scalar curvature of a metric with unit volume the problem of finding riemannian metrics on a closed manifold with prescribed scalar curvature function is now fairly well understood from the works of kazdan and warner in 1970 s 10 and references cited in it in this paper we shall consider the same problem under a constraint on the volume for this purpose it is useful to introduce an invariant p m of a smooth closed manifold m which will be called the yamabe number of m defined as the supremum of m c of all conformal classes c of riemannian metrics on m

large deviations for stochastic pde with levy noise abstract we prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small levy noise we use general large deviations theorems of varadhan and bryc coupled with the techniques of feng and kurtz 2006 15 viscosity solutions of integro partial differential equations in hilbert spaces and deterministic optimal control methods the laplace limit is identified as a viscosity solution of a hamilton jacobi bellman equation of an associated control problem we also establish exponential moment estimates for solutions of stochastic evolution equations driven by levy noise general results are applied to stochastic hyperbolic equations perturbed by subordinated wiener process

aubin s lemma for the yamabe constants of infinite coverings and a positive mass theorem aubin s lemma says that if the yamabe constant of a closed conformal manifold m c is positive then it is strictly less than the yamabe constant of any of its non trivial finite conformal coverings we generalize this lemma to the one for the yamabe constant of any m c of its infinite conformal coverings provided that 1 m has a descending chain of finite index subgroups tending to 1 m moreover if the covering m is normal the limit of the yamabe constants of the finite conformal coverings associated to the descending chain is equal to that of m c for the proof of this we also establish a version of positive mass theorem for a specific class of asymptotically flat manifolds with singularities

the prescribed scalar curvature problem for metrics with unit total volume we solve the modified kazdan warner problem of finding metrics with prescribed scalar curvature and unit total volume

computations of the orbifold yamabe invariant we consider the yamabe invariant of a compact orbifold with finitely many singular points we prove a fundamental inequality for the estimate of the invariant from above which also includes a criterion for the non positivity of it moreover we give a sufficient condition for the equality in the inequality in order to prove it we also solve the orbifold yamabe problem under a certain condition we use these results to give some exact computations of the yamabe invariant of compact orbifolds

the yamabe invariant for non simply connected manifolds the yamabe invariant is an invariant of a closed smooth manifold defined using conformal geometry and the scalar curvature recently petean showed that the yamabe invariant is nonnegative for all closed simply connected manifolds of dimension 5 we extend this to show that yamabe invariant is nonnegative for all closed manifolds of dimension 5 with fundamental group of odd order having all sylow subgroups abelian the main new geometric input is a way of studying the yamabe invariant on toda brackets a similar method of proof shows that all closed oriented manifolds of dimension 5 with non spin universal cover with finite fundamental group having all sylow subgroups elementary abelian admit metrics of positive scalar curvature once one restricts to the complement of manifolds whose homology classes are toral the exceptional toral homology classes only exist in dimensions not exceeding the rank of the fundamental group so this proves important cases of the gromov lawson rosenberg conjecture once the dimension is sufficiently large

analyticity on rotation invariant families of curves let be a rotation invariant family of smooth jordan curves contained in a the open unit disc in c for each t let dt be the simply connected domain bounded by t we present various conditions which imply that if is a continuous function on a such that for every fe the function f t has a continuous extension to dt which is analytic in dr then is analytic in a

testing families of complex lines for the unit ball abstract let p 1 and p 2 be two points in c 2 such that the complex line joining them is tangent to the unit sphere b 2 we prove that if a real analytic function f on b 2 extends holomorphically on every complex line through p 1 and p 2 then f extends holomorphically to the whole ball b 2

manifolds of positive scalar curvature and conformal cobordism theory for a supergoup we study closed manifolds with positive conformal classes we use the relative yamabe invariant from 2 to define the conformal cobordism relation on the category of such manifolds we prove that the corresponding conformal cobordism groups are isomorphic to the cobordism groups defined by stolz in 19 as a corollary we show that the conformal concordance relation on positive conformal classes coincides with the standard concordance relation on positive scalar curvature metrics our main technical tools come from analysis and conformal geometry

