the schwarz lemma at the boundary of the symmetrized bidisc abstract the symmetrized bidisc g 2 is defined by g 2 z 1 z 2 z 1 z 2 c 2 z 1 1 z 2 1 z 1 z 2 c it is a bounded inhomogeneous pseudoconvex domain without c 1 boundary and especially the symmetrized bidisc hasn t any strongly pseudoconvex boundary point and the boundary behavior of both caratheodory and kobayashi metrics over the symmetrized bidisc is hard to describe precisely in this paper we study the boundary schwarz lemma for holomorphic self mappings of the symmetrized bidisc g 2 and our boundary schwarz lemma in the paper differs greatly from the earlier related results

rigidity of holomorphic mappings and a new schwarz lemma at the boundary a rigidity theorem for holomorphic mappings in the nature of the uniqueness statement of the classical one variable schwarz lemma is proved at the boundary of a strongly pseudoconvex domain the result reduces to an interesting and apparently new result even in one complex dimension the theorem has a variety of geometric and analytic interpretations department of mathematics university of michigan ann arbor michigan 48109 department of mathematics box 1146 washington university in st louis st louis missouri 63130 this content downloaded from 40 77 167 14 on wed 15 jun 2016 05 36 04 utc all use subject to http about jstor org terms

stationary discs and finite jet determination for non degenerate generic real submanifolds abstract let m c n be a generic real submanifold of class c 4 in case m is levi non degenerate in the sense tumanov we construct stationary discs for m if furthermore m satisfies an additional non degeneracy condition we apply the method of stationary discs to obtain 2 jet determination of cr automorphisms of m

adaptive stabilization for a kirchhoff type nonlinear beam under boundary output feedback control abstract in this paper the boundary stabilization for a kirchhoff type nonlinear beam with one end fixed and control at the other end is considered a gain adaptive controller is designed in terms of measured end velocity the existence and uniqueness of the classical solution of the closed loop system are justified the exponential stability of the system is obtained

noncyclic geometric phase and its non abelian generalization we use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the abelian and non abelian geometric phases this derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the abelian noncyclic geometric phase we also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non abelian noncyclic geometric phase for a spin 1 magnetic or electric quadrupole interacting with a precessing magnetic electric field

a boundary condition with memory for kirchhoff plates equations in this paper we study the stability of solutions for kirchhoff plates equations with a memory condition working at the boundary we show that such dissipation is strong enough to produce exponential decay to the solution provided the relaxation functions also decays exponentially when the relaxation functions decays polynomially we show that the solution decays polynomially and with the same rate

on the real and complex geometric phases some of the existing definitions of the real and complex geometric phases are unified and generalized the relationship of the real to the complex geometric phase is discussed it is shown why these two phases are in general completely independent in particular a simple explicit example is given which shows that the real geometric phase is not in general the real part of the complex geometric phase

non cyclic geometric phases in a proposed two photon interferometric experiment an experimental arrangement is proposed which combines the idea of two photon interferometry with that of geometric phases a pair of photons generated by parametric down conversion is fed into a pair of mach zehnder interferometers in each of which there is phase shift produced by the helically wound optical fibre in the manner of tomita and chiao 1986 if the winding angle is a multiple of 2 pi there is a closed path in parameter space in accordance with the pioneering analysis of berry 1984 more general windings are also considered in which the path in parameter space is not closed and the geometric phase for such open paths is calculated in accordance with the gauge invariant prescription of aitchison and wanelik if the proposed experiment is realized it will provide a test for the noncyclic geometric phase for photons and for the set of prescriptions equivalent to that of aitchison and wanelik 1992 furthermore the geometric phase in this arrangement will be unequivocally quantum mechanical because of the non classical character of two photon interferometry

on the perturbation of the stationary discs in almost complex manifolds abstract we study the neighborhood of an embedded j holomorphic disc f attached along its boundary to a strictly pseudoconvex domain r 4 j in particular we give sufficient conditions for the set of all nearby attached j holomorphic discs to be a manifold

