on the uniqueness of static perfect fluid solutions in general relativity following earlier work of masood ul alam we consider a uniqueness problem for non rotating stellar models given a static asymptotically flat perfectfluid spacetime with barotropic equation of state p and given another such spacetime which is spherically symmetric and has the same p and the same surface potential we prove that both are identical provided p satisfies a certain differential inequality this inequality is more natural and less restrictive than the conditions required by masood ul alam

formulae for the derivatives of heat semigroups abstract formulae for the derivatives of solutions of diffusion equations are derived which clearly exhibit and allow estimation of the equations smoothing properties these also give formulae for the logarithmic gradient of the corresponding heat kernels extending and giving a very elementary proof of bismut s well known formula corresponding formulae are derived for solutions of heat equations for differential forms and their exterior derivatives

on the construction of global coordinate systems in euclidean spaces abstract several techniques for constructing global coordinate systems in r n adapted to functions are given some of these techniques are based on geometrical assumptions and other ones on analytical assumptions

a geometric approach to the classification of the equilibrium shapes of self gravitating fluids the classification of the equilibrium shapes that a self gravitating fluid can take in a riemannian manifold is a classical problem in mathematical physics in this paper it is proved that the equilibrium shapes are isoparametric submanifolds some geometric properties of them are also obtained e g classification and existence for some riemannian spaces and relationship with the isoperimetric problem and the group of isometries of the manifold our approach to the problem is geometrical and allows to study the equilibrium shapes on general riemannian spaces

classifications of bundle connection pairs by parallel translation and lassos let m be a connected manifold and g be a closed lie subgroup of gl v the general linear group on a finite dimensional vector space v denote by m the space of h1 loops in m starting at a fixed point m let m be the set of p c m g modulo conjugation by an element of g such that p p p for m is the concatenation of the loops and p p if is reparameterization of and the differential of p satisfies a locality condition it is shown that the bundle connection pairs e up to equivalence with structure group g and fiber v are in one to one correspondence with m a similar result has been announced by kobayashi the correspondence is induced by the parallel translation operators of connections furthermore if the manifold m is simply connected then the space of bundle connection pairs can be classified by a collection of lie algebra valued 1 forms on the manifold m called integrated lassos these 1 forms are related to the differentials of elements of m this last result generalizes weil s characterization of u 1 line bundle connection pairs by the curvature 2 form it is also a generalization of gross results to base manifolds m other than rn

ym2 continuum expectations lattice convergence and lassos the two dimensional yang mills theory ym2 is analyzed in both the continuum and the lattice in the complete axial gauge the continuum theory may be defined in terms of a lie algebra valued white noise and parallel translation may be defined by stochastic differential equations this machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesc the expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group the time parameters of the heat kernels are determined by the areas enclosed by the collectionc and the arguments are determined by the crossing topologies of the curves inc the expectations for the wilson lattice models have a similar structure and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations it is also shown that the lasso variables advocated by l gross 36 exist and are sufficient to generate all the measurable functions on the ym2 measure space

on the formulation of the feynman path integral through broken line paths we study the formulation of the feynman path integral through broken line paths in non relativistic quantum mechanics this formulation is very familiar to us and well known to be useful but its rigorous meaning is given little except for special cases in the present paper using the ideas in the theory of difference methods and the theory of pseudo differential operators we show rigorously for some class of potentials that this formulation is well defined and that this feynman path integral gives the probability amplitude i e the solution of the schrodinger equation

lp contractive projections and the heat semigroup for differential forms a projection operator that is contractive on lp for two distinct values of p is shown to be contractive for all values of p and the range must be of a special form this result is used to show that the heat semigroup for k forms on many manifolds with nontrivial cohomology in dimension k cannot be contractive on any lp for p 2

conditioning and initial enlargement of filtration on a riemannian manifold we extend to riemannian manifolds the theory of conditioned stochastic differential equations we also provide some enlargement formulas for the brownian filtration in this nonflat setting

path integrals and the essential self adjointness of differential operators on noncompact manifolds we consider schrodinger operators on possibly noncompact riemannian manifolds acting on sections in vector bundles with locally square integrable potentials whose negative part is in the underlying kato class using path integral methods we prove that under geodesic completeness these differential operators are essentially self adjoint on mathsf c infty 0 and that the corresponding operator closures are semibounded from below these results apply to nonrelativistic pauli dirac operators that describe the energy of hydrogen type atoms on riemannian 3 manifolds

isoparametric functions on riemannian manifolds i on etablit les proprietes de base des fonctions transnormales et on demontre le lemme du tube on demontre que soit m une variete de riemann lisse complete connexe et f une fonction transnormale sur m alors a les varietes focales de f sont des sous varietes lisses de m b chaque ensemble de niveau regulier de f est un tube sur d autres des varietes focales