analyticity of functions analytic on circles abstract let be the open unit disc in c let p b and let f be a continuous function on which extends holomorphically from each circle in centered at the origin and from each circle in which passes through p then f is holomorphic on

classification of prime 3 manifolds with σ invariant greater than rp 3 in this paper we compute the invariants sometimes also called the smooth yamabe invariants of rp 3 and rp 2 s 1 which are equal and show that the only prime 3 manifolds with larger invariants are s 3 s 2 s 1 and s 2 s 1 the nonorientable s 2 bundle over s 1 more generally we show that any 3 manifold with invariant greater than rp 3 is either s 3 a connect sum with an s 2 bundle over s 1 or has more than one nonorientable prime compo

separate holomorphic extension along lines and holomorphic extension from the sphere to the ball we give positive answer to a conjecture by agranovsky a continuous function on the sphere which has separate holomorphic extension along the complex lines which pass through three non aligned interior points is the trace of a holomorphic function in the ball

testing analyticity on circles consider a continuous one parameter family of circles in a complex plane that contains two circles lying in the exterior of one another we prove that if a continuous function on the union of the circles extends holomorphically into each circle then the function is holomorphic in the interior of the union of the circles

multiplicity results for the yamabe equation by lusternik schnirelmann theory abstract let m g be any closed riemannianan manifold and n h be a riemannian manifold of constant positive scalar curvature we prove that the yamabe equation on the riemannian product m n g h has at least cat m 1 solutions for small enough where cat m denotes the lusternik schnirelmann category of m the solutions obtained are functions of m and c a t m of them have energy arbitrarily close to the minimum

parametric argument principle and its applications to cr functions and manifolds it follows from the classical argument principle that if a holomor phic mapping f from the unit discto c or more generally to c n smooth in the closed disc is homologicaly trivial on the unit circle i e h1 0 f s 1 which is equivalent to either being a point or having endpoints i e 6 then f const in other words the image of the unit disc degenerates to a point we establish a parametric version of this fact for a variety of holomorphic map pings fromto c n in place of a single mapping we find conditions for a holomorphic mapping of the unit disc depending on additional real parameters under which homological triviality of the boundary image implies collapse of the dimension of the image of the interior as an application we obtain estimates of dimensions of complex tan gent bundles of real submanifolds in c n in terms of zero moment conditions on families of closed curves covering the manifold apply ing this result to the graphs of functions we obtain solution of several known problems about characterization of holomorphic cr functions in terms of moment conditions on families of curves

an obstruction to the positivity of relative yamabe invariants for a supergroup gamma we describe an obstruction to the existence of positive scalar curvature metrics with minimal boundary condition on a compact n dimensional gamma manifold w with nonempty boundary m 3 leq n leq 8 in terms of the bordism class m in the stolz obstruction group r n 1 gamma associated to gamma st2 in par ticular when w is a 5 dimensional spin manifold and the alpha invariant of a connected component of m is nonzero we prove that w does not admit a positive scalar curvature metric with minimal boundary condition

holomorphic extensions from open families of circles for a circle z c z c p write z w z a w a 2 0 z a p a continuous function f on extends holomorphically from into the disc bounded by if and only if the function f z z f z defined on z z z e r has a bounded holomorphic extension into a r in the paper we consider open connected families of circles c write u u e c and assume that a continuous function on u extends holomorphically from each e c we show that this happens if and only if the function f z z f z defined on z z z e u has a bounded holomorphic extension into the domain u r e q for each open family q compactly contained in c this allows us to use known facts from several complex variables in particular we use the edge of the wedge theorem to prove a theorem on real analyticity of such functions

generalized mehler semigroups and applications we construct and study generalized mehler semigroups p t t 0 and their associated markov processesm the construction methods for p t t 0 are based on some new purely functional analytic results implying in particular that any strongly continuous semigroup on a hilbert spaceh can be extended to some larger hilbert spacee with the embeddingh e being hilbert schmidt the same analytic extension results are applied to construct strong solutions to stochastic differential equations of typedx t c dw t ax t dt with possibly unbounded linear operatorsa andc onh on a suitably chosen larger spacee for gaussian generalized mehler semigroups p t t 0 with corresponding markov processm the associated non symmetric dirichlet forms ed e are explicitly calculated and a necessary and sufficient condition for path regularity ofm in terms of e d e is proved then using dirichlet form methods it is shown thatm weakly solves the above stochastic differential equation if the state spacee is chosen appropriately finally we discuss the differences between these two methods yielding strong resp weak solutions