a burns krantz type theorem for domains with corners in their pioneering paper bk94 burns andkrantzhave established a boundary version of the classical cartan s uniqueness theorem if f is holomorphic self map of a smoothly bounded strongly pseudoconvex domain d c such that f z z o z p 3 as z p for some p d then f z z furthermore they give an example showing that the exponent 3 in the above statement cannot be decreased the goal of this paper is twofold first to give purely local boundary uniqueness results for maps defined only on one side as germs at a boundary point and hence not necessarily sending any domain to itself and also under the weaker assumption that f z z o z p 3 holds only for z in a proper cone in d with vertexp such results have no analogues in one complex variable in contrast with the situation when a domain is preserved our second goal is to extend the above results from boundaries of domains to real submanifolds m c of higher codimension recall that m is called generic if its tangent space tpm at every point p m spans tpc over c i e tpc n tpm itpm here the usual replacement for a one sided neighborhood of a boundary point is a wedge with edgem at a pointp m in the direction of an open cone npm tpc tpm i e a domain w c such that for any pair of open cones npm with 0 and 0 there exists a neighborhood u of p in c such that m u u w and m u u contains a neighborhood of p in w where we identify npm with a fixed complement of tpm in tpc c furthermore we extend the notion of strong pseudoconvexity to wedges as follows we call a wedge w with edge m at p in the direction of strongly

julia theory for slice regular functions the theory of slice regular functions is nowadays widely studied and has found elegant applications to functional calculus for quaternionic linear operators and schur analysis however much less is known about their boundary behaviors in this paper we initiate the study of julia theory for slice regular functions more precisely we establish the quaternionic versions of the julia lemma the julia caratheodory theorem the boundary schwarz lemma the hopf lemma and the burns krantz rigidity theorem for slice regular self mappings of the open unit ball b h and of the right half space h we provide an explicit example to show that in quaternionic hopf lemma the slice derivative of a slice regular function f at the boundary fixed point may not be real in contrast to the complex version our result implies that the commonly believed version of the hopf lemma turns out to be totally wrong this new quaternionic version of the hopf lemma also improves osserman estimate even in the complex setting and is essentially the first significant theorem belonging to the theory of quaternions itself other than the theory of complex analysis since the result involves the lie brackets which reflect the non commutative feature of quaternions

asymptotic behavior to a von karman plate with boundary memory conditions in this paper we study the stability of solutions to a von karman system for kirchhoff plate equations with a memory condition working at the boundary we show that such dissipation is strong enough to produce exponential decay of the solution provided the relaxation functions also decay exponentially when the relaxation functions decay polynomially we show that the solution decays polynomially

stationary holomorphic discs and finite jet determination problems we construct a family of small analytic discs attached to levi non degenerate hypersurfaces in mathbb c n 1 which is globally biholomorphically invariant we then apply this technique to study unique determination problems along levi non degenerate hypersurfaces that are merely of class mathcal c 4 this method gives 2 jet determination results for germs of biholomorphisms cr diffeomorphisms as well as in the almost complex setting

an existence theorem for stationary discs in almost complex manifolds an existence theorem for stationary discs of strongly pseudo convex domains in almost complex manifolds is proved more precisely it is shown that for all points of a suitable neighborhood of the boundary and for any vector belonging to certain open subsets of the tangent spaces there exists a unique stationary disc passing through that point and tangent to the given vector this result gives a generalization of a theorem of b coupet h gaussier and the second author originally proved only for almost complex structures which are small deformations of an integrable one

extremal discs and the regularity of cr mappings in higher codimension we develop a local theory of extremal analytic discs attached to a strictly pseudoconvex manifold of higher codimension in complex space we then apply the results to prove the regularity of cr homeomorphisms of such manifolds introduction in his celebrated paper lempert li introduced extremal analytic discs as geodesies of the kobayashi metric on a convex domain d c cn let a be the unit disc in the complex plane an analytic disc a d is called extremal if 0 attains its maximum among all analytic discs in d with the same values and directions at 0 g a extremal discs are important holomorphic invariants of d their boundaries lie in the boundary bd of the domain and form an invariant family of curves for the cr structure of bd lempert li l2 applies the extremal discs to the regularity of biholomorphic mappings monge amp re equations and normal forms of convex domains other applications include deformations and embeddibility of cr structures bl bd and polynomial hulls s

