on witten s proof of the positive energy theorem this paper gives a mathematically rigorous proof of the positive energy theorem using spinors this completes and simplifies the original argument presented by edward witten we clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant dirac operator

deforming three manifolds with positive scalar curvature in this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact three manifold is path connected the proof uses the ricci flow with surgery the conformal method and the connected sum construction of gromov and lawson the work of perelman on hamilton s ricci flow is fundamental as one of the applications we prove the path connectedness of the space of trace free asymptotically flat solutions to the vacuum einstein constraint equations on r 3

subcritical approach to conformally invariant extension operators on the upper half space abstract in this work we obtain sharp embedding inequalities for a family of conformally invariant integral extension operators this family includes among others the classical poisson extension operator and the extension operator with riesz kernel we show that the sharp constants in these inequalities are attained and classify the corresponding extremal functions we also compute the limiting behavior at the boundary of the extensions of the extremal functions

a class of fully nonlinear equations arising from conformal geometry in this paper we study the long time existence of solutions for a class of fully nonlinear parabolic equations arising from conformal geometry in particular we prove that every smooth compact n dimensional manifold n ge3 admits a riemannian metric g with its ricci curvature ric and scalar curvature r satisfying det ric r cdot g const

on the singular second order elliptic equation abstract let m g be a smooth and compact riemannian manifold with dimensions n 3 this paper deals with the existence of nontrivial solutions for a nonlinear second order elliptic equation with a singular term and critical sobolev growth we provide a geometric application of this singular equation

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

the increasing cost tree search for optimal multi agent pathfinding we address the problem of optimal pathfinding for multiple agents given a start state and a goal state for each of the agents the task is to find minimal paths for the different agents while avoiding collisions previous work on solving this problem optimally used traditional single agent search variants of the a algorithm we present a novel formalization for this problem which includes a search tree called the increasing cost tree ict and a corresponding search algorithm called the increasing cost tree search icts that finds optimal solutions icts is a two level search algorithm the high level phase of icts searches the increasing cost tree for a set of costs cost per agent the low level phase of icts searches for a valid path for every agent that is constrained to have the same cost as given by the high level phase we analyze this new formalization compare it to the a search formalization and provide the pros and cons of each following we show how the unique formalization of icts allows even further pruning of the state space by grouping small sets of agents and identifying unsolvable combinations of costs experimental results on various domains show the benefits and limitations of our new approach a speedup of up to 3 orders of magnitude was obtained in some cases

conformal deformation to constant negative scalar curvature on noncompact riemannian manifolds si m g est une variete de riemann complete de courbure scalaire non positive s satisfaisant s x e 0 pour x m m 0 ou m 0 est un ensemble compact alors il y a une metrique conforme complete g de courbure scalaire s 1

newton okounkov bodies semigroups of integral points graded algebras and intersection theory generalizing the notion of newton polytope we dene the newtonokounkov body respectively for semigroups of integral points graded algebras and linear series on varieties we prove that any semigroup in the lattice z n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone applying this we obtain several results we show that for a large class of graded algebras the hilbert functions have polynomial growth and their growth coecients satisfy a brunn minkowski type inequality we prove analogues of the fujita approximation theorem for semigroups of integral points and graded algebras which imply a generalization of this theorem for arbitrary linear series applications to intersection theory include a far reaching generalization of the kushnirenko theorem from newton polytope theory and a new version of the hodge inequality we also give elementary proofs of the alexandrov fenchel inequality in convex geometry and its analogue in algebraic geometry

on the positive mass theorem for manifolds with corners we study the positive mass theorem for certain non smooth metrics following p miao s work our approach is to smooth the metric using the ricci flow as well as improving some previous results on the behaviour of the adm mass under the ricci flow we extend the analysis of the zero mass case to higher dimensions

invariants of conformal laplacians the conformal laplacian d d d n 2 s 4 n 1 acting on functions on a riemannian manifold mn with scalar curvature s is a conformally invariant operator in this paper we will use d to construct new conformal invariants one of these is a pointwise invariant one is the integral of a local expression and one is a nonlocal spectral invariant derived from functional determinants we begin in 1 by describing the laplacian d and its green function in the context of conformal geometry we then derive a basic formula giving the variation in the heat kernel of d this formula is strikingly simpler than the corresponding formula for the ordinary laplacian given by ray and singer 15 the heat kernel of d has an asymptotic expansion k t x x 4 t n 2 ak x tk in 2 we prove that a n 2 2 is a pointwise conformal invariant of weight 2 i e it satisfies a n 2 2 x 2g 2a n 2 2 x g where g is the metric and is any smooth positive function in particular this shows the existence of a nontrivial locally computable conformally invariant density naturally associated to the conformal structure of an even dimensional manifold the key to the proof is to consider the parametrix of the green s function which is obtained from the heat kernel by an integral transform one finds that a n 2 2 occurs as the coefficient of the first log term in this parametrix and its conformal invariance then follows directly from the conformal invariance of the green s function in 3 we show that an 2 is a global conformal invariant the calculations in 4 show that it is not a pointwise invariant the proof is a direct calculation of the invariant of an 2 using equation 1 10

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

conformal structures of static vacuum data in the cauchy problem for asymptotically flat vacuum data the solution jets along the cylinder at space like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future past null infinity the tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity for the solution jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the cotton tensor of the initial three metric h satisfies a certain conformally invariant condition at space like infinity it is sufficient that h be asymptotically static at space like infinity the purpose of this article is to characterize the gap between these conditions we show that with the class of metrics which satisfy condition on the cotton tensor and a certain non degeneracy requirement is associated a one form with conformally invariant differential d we provide two criteria if h is real analytic is closed and one of its integrals satisfies a certain equation then h is conformal to static data near space like infinity if h is smooth is asymptotically closed and one of its integrals satisfies a certain equation asymptotically then h is asymptotically conformal to static data at space like infinity

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

moduli spaces of critical riemannian metrics with l n 2 norm curvature bounds abstract we consider the moduli space of a class of critical metrics on compact manifolds that includes extremal kahler metrics we show that under the conditions of volume and diameter bounds l n 2 curvature bounds and sobolev constant bounds this moduli space can be compactified by including orbifolds with finitely many orbifold points in dimension 4 these orbifold points are c riemannian orbifold points and in the 4 dimensional extremal kahler case they are also of class c

an integral type brezis nirenberg problem on the heisenberg group abstract this paper is devoted to a class of integral type brezis nirenbreg problems on the heisenberg group they are a class of nonlinear integral equations on the bounded domains of heisenberg group and related to the cr yamabe problems on cr manifold based on the sharp hardy littlewood sobolev inequalities on the heisenberg group the nonexistence and existence results are obtained