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

multiple positive solutions for nth order impulsive integro differential equations in banach spaces abstract in this paper the author obtains the existence of multiple positive solutions for a boundary value problem of n th order nonlinear impulsive integro differential equations on an infinite interval in a banach space by means of fixed point index theory of completely continuous operators

a cameron martin type quasi invariance theorem for pinned brownian motion on a compact riemannian manifold the results in driver 13 for quasi invariance of wiener measure on the path space of a compact riemannian manifold m are extended to the case of pinned wiener measure to be more explicit let h 0 1 tom be a c1 function where m is a compact riemannian manifold o e m is a base point and tom is the tangent space to m at o e m let w m be the space of continuous paths from 0 1 into m v be wiener measure on w m concentrated on paths starting at o e m and hs w denote the stochastic parallel translation operator along a path to e w m up to time s note hs w is only well defined up to v equivalence for c e w m let xh w denote the vector field along a given by xsh c hs cw h s for each s e 0 1 one should interpret xh as a vector field on w m the vector field xh induces a flow sh t w m w m which leaves wiener measure v quasi invariant see driver 13 it is shown in this paper that the same result is valid if h l 0 and the wiener measure v is replaced by a pinned wiener measure ive the measure ve is proportional to the measure v conditioned on the set of paths which start at 0 e m and end at a fixed end point e e m also as in 13 one gets an integration by parts formula for the vector fields xh defined above

existence and uniqueness of solutions for a class of semilinear parabolic pdes with non lipschitz coefficients in this paper the existence and uniqueness of solutions for a class of semilinear parabolic partial differential equations with non lipschitz coefficients on riemannian manifold are obtained two non lipschitz functions are provided to show our results

finite dimensional approximations to wiener measure and path integral formulas on manifolds abstract certain natural geometric approximation schemes are developed for wiener measure on a compact riemannian manifold these approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi group on riemannian manifolds the path space is approximated by finite dimensional manifolds h p m consisting of piecewise geodesic paths adapted to partitions p of 0 1 the finite dimensional manifolds h p m carry both an h 1 and a l 2 type riemannian structures g 1 p and g 0 p respectively it is proved that 1 z i p e 1 2 e d vol g i p i d as mesh p 0 where e is the energy of the piecewise geodesic path h p m and for i 0 and 1 z i p is a normalization constant vol g i p is the riemannian volume form relative to g i p and is wiener measure on paths on m here 1 1 and 0 exp 1 6 1 0 scal s ds where scal is the scalar curvature of m these results are also shown to imply the well known integration by parts formula for the wiener measure

integration by parts formulas and dilatation vector fields on elliptic probability spaces by coupling two arbitrary riemannian connections and on a riemannian manifold m we perform the stochastic calculus of variation on the path space p m of the manifold m the method uses direct calculations on ito s stochastic differential equations in this context we obtain intertwinning formulas with the ito map for first order operators on the path space p m of m by a judicious choice of the second connection in terms of the connection we can prolongate the intertwinning formulas to second order differential operators thus we obtain expressions of heat operators on the path space p m of a riemannian manifold m endowed with an arbitrary connection the integration by parts of the laplacians on p m leads us to the notion of dilatation vector field on the path space

on generalized schrodinger semigroups abstract we prove a feynman kac formula for schrodinger type operators on vector bundles over arbitrary riemannian manifolds where the potentials are allowed to have strong singularities like those that typically appear in atomic quantum mechanical problems this path integral formula is then used to prove several l p type results like bounds on the ground state energy and l 2 l p smoothing properties of the corresponding schrodinger semigroups as another main result we will prove that with a little control on the riemannian structure the latter semigroups are also l 2 bounded continuous smoothing for kato decomposable potentials

lpcontraction semigroups for vector valued functions abstract we discuss semigroups acting on vector valued functions we consider a comparison theorem between two semigroups a semigroup acting on scalar valued functions and a semigroup acting on vector valued functions the criterion is given by the abstract kato theorem by using this we give a sufficient condition for the criterion in the setting of square field operator we also consider the essential self adjointness of a perturbed semigroup