the l p l p cr hartogs separate analyticity theorem for convex domains in this note a very general theorem of the cr hartogs type is proved for almost generic strictly convex domains in mathbf c mathbf n with real analytic boundary given such a domain d and given an l p function f on partial d which has holomorphic extensions on the slices of d by complex lines parallel to the coordinate axes f must be cr i e f has a holomorphic extension to d which is in the hardy space h p d this is the first general result of cr hartogs type which is not for the ball or some other domain with symmetries and holds for l 1 functions as a corollary the szeg kernel is shown to be the strong operator limit of pi 1 pi 2 n where pi i is the projection onto z i holomorphically extendible l 2 partial d functions in mathbf c mathbf 2 with a slightly more complicated formula in mathbf c mathbf n

the spinorial tau invariant and 0 dimensional surgery let m be a compact manifold with a metric g and with a fixed spin structure chi let lambda 1 g be the first non negative eigenvalue of the dirac operator on m g chi we set tau m chi sup inf lambda 1 g where the infimum runs over all metrics g of volume 1 in a conformal class g 0 on m and where the supremum runs over all conformal classes g 0 on m let m chi be obtained from m chi by 0 dimensional surgery we prove that tau m chi geq tau m chi as a corollary we can calculate tau m chi for any riemann surface m

subordinated α stable ornstein uhlenbeck process as a tool for financial data description abstract the classical financial models are based on the standard brownian diffusion type processes however in the exhibition of some real market data like interest or exchange rates we observe characteristic periods of constant values moreover in the case of financial data the assumption of normality is often unsatisfied in such cases the popular vasicek model that is a mathematical system describing the evolution of interest rates based on the ornstein uhlenbeck process seems not to be applicable therefore we propose an alternative approach based on a combination of the popular ornstein uhlenbeck process with a stable distribution and subdiffusion systems that demonstrate such characteristic behavior the probability density function of the proposed process can be described by a fokker planck type equation and therefore it can be examined as an extension of the basic ornstein uhlenbeck model in this paper we propose the parameters estimation method and calibrate the subordinated vasicek model to the interest rate data

maskit combinations of poincare einstein metrics we establish a boundary connected sum theorem for asymptotically hyperbolic einstein metrics and also show that if the two metrics have scalar positive conformal infinities then the same is true for this boundary join this construction is also extended to spaces with a finite number of interior conic singularities and as a result we show that any 3 manifold which is a finite connected sum of quotients of s3 and s2 s1 bounds such a space with conic singularities putatively any 3 manifold admitting a metric of positive scalar curvature is of this form

deformation of scalar curvature and volume the stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the einstein metrics one may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant in this paper we localize a condition satisfied by such stationary points to smooth bounded domains the condition involves a generalization of the static equations and we interpret solutions and their boundary values of this equation variationally on domains carrying a metric that does not satisfy the condition we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric we apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved finally we note that starting from a counterexample of min oo s conjecture such as that of brendle marques neves counterexamples of arbitrarily large volume and different topological types can be constructed

on the cauchy problem for non local ornstein uhlenbeck operators abstract we study the cauchy problem involving non local ornstein uhlenbeck operators in finite and infinite dimensions we prove classical solvability without requiring that the levy measure corresponding to the large jumps part has a first finite moment moreover we determine a core of regular functions which is invariant for the associated transition markov semigroup such a core allows to characterize the marginal laws of the ornstein uhlenbeck stochastic process as unique solutions to fokker planck kolmogorov equations for measures

non gaussian ornstein uhlenbeck based models and some of their uses in financial economics non gaussian processes of ornstein uhlenbeck ou type offer the possibility of capturing important distributional deviations from gaussianity and for flexible modelling of dependence structures this paper develops this potential drawing on and extending powerful results from probability theory for applications in statistical analysis their power is illustrated by a sustained application of ou processes within the context of finance and econometrics we construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive ou processes and we study these models in relation to financial data and theory

