isospectral conformal metrics on 3 manifolds theorem let g uj go be a sequence of conformal metrics satisfying the following conditions i vol m gj ao for some positive constant ao ii f r2 gj lp gj 12 dvj a 0 i e for each q defined on m we have

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

sharp constants for moser trudinger inequalities on spheres in complex space ℂn the main results of this paper concern sharp constants for the moser trudinger inequalities on spheres in complex space c n we derive moser trudinger inequalities for smooth functions and holomorphic functions with different sharp constants see theorem 1 1 the sharp moser trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the cr setting though there is a close connection in spirit between inequalities proven here on complex spheres and those on the heisenberg group for functions with compact support in any finite domain proven earlier by the same authors 17 derivation of the sharp constants for moser trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the heisenberg group variants of moser onofri type inequalities are also given on complex spheres as applications of our sharp inequalities see theorems 1 2 and 1 3 one of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients see theorem 1 4 c 2004 wiley periodicals inc

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

convergence behavior of interior point algorithms we show that most interior point algorithms for linear programming generate a solution sequence in which every limit point satisfies the strict complementarity condition these algorithms include all path following algorithms and some potential reduction algorithms the result also holds for the monotone complementarity problem if a strict complementarity solution exists in general the limit point is a solution that maximizes the number of its nonzero components among all solutions

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

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

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

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

compactification of a class of conformally flat 4 manifold in this paper we generalize huber s result on complete surfaces of finite total curvature for complete locally conformally flat 4 manifolds of positive scalar curvature with q curvature integrable where q is a variant of the chern gauss bonnet integrand we first derive the cohn vossen inequality we then establish finiteness of the topology this allows us to provide conformal compactification of such manifolds

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

an extension of karmarkar projective algorithm for convex quadratic programming we present an extension of karmarkar s linear programming algorithm for solving a more general group of optimization problems convex quadratic programs this extension is based on the iterated application of the objective augmentation and the projective transformation followed by optimization over an inscribing ellipsoid centered at the current solution it creates a sequence of interior feasible points that converge to the optimal feasible solution in o ln iterations each iteration can be computed in o ln3 arithmetic operations wheren is the number of variables andl is the number of bits in the input in this paper we emphasize its convergence property practical efficiency and relation to the ellipsoid method

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

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

a polynomial arc search interior point algorithm for convex quadratic programming abstract arc search is developed for linear programming in 24 25 the algorithms search for optimizers along an ellipse that is an approximation of the central path in this paper the arc search method is applied to primal dual path following interior point method for convex quadratic programming a simple algorithm with iteration complexity o n log 1 is devised several improvements on the simple algorithm which improve computational efficiency increase step length and further reduce duality gap in every iteration are then proposed and implemented it is intuitively clear that the iteration with these improvements will reduce the duality gap more than the iteration of the simple algorithm without the improvements though it is hard to show how much these improvements reduce the complexity bound the proposed algorithm is implemented in matlab and tested on quadratic programming problems originating from 13 the result is compared to the one obtained by loqo in 22 the proposed algorithm uses fewer iterations in all these problems and the number of total iterations is 27 fewer than the one obtained by loqo this preliminary result shows that the proposed algorithm is promising

the nonlinear geometry of linear programming i affine and projective scaling trajectories this series of papers studies a geometric structure underlying karmarkar s projective scaling algorithm for solving linear programming problems a basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program the geometric structure studied is the set of trajectories obtained by integrating this vector field which we call ptrajectories we also study a related vector field the affine scaling vector field and its associated trajectories called a trajectories the affine scaling vector field is associated to another linear programming algorithm the affine scaling algorithm affine and projective scaling vector fields are each defined for linear programs of a special form called strict standard form and canonical form respectively this paper derives basic properties of p trajectories and a trajectones it reviews the projective and affine scaling algorithms defines the projective and affine scaling vector fields and gives differential equations for p trajectories and a trajectories it shows that projective transformations map p trajectories into p trajectories it presents karmarkar s interpretation of a trajectories as steepest descent paths of the objective function c x with respect to the riemannian geometry ds2 z dx dx x2 restricted to the relative interior of the polytope of feasible solutions p trajectories of a canonical form linear program are radial projections of a trajectories of an associated standard form linear program as a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program this algorithm is essentially karmarkar s algorithm these trajectories are studied in subsequent papers by two nonlinear changes of variables called legendre transform coordinates and projective legendre transform coordinates respectively it will be shown that p trajectories have an algebraic and a geometric interpretation they are algebraic curves and they are geodesics actually distinguished chords of a geometry isometric to a hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions the a trajectories of strict standard form linear programs have similar interpretations they are algebraic curves and are geodesics of a geometry isometric to euclidean geometry received by the editors july 28 1986 and in revised form september 28 1987 and march 21 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 research of the first author partially supported by onr contract n00014 87 k0214 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

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 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

regularity of a fourth order nonlinear pde with critical exponent in this paper we demonstrate the regularity of minimizers for a variational problem a special case of which arises in spectral theory and conformal geometry the associated euler lagrange equation is fourth order semilinear the leading term is the bilaplacian and lower order terms appear at critical powers