zero sets of solutions to semilinear elliptic systems of first order consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over an n dimensional manifold assume the operator has the strong unique continuation property we show that the zero set of the solution is contained in a countable union of smooth n 2 dimensional submanifolds hence it is countably n 2 rectifiable and its hausdorff dimension is at most n 2 moreover it has locally finite n 2 dimensional hausdorff measure we show by example that every real number between 0 and n 2 actually occurs as the hausdorff dimension for a suitable choice of operator we also derive results for scalar elliptic equations of second order

positive solutions for nth order nonlinear impulsive singular integro differential equations on infinite intervals in banach spaces abstract in this paper the cone theory and monch fixed point theorem combined with a monotone iterative technique are used to investigate the positive solutions of a class of boundary problems for n th order nonlinear impulsive singular integro differential equations of mixed type on an infinite interval in banach spaces the conditions for the existence of a positive solution are established in addition an explicit iterative approximation of the solution for the boundary value problem is derived

path integrals on manifolds with boundary we give time slicing path integral formulas for solutions to the heat equation corresponding to a self adjoint laplace type operator acting on sections of a vector bundle over a compact riemannian manifold with boundary more specifically we show that such a solution can be approximated by integrals over finite dimensional path spaces of piecewise geodesics subordinated to increasingly fine partitions of the time interval we consider a subclass of mixed boundary conditions which includes standard dirichlet and neumann boundary conditions

path integrals supersymmetric quantum mechanics and the atiyah singer index theorem for twisted dirac feynman s time slicing construction approximates the path integral by a product determined by a partition of a finite time interval of approximate propagators this paper formulates general conditions to impose on a short time approximation to the propagator in a general class of imaginary time quantum mechanics on a riemannian manifold which ensure that these products converge the limit defines a path integral which agrees pointwise with the heat kernel for a generalized laplacian the result is a rigorous construction of the propagator for supersymmetric quantum mechanics with potential as a path integral further the class of laplacians includes the square of the twisted dirac operator which corresponds to an extension of n 1 2 supersymmetric quantum mechanics general results on the rate of convergence of the approximate path integrals suffice in this case to derive the local version of the atiyah singer index theorem

on the existence of solutions for nonlinear first order implicit impulsive integro differential equations abstract by using the monch fixed point theorem the purpose of this paper is to obtain the new existence theorems of solutions for a new class of initial value problems of nonlinear first order implicit impulsive integro differential equations in banach spaces under some weaker conditions

finite dimensional approximation of riemannian path space geometry abstract in this paper we construct a finite dimensional approximation for the geometry on the path space over a compact riemannian manifold this approximation allows to construct the horizontal lift of the ornstein uhlenbeck process on the path space through the markovian connection we also prove a representation formula for the heat semigroup on adapted vector fields as well as a commutation formula for its derivative

index theorem and equivariant cohomology on the loop space in this paper extending ideas of witten and atiyah we describe some relations of equivariant cohomology on the loop space of a manifold to the path integral representation of the index of the dirac operator on a twisted spin complex in particular the natural extension of the chern character to the whole loop space is described also it is shown that the non zero free homotopy classes of the loop space have 0 measure for the index measures associated to the index problem

gradient estimates for harmonic functions on regular domains in riemannian manifolds abstract derivative formulae for heat semigroups are used to give gradient estimates for harmonic functions on regular domains in riemannian manifolds this probabilistic method provides an alternative to coupling techniques as introduced by cranston and allows us to improve some known estimates we discuss two slightly different ways to exploit derivative formulae where each one should be interesting by itself

the witten complex and the degenerate morse inequalities on utilise le complexe introduit par witten pour demontrer des inegalites de morse dans le cas ou les points critiques de la fonction de morse sont isoles et aussi les inegalites de morse degenerees de bott

path integral formulae for heat kernels and their derivatives the heat kernel and its derivatives of a vector laplacian on the sections of a bundle over a compact riemannian manifold are expressed as products of the scalar heat kernel of the manifold and path integrals over the brownian bridge the small time asymptotics of these integrals are computed

strong p completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds here we discuss the regularity of solutions of sde s and obtain conditions under which a sde on a complete riemannian manifoldm has a global smooth solution flow in particular improving the usual global lipschitz hypothesis whenm r n there are also results on non explosion of diffusions

stochastic differential equations on noncompact manifolds moment stability and its topological consequences in this paper we discuss the stability of stochastic differential equations and the interplay between the moment stability of a sde and the topology of the underlying manifold sufficient and necessary conditions are given for the moment stability of a sde in terms of the coefficients finally we prove a vanishing result for the fundamental group of a complete riemannian manifold in terms of purely geometrical quantities

