phase ordering from off critical quenches and the measurement of the dynamic exponent lambda the ordering dynamics of a system with a nonconserved order parameter is considered following a quench into the ordered phase from high temperature the correlation of the order parameter field with its initial condition or more generally the correlation between the fields at different times involves the dynamic exponent lambda a non trivial exponent associated with the t 0 fixed point that drives phase ordering it is shown that lambda can be determined from the growth of the mean order parameter m t in a system which has a small but non zero mean order parameter m0 at t 0 since m t approximately m0l t lambda where l t approximately t1 2 is the characteristic length scale at time t the role of a weak external field h is also considered for h not 0 the growth of m t involves two power law terms m t approximately h at3 btlambda 2 for m0 0 with x 1 2 or 1 for a scalar or vector order parameter respectively the results are illustrated by the exact solution of the o n model at large n for general values of m0 and h

on the decomposition of global conformal invariants i this is the first of two papers where we address and partially confirm a conjecture of deser and schwimmer originally postulated in high energy physics the objects of study are scalar riemannian quantities constructed out of the curvature and its covariant derivatives whose integrals over compact manifolds are invariant under conformal changes of the underlying metric our main conclusion is that each such quantity that locally depends only on the curvature tensor without covariant derivatives can be written as a linear combination of the chern gauss bonnet integrand and a scalar conformal invariant

on conformal deformations of metrics onsn abstract on s n there is a naturally metric defined n th order conformal invariant operator p n associated with this operator is a so called q curvature quantity when two metrics are pointwise conformally related their associated operators together with their q curvatures satisfy the natural differential equations this paper is devoted to the question of which function can be a q curvature candidate this is the so called prescribing q curvature problem our main result is that if q is positive nondegenerate and the naturally defined mapping associated with q has nonzero degree then our problem has a solution this is the natural generalization of prescribing gaussian curvature on s 2 into s n

the profile of bubbling solutions of a class of fourth order geometric equations on 4 manifolds abstract we study a class of fourth order geometric equations defined on a 4 dimensional compact riemannian manifold which includes the q curvature equation we obtain sharp estimates on the difference near the blow up points between a bubbling sequence of solutions and the standard bubble

algebraic topological methods for the supercritical q curvature problem abstract we study the problem of existence of conformal metrics with prescribed q curvature on closed four dimensional riemannian manifolds this problem has a variational structure and in the case of interest here it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated euler lagrange functional the so called true critical points at infinity of the associated variational problem occur using the characterization of the critical points at infinity of the associated variational problem which is established in 42 combined with some arguments from morse theory some algebraic topological methods and some tools from dynamical system originating from conley s isolated invariant sets and isolated blocks theory we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold stable and unstable manifolds of some of the critical points at infinity of the associated euler lagrange functional

global existence and convergence for a higher order flow in conformal geometry an important problem in conformal geometry is the construction of conformal metrics for which a certain curvature quantity equals a prescribed function e g a constant in two dimensions the uniformization theorem assures the existence of a conformal metric with constant gauss curvature moreover j moser 20 proved that for every positive function f on s2 satisfying f x f x for all x e s2 there exists a conformal metric on s2 whose gauss curvature is equal to f a natural conformal invariant in dimension four is 1 2 q ar r2 3 irici2 where r denotes the scalar curvature and ric the ricci tensor this formula can also be written in the form

constant q curvature metrics in arbitrary dimension working in a given conformal class we prove existence of constant q curvature metrics on compact manifolds of arbitrary dimension under generic assumptions the problem is equivalent to solving a nth order non linear elliptic differential or integral equation with variational structure where n is the dimension of the manifold since the corresponding euler functional is in general unbounded from above and below we use critical point theory jointly with a compactness result for the above equation

a new conformal invariant on 3 dimensional manifolds abstract by improving the analysis developed in the study of k yamabe problem we prove in this paper the following new inequality for any 3 dimensional riemannian manifold m 3 g of non negative scalar curvature m 1 g d v g 2 3 vol g m 2 g d v g equality holds if and only if m 3 g is a space form

volume comparison and the σk yamabe problem abstract in this paper we study the problem of finding a conformal metric with the property that the k th elementary symmetric polynomial of the eigenvalues of its weyl schouten tensor is constant a new conformal invariant involving maximal volumes is defined and this invariant is then used in several cases to prove existence of a solution and compactness of the space of solutions provided the conformal class admits an admissible metric in particular the problem is completely solved in dimension four and in dimension three if the manifold is not simply connected