ornstein uhlenbeck cauchy process we combine earlier investigations of linear systems subject to levy fluctuations with recent attempts to give meaning to so called levy flights in external force fields we give a complete construction of the ornstein uhlenbeck cauchy process as a fully computable paradigm example of doob s stable noise supported ornstein uhlenbeck process despite the nonexistence of all moments we determine local characteristics forward drift of the process generators of forward and backward dynamics and relevant pseudodifferential evolution equations the induced nonstationary spatial process is proved to be markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense

diffusion semigroups in spaces of continuous functions with mixed topology abstract we study transition semigroups and kolmogorov equations corresponding to stochastic semilinear equations on a hilbert space h it is shown that the transition semigroup is strongly continuous and locally equicontinuous in the space of polynomially increasing continuous functions on h when endowed with the so called mixed topology as a result we characterize cores of certain second order differential operators in such spaces and show that they have unique extensions to generators of strongly continuous semigroups

testing analyticity on rotation invariant families of curves let r c c be a piecewise smooth jordan curve symmetric with respect to the real axis which contains the origin in its interior and which is not a circle centered at the origin let q be the annulus obtained by rotating r around the origin we characterize the curves r with the property that if f e c q is analytic on sr for every 181 1 then f is analytic in int q

propagation of boundary cr foliations and morera type theorems for manifolds with attached analytic discs abstract we prove that homologically nontrivial generic smooth 2 n 1 parameter families of analytic discs in c n n 2 attached by their boundaries to a cr manifold test cr functions in the following sense if a smooth function on analytically extends into any analytic discs from the family then the function satisfies tangential cr equations on in particular we give an answer theorem 1 to the following long standing open question so called strip problem earlier solved only for special families mainly for circles given a smooth one parameter family of jordan curves in the plane and a function f admitting holomorphic extension inside each curve must f be holomorphic on the union of the curves we prove for real analytic functions and arbitrary generic real analytic families of curves that the answer is yes if no point is surrounded by all curves from the family the latter condition is essential we generalize this result to characterization of complex curves in c 2 as real 2 manifolds admitting nontrivial families of attached analytic discs theorem 4 the main result implies fairly general morera type characterization of cr functions on hypersurfaces in c 2 in terms of holomorphic extensions into three parameter families of attached analytic discs theorem 2 one of the applications is confirming in real analytic category the globevnik stout conjecture theorem 3 on boundary values of holomorphic functions it is proved that a smooth function on the boundary of a smooth strictly convex domain in c n extends holomorphically inside the domain if it extends holomorphically into complex lines tangent to a given strictly convex subdomain the proofs are based on a universal approach namely on the reduction to a problem of propagation from the boundary to the interior of degeneracy of cr foliations of solid torus type manifolds theorem 2 2

linking numbers and boundaries of varieties called a cornerstone of algebraic geometry gh p 62 is a simple consequence of the fact that analytic varieties carry a natural orientation recast in terms of linking numbers it is our principal motivation it implies the following let m be a smooth oriented compact 3 manifold in c3 suppose that m bounds a bounded complex 2 variety v here bounds means in the sense of stokes theorem i e that b v m as currents let a be an algebraic curve in c3 which is disjoint from m consider the linking number link m a of m and a since this linking number is equal to the intersection number i e the sum of the intersection indices of v and a by the positivity

thin discs and a morera theorem for cr functions f z dz 0 for every closed curve in d there are generalizations of this theorem to domains and real hypersurfaces in cn see a and gs and references there in this paper we obtain a version of the morera theorem for cr functions on manifolds of higher codimension let m be a smooth real manifold in cn recall that m is generic if the tangent space tz m spans the whole space cn for z m a cr function on m is a continuous function that satisfies the weak tangential cauchy riemann equations an analytic disc is a continuous mapping a cn holomorphic in the standard disc c 1 we say that the disc a is attached to m if it maps the circle b to m that is a b m let f be a continuous function on a real manifold m cn we say that f has the morera property with respect to an analytic disc a attached to m if