boundary schwarz lemma for solutions to poisson s equation abstract let d be the closure of the unit disk d in c and g a continuous function of d in this paper we establish the schwarz lemma at the boundary for solutions to the poisson s equation f g

polynomial hulls with convex fibers and complex geodesics let x be a compact subset of z 1 cn with convex fibers several equivalent conditions are obtained that characterize polynomially convex hulls y x with nonempty interior under the latter assumption it is shown that every z0 w0 y and such that z0 1 is contained in a closed n dimensional complex submanifold of d cn which is tangent to y along an analytic disc we show as an application that some results of lempert on complex geodesies in convex domains are direct consequences of properties of polynomial hulls

global attractors for von karman evolutions with a nonlinear boundary dissipation abstract dynamic von karman equations with a nonlinear boundary dissipation are considered questions related to long time behaviour existence and structure of global attractors are studied it is shown that a nonlinear boundary dissipation with a large damping parameter leads to an existence of global compact attractor for all weak finite energy solutions this result has been known in the case of full interior dissipation but it is new in the case when the boundary damping is the main dissipative mechanism in the system in addition we prove that fractal dimension of the attractor is finite the proofs depend critically on the infinite speed of propagation associated with the von karman model considered

fefferman s mapping theorem on almost complex manifolds in complex dimension two we give a necessary and sufficient condition for the smooth extension of a diffeomorphism between smooth strictly pseudoconvex domains in four real dimensional almost complex manifolds see theorem 1 1 the proof is mainly based on a reflection principle for pseudoholomorphic discs on precise estimates of the kobayashi royden infinitesimal pseudometric and on the scaling method in almost complex manifolds

jet determination of smooth cr automorphisms and generalized stationary discs we prove finite jet determination for finitely smooth cr diffeomorphisms of finitely smooth levi degenerate hypersurfaces in mathbb c n 1 by constructing generalized stationary discs glued to such hypersurfaces

identity principles for commuting holomorphic self maps of the unit disc abstract let f g be two commuting holomorphic self maps of the unit disc if f and g agree at the common wolff point up to a certain order of derivatives no more than 3 if the wolff point is on the unit circle then f g

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

comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains we prove that for a strongly pseudoconvex domain d subset mathbb c n the infinitesimal caratheodory metric g c z v and the infinitesimal kobayashi metric g k z v coincide if z is sufficiently close to bd and if v is sufficiently close to being tangential to bd also we show that every two close points of d sufficiently close to the boundary and whose difference is almost tangential to bd can be joined by a unique up to reparameterization complex geodesic of d which is also a holomorphic retract of d the same continues to hold if d is a worm domain as long as the points are sufficiently close to a strongly pseudoconvex boundary point we also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed this has consequences for the behavior of the squeezing function

stationary discs and finite jet determination for cr mappings in higher codimension abstract we discuss stationary discs for generic cr manifolds and apply them to the problem of finite jet determination for cr mappings we prove that a c 2 smooth cr diffeomorphism of two c 4 smooth strictly pseudoconvex levi generating cr manifolds is uniquely determined by its 2 jet at a given point a new key element of the proof is the existence of non defective stationary discs

a property of the functions and distributions annihilated by a locally integrable system of complex vector fields page introduction 387 1 basic notation and ingredients 389 2 approximation and constancy on the fibers of the classical solutions of the homogeneous equations 393 3 first consequences and interpretation of the fundamental theorem 398 4 distribution solutions of the homogeneous equations 402 5 regularity of the solutions of the homogeneous equations 409 6 example of a system that is analytic hypo elliptic but not co hypo elliptic 413 references 420