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

compactness of solutions to some geometric fourth order equations we prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds the proof is based on a refined bubbling analysis for which the main estimates are given in integral form our result is used in a subsequent paper to find critical points via minimax arguments of some geometric functional which give rise to conformal metrics of constant q curvature as a byproduct of our method we also obtain compactness of such metrics

the q curvature on a 4 dimensional riemannian manifold m g with mqdvg 8π2 abstract we deal with the q curvature problem on a 4 dimensional compact riemannian manifold m g with m q g d v g 8 2 and positive paneitz operator p g let q be a positive smooth function the question we consider is when can we find a metric g which is conformal to g such that q is just the q curvature of g a sufficient condition to this question is given in this paper

new exponent for dynamic correlations in domain growth the dynamics of the n component ginzburg landau model with non conserved order parameter model a are considered following a quench to zero temperature the correlation function of the order parameter field is found in the 1 n expansion to have the asymptotic scaling form ck t t t d2 t t lambda 2 f k2t k2t for t t with f 0 0 constant the new exponent lambda is calculated to o 1 n for general space dimension d and has a non trivial dependence on n and d

on the prescribed q curvature problem on sn in this paper we prescribe a fourth order curvature the q curvature on the standard n sphere n 5 under flatness condition of order n 4 n near each critical point of the prescribed q curvature function we characterize the critical points at infinity of the associated variational problem and we prove new existence results through euler hopf formulae type our argument gives an upper bound on the morse index of the obtained solutions we also give a lower bound on the number of conformal metrics having the same q curvature

non equilibrium dynamics of the ising model for t less than equal totc the growth of order in ising models with nonconserved order parameter is considered for quenches to final temperatures tf 0 and tf tc the results of numerical simulations in spatial dimension d 2 are presented in all cases a scaling regime is entered for sufficiently long times where the characteristic length scale is the domain size l t approximately t12 for tf 0 and the nonequilibrium correlation length xi t approximately t1z for tf tc the equal time correlation function has the expected scaling forms f r l t and r d 2 eta fc r xi t for tf 0 and tc respectively the scaling function fc x has interesting short distance behaviour which is elucidated using scaling arguments and by in and 1 n expansions the t 0 scaling function f x depends on whether the spin correlations present in the initial conditions are of long or short range as does the exponent which describes the decay of the autocorrelation function a t si t si o approximately l t results for a quench from the equilibrium critical state to tf 0 are consistent with theoretical predictions

generic multiplicity for a scalar field equation on compact surfaces we prove generic multiplicity of solutions for a scalar field equation on compact surfaces via morse inequalities in particular our result improves significantly the multiplicity estimate which can be deduced from the degree counting formula in chen and lin 2003 12 related results are derived for the prescribed q curvature equation

a fully nonlinear conformal flow on locally conformally flat manifolds we study a fully nonlinear flow for conformal metrics the long time existence and the sequential convergence of flow will be established for locally conformally flat manifolds as an application we solve the k yamabe problem for locally conformal flat manifolds when k n 2

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

existence of conformal metrics with constant q curvature given a compact four dimensional manifold we prove existence of conformal metrics with constant q curvature under generic assumptions the problem amounts to solving a fourth order nonlinear elliptic equation with variational structure since the corresponding euler functional is in general unbounded from above and from below we employ topological methods and min max schemes jointly with the compactness result of 35

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

on the decomposition of global conformal invariants ii abstract this paper is a continuation of s alexakis the decomposition of global conformal invariants i submitted for publication see also math dg 0509571 where we complete our partial proof of the deser schwimmer conjecture on the structure of global conformal invariants our theorem deals with such invariants p g n that locally depend only on the curvature tensor r i j k l without covariant derivatives in s alexakis the decomposition of global conformal invariants i ann of math in press we developed a powerful tool the super divergence formula which applies to any riemannian operator that always integrates to zero on compact manifolds in particular it applies to the operator i g n that measures the non conformally invariant part of p g n this paper resolves the problem of using this information we have obtained on the structure of i g n to understand the structure of p g n

second eigenvalue of paneitz operators and mean curvature for n 7 we give the optimal estimate for the second eigenvalue of paneitz operators for compact n dimensional submanifolds in an n p dimensional space form in terms of the mean curvature and the q curvature

