last decade in harmonic analysis on lie groups relative to a heat kernel measure these include analogs of the hermite expansion the segal bargmann transform and the taylor ex pansion some of the results can be understood from the standpoint of geo metric quantization others are intimately related to stochastic analysis