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

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

extremal metrics for spectral functions of dirac operators in even and odd dimensions abstract let m n g be a closed smooth riemannian spin manifold and denote by its atiyah singer dirac operator we study the variation of riemannian metrics for the zeta function and functional determinant of 2 and prove finiteness of the morse index at stationary metrics and local extremality at such metrics under general i e not only conformal change of metrics in even dimensions which is also a new case for the conformal laplacian the relevant stability operator is of log polyhomogeneous pseudodifferential type and we prove new results of independent interest on the spectrum for such operators we use this to prove local extremality under variation of the riemannian metric which in the important example when m n g is the round n sphere gives a partial verification of branson s conjecture on the pattern of extremals thus det 2 has a local max max min min when the dimension is 4 k 4 k 1 4 k 2 4 k 3 respectively

convergence of the q curvature flow on s4 in this paper we study the q curvature flow on the standard sphere s4 we show that the flow converges exponentially for all initial data

a polynomial time primal dual affine scaling algorithm for linear and convex quadratic programming and its power series extension we describe an algorithm for linear and convex quadratic programming problems that uses power series approximation of the weighted barrier path that passes through the current iterate in order to find the next iterate if r 1 is the order of approximation used we show that our algorithm has time complexity o n 1 1 r l 1 1 r iterations and o n3 n2r arithmetic operations per iteration where n is the dimension of the problem and l is the size of the input data when r 1 we show that the algorithm can be interpreted as an affine scaling algorithm in the primal dual setup

topological degree for solutions of fourth order mean field equations we consider the following fourth order mean field equation with navier boundary condition delta 2 u rho frac h x e u int omega h e u rm in omega quad u delta u 0 rm on partial omega quad quad quad quad quad quad quad quad quad quad where h is a c2 positive function is a bounded and smooth domain in mathbb r 4 we prove that for rho in 32m sigma 3 32 m 1 sigma 3 the degree counting formula for is given by d rho left begin array ll frac 1 m chi omega 1 cdot cdot cdot chi omega m rm for m 0 1 rm for m 0 end array right where is the euler characteristic of similar result is also proved for the corresponding dirichlet problem delta 2 u rho frac h x e u int omega h e u quad rm in omega quad u nabla u 0 quad rm on partial omega quad quad quad quad quad quad quad quad quad quad

bicr variants of the hybrid bicg methods for solving linear systems with nonsymmetric matrices we propose bi conjugate residual bicr variants of the hybrid bi conjugate gradient bicg methods referred to as the hybrid bicr variants for solving linear systems with nonsymmetric coefficient matrices the recurrence formulas used to update an approximation and a residual vector are the same as those used in the corresponding hybrid bicg method but the recurrence coefficients are different they are determined so as to compute the coefficients of the residual polynomial of bicr from our experience it appears that the hybrid bicr variants often converge faster than their bicg counterpart numerical experiments show that our proposed hybrid bicr variants are more effective and less affected by rounding errors the factor in the loss of convergence speed is analyzed to clarify the difference of the convergence between our proposed hybrid bicr variants and the hybrid bicg methods

a critical elliptic problem for polyharmonic operators abstract in this paper we study the existence of solutions for a critical elliptic problem for polyharmonic operators we prove the existence result in some general domain by minimizing on some infinite dimensional finsler manifold for some suitable perturbation of the critical nonlinearity when the dimension of domain is larger than critical one for the critical dimensions we prove also the existence of solutions in domains perforated with the small holes some unstable solutions are obtained at higher level sets by coron s topological method provided that the minimizing solution does not exist

heat kernel expansions ambient metrics and conformal invariants abstract the conformal powers of the laplacian of a riemannian metric which are known as the gjms operators admit a combinatorial description in terms of the taylor coefficients of a natural second order one parameter family h r g of self adjoint elliptic differential operators h r g is a non laplace type perturbation of the conformal laplacian p 2 g h 0 g it is defined in terms of the metric g and covariant derivatives of the curvature of g we study the heat kernel coefficients a 2 k r g of h r g on closed manifolds we prove general structural results for the heat kernel coefficients a 2 k r g and derive explicit formulas for a 0 r g and a 2 r g in terms of renormalized volume coefficients the taylor coefficients of a 2 k r g as functions of r interpolate between the renormalized volume coefficients of a metric g k 0 and the heat kernel coefficients of the conformal laplacian of g r 0 although h r g is not conformally covariant there is a beautiful formula for the conformal variation of the trace of its heat kernel its proof rests on the combinatorial relations between taylor coefficients of h r g and gjms operators as a consequence we give a heat equation proof of the conformal transformation law of the integrated renormalized volume coefficients by refining these arguments we also give a heat equation proof of the conformal transformation law of the renormalized volume coefficients itself the taylor coefficients of a 2 r define a sequence of higher order riemannian curvature functionals with extremal properties at einstein metrics which are analogous to those of integrated renormalized volume coefficients among the various additional results the reader finds a polyakov type formula for the renormalized volume of a poincare einstein metric in terms of q curvature of its conformal infinity and additional holographic terms the present study of relations between spectral theoretic heat kernel coefficients and geometric quantities like renormalized volume coefficients is stimulated by the holographic perspective of the ads cft duality proposing relations between functional determinants and renormalized volumes

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

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