transverse ordering in off critical quenches of a system with continuous symmetry the dynamics of a system quenched below the critical point with explicitly broken symmetry off critical quench is considered in the framework of an o n vector model in the large n limit with non conserved order parameter considering the behaviour of fluctuations in the transverse directions the author finds ordering and scaling of the transverse structure factor in the intermediate time regime between the usual early and late stages

the principal eigenvalue of a conformally invariant differential operator with an application to semilinear elliptic pde in this paper we give conditions under which the paneitz operator a conformally invariant fourth order differential operator is positive we also prove a sharp estimate for a related conformal invariant these results are combined to study the existence of a semilinear pde with critical sobolev growth

some properties of the schouten tensor and applications to conformal geometry the riemannian curvature tensor decomposes into a conformally invariant part the weyl tensor and a non conformally invariant part the schouten tensor a study of the kth elementary symmetric function of the eigenvalues of the schouten tensor was initiated in an earlier paper by the second author and a natural condition to impose is that the eigenvalues of the schouten tensor are in a certain cone k we prove that this eigenvalue condition for k n 2 implies that the ricci curvature is positive we then consider some applications to the locally conformally flat case in particular to extremal metrics of k curvature functionals and conformal quermassintegral inequalities using the results of the first and third authors

multiple solutions for a nonlinear and non homogeneous problem in orlicz sobolev spaces abstract we study a non homogeneous boundary value problem in a smooth bounded domain in r n we prove the existence of at least two non negative and non trivial weak solutions our approach relies on orlicz sobolev spaces theory combined with adequate variational methods and a variant of mountain pass lemma

geometric inequalities on locally conformally flat manifolds through the study of some elliptic and parabolic fully nonlinear pdes we establish conformal versions of the quermassintegral inequality the sobolev inequality and the moser trudinger inequality for the geometric quantities associated to the schouten tensor on locally conformally flat manifolds

classification of singular radial solutions to the σk yamabe equation on annular domains the study of the kth elementary symmetric function of the weyl schouten curvature tensor of a riemannian metric the so called k curvature has produced many fruitful results in conformal geometry in recent years in these studies in conformal geometry the deforming conformal factor is considered to be a solution of a fully nonlinear elliptic pde important advances have been made in recent years in the understanding of the analytic behavior of solutions of the pde however the singular behavior of these solutions which is important in describing many important questions in conformal geometry is little understood this note classifies all possible radial solutions in particular the singular solutions of the k yamabe equation which describes conformal metrics whose k curvature equals a constant although the analysis involved is of elementary nature these results should provide useful guidance in studying the behavior of singular solutions in the general situation

non uniqueness results for critical metrics of regularized determinants in four dimensions the regularized determinant of the paneitz operator arises in quantum gravity see connes in noncommutative geometry 1994 iv 4 an explicit formula for the relative determinant of two conformally related metrics was computed by branson in commun math phys 178 301 309 1996 a similar formula holds for cheeger s half torsion which plays a role in self dual field theory see juhl in families of conformally covariant differential operators q curvature and holography progress in mathematics vol 275 2009 and is defined in terms of regularized determinants of the hodge laplacian on p forms p n 2 in this article we show that the corresponding actions are unbounded above and below on any conformal four manifold we also show that the conformal class of the round sphere admits a second solution which is not given by the pull back of the round metric by a conformal map thus violating uniqueness up to gauge equivalence these results differ from the properties of the determinant of the conformal laplacian established in commun math phys 149 241 262 1992 ann math 142 171 212 1995 commun math phys 189 655 665 1997

on fractional gjms operators we describe a new interpretation of the fractional gjms operators as generalized dirichlet to neumann operators associated to weighted gjms operators on naturally associated smooth metric measure spaces this gives a geometric interpretation of the caffarelli silvestre extension for when 0 1 and both a geometric interpretation and a curved analogue of the higher order extension found by r yang for when 1 we give three applications of this correspondence first we exhibit some energy identities for the fractional gjms operators in terms of energies in the compactified poincare einstein manifold including an interpretation as a renormalized energy second for 1 2 we show that if the scalar curvature and the fractional q curvature q2 of the boundary are nonnegative then the fractional gjms operator p2 is nonnegative third by assuming additionally that q2 is not identically zero we show that p2 satisfies a strong maximum principle 2014 wiley periodicals inc