extremal solutions for nth order impulsive integro differential equations on the half line in banach spaces abstract by establishing a comparison result and using the upper and lower solutions the author obtains the existence of maximal and minimal solutions for a boundary value problem of n th order nonlinear impulsive integro differential equations on the half line in banach spaces

quasi invariance of the wiener measure on the path space over a complete riemannian manifold we prove a generalization of the cameron martin theorem for a geometrically and stochastically complete riemannian manifold namely the wiener measure on the path space over such a manifold is quasiinvariant under the flow generated by a cameron martin vector field

isoparametric hypersurfaces in complex hyperbolic spaces we review some results of an ongoing research on isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in the complex hyperbolic space in order to motivate this topic we recall first the main steps of its long history

a cameron martin type quasi invariance theorem for brownian motion on a compact riemannian manifold abstract let m be a compact manifold without boundary o be a fixed base point in m g be a riemannian metric on m and be a g compatible covariant derivative on tm the tangent space to m assume the torsion t of satisfies the skew symmetry condition g t x y y 0 for all vector fields x and y on m for example take to be the levi civita covariant derivative on m g also let denote the wiener measure on w0 m c 0 1 m 0 o and let h s denote stochastic parallel translation relative to along the path w0 m up to time s 0 1 given a c1 function h 0 1 t0m it is shown that the differential equation t h t h with initial condition 0 id w m w m has a solution r maps w m w m the measurable maps from w m to w m this function is a flow on w m i e for all t r t t furthermore t has the quasi invariance property the law t of t with respect to the wiener measure is equivalent to for all t r this result is used to prove an integration by parts formula for the h derivative hf defined by h f d dt 0 f t where f is a c2 cylinder function on w m

convexity of solutions of semilinear elliptic equations on considere le probleme elliptique suivant u f u dans u m sur ou est un domaine convexe de r 2 et m est soit une constante reelle soit on etablit que pour une bonne fonction monotone g t pout toute solution u on a g u est strictement convexe dans

minimal nonnegative solutions for nth order integro differential equations in banach spaces1this research work was supported by the national natural science foundation of china and the doctoral program foundation of the state education commission of china 1 in this paper the cone theory and monotone iterative technique are used to investigate the minimal nonnegative solutions of initial value problems ivps for nth order nonlinear integro differential equations of mixed type on an infinite interval in a banach space

an approximation to wiener measure and quantization of the hamiltonian on manifolds with non positive sectional curvature this paper gives a rigorous interpretation of a feynman path integral on a riemannian manifold m with non positive sectional curvature an l2 riemannian metric gp is given on the space of piecewise geodesic paths hp m adapted to the partition p of 0 1 whence a finite dimensional approximation of wiener measure is developed it is proved that as mesh p 0 the approximate wiener measure converges in an l1 sense to the measure exp 2 3203 01scal s ds d on the wiener space w m with wiener measure this gives a possible prescription for the path integral representation of the quantized hamiltonian as well as yielding such a result for the natural geometric approximation schemes originating in andersson and driver 1999 3 and followed by lim 2007 34

short time asymptotics of a rigorous path integral for n 1 supersymmetric quantum mechanics on a riemannian manifold following feynman s prescription for constructing a path integral representation of the propagator of a quantum theory a short time approximation to the propagator for imaginary time n 1 supersymmetric quantum mechanics on a compact even dimensional riemannian manifold is constructed the path integral is interpreted as the limit of products determined by a partition of a finite time interval of this approximate propagator the limit under refinements of the partition is shown to converge uniformly to the heat kernel for the laplace de rham operator on forms a version of the steepest descent approximation to the path integral is obtained and shown to give the expected short time behavior of the supertrace of the heat kernel

existence of solutions for nth order integro differential equations in banach spaces abstract this paper discusses the existence of solutions of initial value problems for n th order nonlinear integro differential equations of mixed type on an infinite interval in a banach space