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

parametrization of local cr automorphisms by finite jets and applications the goal of this work is to study the structure of the group of all local realanalytic cr automorphisms of a germ of a real analytic hypersurface m p in c or equivalently of all local biholomorphisms h c p c p mapping the germ m p into itself we also deal with the more general situation of a realanalytic cr manifold of arbitrary codimension see 2 for every fixed point p m we refer to this group of local biholomorphisms as the stability group of m p and denote it by aut m p endowed with the topology of uniform convergence on compact neighbourhoods of p it becomes a topological group this natural inductive limit topology can be described in the following way a sequence of germs of biholomorphisms hj aut m p converges to another such germ h aut m p if there exists a neighbourhood u of p in c to which all mappings hj extend holomorphically and such that hj h uniformly on u this group arises in a variety of different circumstances one of the most important ones where it naturally appears is the biholomorphic equivalence problem which consists in deciding whether two given germs of real analytic hypersurfaces are biholomorphically equivalent the biholomorphic equivalence problem has been studied since the beginning of the last century when poincare 37 observed probably for the first time that two real hypersurfaces in c for n 2 are not necessarily locally equivalent via a holomorphic transformation though they are via a c transformation today the biholomorphic equivalence problem for strongly pseudoconvex or levi nondegenerate hypersurfaces is pretty well understood based on the works of e cartan 14 15 in dimension two and the results of tanaka 42 and chern moser 17 in arbitrary dimension thanks to the works initiated in the 1970s a number of remarkable properties of the stability group of strongly pseudoconvex real analytic hypersurfaces have been discovered see e g the surveys by vitushkin 45 huang 30 and baouendi ebenfelt and rothschild 5 for complete discussions on this matter these properties depend on whether the germ m p can be biholomorphically mapped onto a piece of the unit sphere if there exists such a biholomorphism one says that m p is spherical for instance if m p is not spherical the group aut m p is

stationary discs glued to a levi non degenerate hypersurface we obtain an explicit parametrization of stationary discs glued to some levi non degenerate hypersurfaces these discs form a family which is invariant under the action of biholomorphisms we use this parametrization to construct a local circular representation of these hypersurfaces as a corollary we get the uniqueness of biholomorphisms with given 1 jet at some convenient point

admissible boundary values of bounded holomorphic functions in wedges if m c n is a generic cauchy riemann manifold and w c n is a wedge domain with edge m then every bounded holomorphic function on w has an admissible limit at almost every point of m moreover if a holomorphic function f on w has a distribution boundary value bv f on m that is a bounded measurable function then f is bounded on every finer wedge near m and its admissible limit equals bv f p at almost every point p m

on the geometry of model almost complex manifolds with boundary we study some special almost complex structures on strictly pseudoconvex domains in 2n they appear naturally as limits under a nonisotropic scaling procedure and play a role of model objects in the geometry of almost complex manifolds with boundary we determine explicitely some geometric invariants of these model structures and derive necessary and sufficient conditions for their integrability as applications we prove a boundary extension and a compactness principle for some elliptic diffeomorphisms between relatively compact domains

second angular derivatives and parabolic iteration in the unit disk in this paper we deal with second angular derivatives at denjoy wolff points for parabolic functions in the unit disc namely we study and analyze the existence and the dynamical meaning of this second angular derivative for instance we provide several characterizations of that existence in terms of the so called koenigs function it is worth pointing out that there are two quite different classes of parabolic iteration those with positive hyperbolic step and those with zero hyperbolic step in the first case the koenigs function is in the caratheodory class but in the second case it is even unknown if it is normal therefore the ideas and techniques to approach these two cases are really different in the end we also present several rigidity results related to the second angular derivatives at denjoy wolff points

adaptive stabilization of a von karman plate equation with a boundary output feedback control we consider a von karman plate equation with boundary memory condition and output feedback control we prove the existence of solutions using the galerkin method and then investigate the stabilization of the corresponding solutions by choosing a suitable lyapunov functional

moduli for pointed convex domains a moduli space for the class of pointed strictly linearly convex domains in n is obtained it is shown that the space of pointed smoothly bounded strictly linearly convex domains with a fixed indicatrix is parameterized by a class of deformations of the cr structure of the boundary of the indicatrix these deformations are constructed by using the circular representation of a domain to pull back its complex structure tensor to the indicatrix a careful study of the pull back structure shows that the allowable deformations are parameterized by a class of complex hamiltonian vector fields the proof of this fact is based on the folland stein estimates for the bar partial b complex of the boundary of the indicatrix