semi classical limit of the bottom of spectrum of a schrodinger operator on a path space over a compact riemannian manifold abstract we determine the limit of the bottom of spectrum of schrodinger operators with variable coefficients on wiener spaces and path spaces over finite dimensional compact riemannian manifolds in the semi classical limit these are extensions of the results in s aida semiclassical limit of the lowest eigenvalue of a schrodinger operator on a wiener space j funct anal 203 2 2003 401 424 the problem on path spaces over riemannian manifolds is considered as a problem on wiener spaces by using ito s map however the coefficient operator is not a bounded linear operator and the dependence on the path is not continuous in the uniform convergence topology if the riemannian curvature tensor on the underling manifold is not equal to 0 the difficulties are solved by using unitary transformations of the schrodinger operators by approximate ground state functions and estimates in the rough path analysis

stochastic analysis on the path space of a riemannian manifold i markovian stochastic calculus abstract on the brownian flow of a compact riemannian manifold an intrinsic stochastic calculus is defined this calculus is markovian the stochastic calculus of variation on the bundle of orthonormal frames appears only as an intermediate tool the bismut connection plays a paramount role in the final construction

renormalized differential geometry on path space structural equation curvature abstract the theory of integration in infinite dimensions is in some sense the backbone of probability theory on this backbone the stochastic calculus of variations has given rise to the flesh of differential calculus its first step is the construction at each point of the probability space of a cameron martin like tangent space in which the desired differential calculus can be developed this construction proceeds along the lines of first order differential geometry in this paper we address the following questions what could be the meaning of curvature of the probability space how and why how can curvatures be defined and computed why could a second order differential geometry be relevant to stochastic analysis we try to answer these questions for the probability space associated to the brownian motion of a compact riemannian manifold why a basicenergy identity for anticipative stochastic integralswill be obtained as a byproduct of our computation of curvature how there are essentially four bottlenecks in the development of differential geometry on wiener riemann manifolds i the difficulty of finding an atlas of local charts such that the changes of charts preserve the class of the wiener like measures together with their associated cameron martin like tangent spaces ii the difficulty of finding cylindrical approximations preserving the natural geometrical objects iii the difficulty of renormalizing the divergent series to which the summation operations of finite dimensional differential geometry give rise in the non intrinsic context of local charts iv the nonavailability of the computational procedures analogous to the local coordinates systems of the classical differential geometry in the context of path space the ito filtration provides a much richer structure than that available in the framework of an abstract wiener riemann manifold our work is a systematic attempt to replace the machinery of local charts with amethodology of moving frames in our context stochastic parallel transport provides a canonical moving frame on the path space the concept of a cylindrical approximation has to be reshaped in our new situation into some geometric limit theorems establishing that the riemannian geometric objects of the cylindrical approximations induce by a limiting procedure geometric objects on the path space those limit theorems are reminiscent of the classical theorems which say that a stratonovich sde is the limit of an appropriate sequence of ode the canonical coordinate system provided by the moving frame will make it possible to proceed to the needed renormalizations byintrinsicstochastic integrals in this context the anticipative stochastic integral theory of nualart and pardoux will play a decisive role finally the moving frame will provide aneffective algorithm of computation for this differential geometry in infinite dimensions in our study we encounter a new type of renormalization thehypoelliptic renormalization which corresponds to the fact that the bracket of smooth vector fields taking their values in the cameron martin space can get out of this cameron martin space this hypoelliptic problem induces the nonrenormalizability of some geometrical objects it leads also to a concept oftangent processesto probability spaces extending that based on the cameron martin theorem for tangent processes a formula of integration by parts still holds furthermore the tangent processes form a lie algebra under the bracket on the other hand tangent processes cannot be stochastically integrated this operation is well defined only for cameron martin type vector fields

a class of second order impulsive integro differential equations on unbounded domain in a banach space by establishing a comparison result the author obtains the existence of maximal and minimal solutions of the initial value problem ivp for a class of second order impulsive integro differential equations in a banach space

heat equation and compactifications of complete riemannian manifolds soit m une variete de riemann complete dont la courbure de ricci est bornee inferieurement soit m une compactification de m soit f c m on cherche des fonctions f x t c m 0 satisfaisant t f x t 0 x t m 0 f x 0 f x x m f x t f x x t m 0

integration by parts and quasi invariance for heat kernel measures on loop groups abstract integration by parts formulas are established both for wiener measure on the path space of a loop group and for the heat kernel measures on the loop group the wiener measure is defined to be the law of a certain loop group valued brownian motion and the heat kernel measures are time t t 0 distributions of this brownian motion a corollary of either of these integrations by parts formulas is the closability of the pre dirichlet form considered by b k driver and t lohrenz 1996 j functional anal 140 381 448 we also show that the heat kernel measures are quasi invariant under right under right and left translations by finite energy loops

the chern gauss bonnet theorem via supersymmetric euclidean field theories we prove the chern gauss bonnet theorem using sigma models whose source supermanifolds have super dimension 0 2 along the way we develop machinery for understanding manifold invariants encoded by families of 0 dimensional euclidean field theories and their quantization

discrete approximations to integrals over unparametrized paths we discuss measures on spaces of unparametrized paths related to the wiener measure these measures arise naturally in the study of one dimensional gravity coupled to scalar fields two kinds of discrete approximations are defined the piecewise linear and the hypercubic approximations the convergence of these approximations in the sense of weak convergence of measures is proven we describe a family of sets of unparametrized paths that are analogous to cylinder sets of parametrized paths integrals over some of these sets are evaluated in terms of dirichlet propagators in bounded regions

infinite boundary value problems for nth order nonlinear impulsive integro differential equations in banach spaces abstract in this paper the existence of unbounded nonnegative solutions of a boundary value problem for n th order impulsive integro differential equations on an infinite interval in banach spaces is obtained by means of the monch fixed point theory in a banach space

a dynamical zeta function for pseudo riemannian foliations we investigate a generalization of geodesic random walks to pseudo riemannian foliations the main application we have in mind is to consider the logarithm of the associated zeta function as grand canonical partition function in a theory unifying aspects of general relativity quantum mechanics and dynamical systems

stochastic jacobi fields and vector fields induced by varying area on path spaces we study two classes of vector fields on the path space over a closed manifold with a wiener riemannian measure by adopting the viewpoint of yang mills field theory we study a vector field defined by varying a metric connection we prove that the vector field obtained in this way satisfies a jacobi field equation which is different from that of classical one by taking in account that a brownian motion is invariant under the orthogonal group action so that it is a geometric vector field on the space of continuous paths and induces a quasi invariant solution flow on the path space the second object of this paper is vector fields obtained by varying area here we follow the idea that a continuous semimartingale is indeed a rough path consisting of not only the path in the classical sense but also its levy area we prove that the vector field obtained by parallel translating a curve in the initial tangent space via a connection is just the vector field generated by translating the path along a direction in the cameron martin space in the malliavin calculus sense and at the same time changing its levy area in an appropriate way this leads to a new derivation of the integration by parts formula on the path space

the topology of asymptotically euclidean static perfect fluid space time it is shown that a geodesically complete asymptotically euclidean static perfect fluid space time satisfying the time like convergence condition and having a connected fluid region is diffeomorphic to 3

asymptotics of spectral gaps on loop spaces over a class of riemannian manifolds abstract we prove the existence of spectral gaps of ornstein uhlenbeck operators on loop spaces over a class of riemannian manifolds which include hyperbolic spaces this is an alternative proof and an extension of a result in chen et al 13 further we study the asymptotic behavior of the spectral gap

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

a poincare lemma for connection forms denote by p the space of piecewise smooth curves in rn beginning at the origin a path 2 form is a function h on p such that for each element in p h is a 2 form at the endpoint of with values in a lie algebra g for example if a is a smooth g valued connection form on rn with curvature f and parallel translation operator p then the equation la p 1 f 1 p defines la as a path 2 form a necessary and sufficient condition is given to characterize those path 2 forms which arise in this way by way of application it is shown that the birula mandelstam generalization of maxwell s equations to nonabelian gauge fields is equivalent to the yang mills equation

positive solutions of an infinite boundary value problem for nth order nonlinear impulsive singular integro differential equations in banach spaces abstract in this paper the cone theory and monch fixed point theorem combined with a monotone iterative technique are used to investigate the positive solutions of a class of boundary problems for n th order nonlinear impulsive singular integro differential equations of mixed type on an infinite interval in banach spaces the conditions for the existence of a positive solution are established in addition an explicit iterative approximation of the solution for the boundary value problem is derived

method of upper and lower solutions for second order impulsive integro differential equations in a banach space abstract this paper discusses the initial value problem for second order nonlinear impulsive integro differential equations of volterra type in a banach space by means of upper and lower solutions by establishing a comparison result criteria on the existence of maximal and minimal solutions are obtained

stability of hypersurfaces of constant mean curvature in riemannian manifolds hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 display many similarities with minimal hypersurfaces of overline m n 1 they are both solutions to the variational problem of minimizing the area function for certain variations in the first case however the admissible variations are only those that leave a certain volume function fixed for precise definitions see sect 2 this isoperimetric character of the variational problem associated to hypersurfaces of constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces