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

positivity of quasi local mass ii a spacetime is a four manifold with a pseudo metric of signature a hypersurface or a 2 surface in a spacetime is spacelike if the induced metric is pos itive definite a quasi local energy momentum vector is a vector in r3 1 associated to a spacelike 2 surface which depends on its first and second fundamental forms and the connection to its normal bundle in the spacetime the time component of the four vector is called quasi local energy mass similar to 10 8 we require the quasi local energy momentum vector to satisfy the following properties 1 it should be zero for the flat spacetime 2 the quasi local mass should be equivalent to the standard definition if the spacetime is spherically symmetric and the quasi local mass is evaluated on the spheres 7 we say that two masses mi and m2 are equivalent if there is a universal constant c 0 such that c lm area e 167r 6 the quasi local energy momentum vector should be nonspacelike and the quasi local mass should be nonnegative

knots links and 4 manifolds in this paper we investigate the relationship between isotopy classes of knots and links in s and the di eomorphism types of homeomorphic smooth 4 manifolds as a corollary of this initial investigation we begin to uncover the surprisingly rich structure of di eomorphism types of manifolds homeomorphic to the k3 surface in order to state our theorems we need to view the seiberg witten invariant of a smooth 4 manifold as a multivariable laurent polynomial to do this recall that the seiberg witten invariant of a smooth closed oriented 4 manifold x with b2 x 1 is an integer valued function which is de ned on the set of spinc structures over x cf w km ko1 t1 in case h1 x z has no 2 torsion which will be the situation in this paper there is a natural identi cation of the spinc structures of x with the characteristic elements of h2 x z in this case we view the seiberg witten invariant as

a comparison theorem for gromov witten invariants in the symplectic category abstract we exploit the geometric approach to the virtual fundamental class due to fukaya ono and li tian to compare gromov witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order various special cases of the comparison theorem in this paper have long been used in the algebraic category some of them have also appeared in the symplectic setting combined with the inherent flexibility of the symplectic category the main theorem leads to a confirmation of pandharipande s gopakumar vafa prediction for gw invariants of fano classes in 6 dimensional symplectic manifolds the proof of the main theorem uses deformations of the cauchy riemann equation that respect the submanifold and carleman similarity principle for solutions of perturbed cauchy riemann equations in a forthcoming paper we apply a similar approach to relative gromov witten invariants and the absolute relative correspondence in genus 0

deformations of the hemisphere that increase scalar curvature consider a compact riemannian manifold m of dimension n whose boundary m is totally geodesic and is isometric to the standard sphere s n 1 a natural conjecture of min oo asserts that if the scalar curvature of m is at least n n 1 then m is isometric to the hemisphere s n equipped with its standard metric this conjecture is inspired by the positive mass theorem in general relativity and has been verified in many special cases in this paper we construct counterexamples to min oo s conjecture in dimension n 3

a first eigenvalue estimate for minimal hypersurfaces on obtient une borne inferieure pour la premiere valeur propre d une hypersurface orientable compacte plongee de facon minimale dans une variete orientable compacte a courbure de ricci positive

local existence of quasispherical space time initial data we consider the system of einstein constraint equations in the special case when the spacelike 3 manifold is assumed to satisfy the quasispherical ansatz of bartnik we introduce two different time slicing conditions under each of which we may view the resulting constraint equations as a parabolic elliptic ode system of partial differential equations we combine recent existence results for parabolic equations in sobolev space with an iterative method to prove local existence results for the quasispherical einstein constraint equations under the two different time slicing conditions

on the riemannian penrose inequality in dimensions less than eight the positive mass theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass and that equality is achieved only for the euclidean metric the riemannian penrose inequality provides a sharp lower bound for the mass when black holes are present more precisely this lower bound is given in terms of the area of an outermost minimal hypersurface and equality is achieved only for schwarzschild metrics the riemannian penrose inequality was first proved in three dimensions in 1997 by g huisken and t ilmanen for the case of a single black hole see hi in 1999 bray extended this result to the general case of multiple black holes using a different technique see br in this article we extend the technique of br to dimensions less than eight part of the argument is contained in a companion article by lee l the equality case of the theorem requires the added assumption that the manifold be spin

analyzing netizens view and reply behaviors on the forum abstract quantitative understanding of human behaviors provides elementary comprehension of the complexity of social system in this paper the netizens behaviors on the bulletin board system bbs are investigated by the statistical analysis of views and replies on some forums the statistical results show that the number of views and replies obeys the power law distribution with different power exponents in addition when the distribution of both views and replies follows power law distribution they are found to have a nonlinear relationship this relationship also obeys the power function when transformed to the log log plot its fit curve appears as a straight line based on the estimation of slopes and intercepts of the lines we can characterize the view and reply behaviors quantitatively the results reveal that the chinese and western netizens have different preferences when replying and viewing the threads at last the time series of reply behaviors are analyzed all series show us with a high burstiness and low memory

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

boundary effect of ricci curvature on a compact riemannian manifold with boundary we study how ricci curvature of the interior affects the geometry of the boundary first we establish integral inequalities for functions defined solely on the boundary and apply them to obtain geometric inequalities involving the total mean curvature then we discuss related rigidity questions and prove ricci curvature rigidity results for manifolds with boundary

total mean curvature scalar curvature and a variational analog of brown york mass we study the supremum of the total mean curvature on the boundary of compact mean convex 3 manifolds with nonnegative scalar curvature and a prescribed boundary metric we establish an additivity property for this supremum and exhibit rigidity for maximizers assuming the supremum is attained when the boundary consists of 2 spheres we demonstrate that the finiteness of the supremum follows from the previous work of shi tam and wang yau on the quasi local mass problem in general relativity in turn we define a variational analog of brown york quasi local mass without assuming that the boundary 2 sphere has positive gauss curvature

quasi spherical metrics and applications in this paper using the idea of bartnik b2 on quasi spherical metrics we continue our study on the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary unlike the previous work st of the authors and the work of liu yau ly we only assume each boundary component has nonnegative curvature which is not identically zero we also study the case that the boundary is embedded in the quotient of the infinity of the euclidean space over a finite group the regularity of the black hole boundary condition of quasi spherical metrics is also discussed

positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature in this paper we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and nonempty boundary using a general version of positive mass theorem of schoen yau and witten we prove the following theorem for any compact manifold with boundary and nonnegative scalar curvature if it is spin and its boundary can be isometrically embedded into euclidean space as a strictly convex hypersurface then the integral of mean curvature of the boundary of the manifold cannot be greater than the integral of mean curvature of the embedded image as a hypersurface in euclidean space moreover equality holds if and only if the manifold is isometric with a domain in the euclidean space conversely under the assumption that the theorem is true then one can prove the adm mass of an asymptotically flat manifold is nonnegative which is part of the positive mass theorem

the symplectic sum formula for gromov witten invariants in the symplectic category there is a connect sum operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds this paper establishes a formula for the gromov witten invariants of a symplectic sum z x y in terms of the relative gw invariants of x and y several applications to enumerative geometry are given gromov witten invariants are counts of holomorphic maps into symplectic manifolds to define them on a symplectic manifold x w one introduces an almost complex structure j compatible with the symplectic form w and

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

topology of surface diagrams of smooth 4 manifolds surface diagrams are a new way to specify any smooth closed orientable 4 manifold by an orientable surface decorated with simple closed curves these curves are cyclically indexed and each curve has a unique transverse intersection with the next the aim of this paper is to announce a uniqueness theorem for these objects within a fixed homotopy class that turns out to be similar to the reidemeister singer theorem for heegaard splittings of 3 manifolds

insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds we construct a solution to inverse mean curvature flow on an asymptotically hyperbolic 3 manifold which does not have the convergence properties needed in order to prove a penrose type inequality this contrasts sharply with the asymptotically flat case the main idea consists in combining inverse mean curvature flow with work done by shi tam regarding boundary behavior of compact manifolds assuming the penrose inequality holds we also derive a nontrivial inequality for functions on s

stability of the positive mass theorem for rotationally symmetric riemannian manifolds we study the stability of the positive mass theorem using the intrinsic flat distance in particular we consider the class of complete asymptotically flat rotationally symmetric riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty we prove that a sequence of these manifolds whose adm masses converge to zero must converge to euclidean space in the pointed intrinsic flat sense in fact we provide explicit bounds on the intrinsic flat distance between annular regions in the manifold and annular regions in euclidean space by constructing an explicit filling manifold and estimating its volume in addition we include a variety of propositions that can be used to estimate the intrinsic flat distance between riemannian manifolds without rotationally symmetry conjectures regarding the intrinsic flat stability of the positive mass theorem in the general case are proposed in the final section lee is partially supported by a psc cuny research grant and nsf dms 0903467 sormani is partially supported by a psc cuny research grant and nsf dms 1006059 1 ar x iv 1 10 4 26 57 v1 m at h d g 1 4 a pr 2 01 1 2 dan a lee and christina sormani

empirical analysis on temporal statistics of human correspondence patterns abstract recently extensive empirical evidence shows that the timing of human behaviors obeys non possion statistics with heavy tailed interevent time distribution in this paper we empirically study the correspondence pattern of a great chinese scientist named hsue shen tsien both the interevent time distribution and response time distributions deviate from the poisson statistics showing an approximate power law decaying the two power law exponents are more or less the same about 2 1 which strongly support the hypothesis in a vazquez j g oliveira z dezso k i goh i kondor a l barabasi phys rev e 73 2006 036127 that the response time distribution of the tasks could in fact drive the interevent time distribution and both the two distributions should decay with the same exponent our result is against the claim in a vazquez j g oliveira z dezso k i goh i kondor a l barabasi phys rev e 73 2006 036127 which suggests the human correspondence pattern belongs to a universality class with exponent 1 5

relative gromov witten invariants we define relative gromov witten invariants of a symplectic manifold relative to a codimension two symplectic submanifold these invariants are the key ingredients in the symplectic sum formula of ip4 the main step is the construction of a compact space of v stable maps simple special cases include the hurwitz numbers for algebraic curves and the enumerative invariants of caporaso and harris

a note on hang wang s hemisphere rigidity theorem let m g be a compact manifold with boundary and ric g ge n 1 g hang and wang proved that m g is isometric to the standard hemisphere if partial m is convex and isometric to mathbb s n 1 1 we prove some rigidity theorems when partial m is isometric to a product manifold where one factor is the standard sphere

quasi local mass and the existence of horizons in this paper we obtain lower bounds for the brown york quasilocal mass and the bartnik quasilocal mass for compact three manifolds with smooth boundaries as a consequence we derive sufficient conditions for the existence of horizons for a certain class of compact manifolds with boundary and some asymptotically flat complete manifolds the method is based on analyzing hawking mass and inverse mean curvature flow

a structure theorem for the gromov witten invariants of kahler surfaces we prove a structure theorem for the gromov witten invariants of compact kahler surfaces with geometric genus pg 0 under the technical assumption that there is a canonical divisor that is a disjoint union of smooth components the theorem shows that the gw invariants are universal functions determined by the genus of this canonical divisor components and the holomorphic euler characteristic of the surface we compute special cases of these universal functions much of the work on the gromov witten invariants of kahler surfaces has focused on rational and ruled surfaces which have geometric genus pg 0 this paper focuses on surfaces with pg 0 a class that includes most elliptic surfaces and most surfaces of general type in this context we prove a general structure theorem that shows with one technical assumption how the gw invariants are completely determined by the local geometry of a generic canonical divisor the structure theorem is a consequence of a simple fact the image localization lemma of section 3 given a kahler surface x and a canonical divisor d kx this lemma shows that the complex structure j on x can be perturbed to a non integrable almost complex structure jd with the property the image of all jd holomorphic maps lies in the support of d this immediately gives some striking vanishing theorems for the gw invariants of kahler surfaces see section 3 more importantly it implies that the gromov witten invariant of x for genus g and n marked points is a sum gwg n x a gw loc g n dk ak over the connected components dk of d of local invariants that count the contribution of maps whose image lies in or after perturbing to a generic moduli space near dk these local invariants have not been previously defined the proof of their existence relies on using non integrable structures and geometric analysis techniques our structure theorem characterizes the local invariants by expressing them in terms of usual gw invariants of certain standard surfaces for this we make the mild assumption that one can deform the kahler structure on x and choose a canonical divisor d so that all components of d are smooth with this assumption the restriction of kx to each component dk of multiplicity mk is the normal bundle nk to dk and nk k kdk partially supported by the n s f

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

standard surfaces and nodal curves in symplectic 4 manifolds continuing the program of ds and u1 we introduce refinements of the donaldson smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components in cases where a corresponding analogue of the gromov taubes invariant is easy to define our invariants agree with those analogues we also prove a vanishing result for some of the invariants that count nodal curves

3 dimensional seiberg witten invariants and non kahlerian geometry the paper is concerned with seiberg witten invariants of closed 3 manifolds and their relation to the 4 dimensional invariants the main points are 1 the introduction of parameter dependent seiberg witten invariants for 3 manifolds with b1 0 these invariants depend on a complicated chamber structure defined by a differential geometric brill noether locus in the parameter space 2 a comparison theorem which shows that the seiberg witten invariants of 3 manifolds m with b 1 m 1 are simply the restrictions of the 4 dimensional invariants of the product s 1 m to the set of pull back spinc structures as an immediate consequence one gets the wall crossing formula for the invariants of 3 dimensional manifolds with b1 1 from the 4 dimensional wall crossing formula for manifolds with b 1 3 a novel method to compute the seiberg witten invariants of 3 manifolds m using non kahlerian complex geometry on s 1 m provided this product has such a structure which is compatible with a product metric and orientation induced from m this method relies essentially on a kobayashi hitchin correspondence which is new in the non kahler case 4 a complete description of those 3 manifolds to which our method applies these are riemannian manifolds which admit an oriented riemannian foliation by geodesics such manifolds are always diffeomorphic to seifert manifolds but the foliation has not to be seifert it can have non closed leaves 5 explicit computations of seiberg witten invariants for the 3 sphere s 3 using non seifert riemannian foliations we show that the chamber structure induced by the brill noether locus is very complicated there exist countably many dif ferent chambers and the seiberg witten invariant takes on all integer values

yamabe flow and adm mass on asymptotically flat manifolds in this paper we investigate the behavior of adm mass and einstein hilbert functional under the yamabe flow through studying the yamabe flow by weighted spaces for parabolic operators we show that the asymptotically flat property is preserved under the yamabe flow we also obtain that adm mass is invariant under the yamabe flow and yamabe flow is the gradient flow of einstein hilbert functional on n dimensional n 3 asymptotically flat manifolds with order n 22 for n 3 4 or n 3 for n 4 moreover we show that adm mass and einstein hilbert functional are non increasing under the yamabe flow on n dimensional asymptotically flat manifolds if we only assume the order n 22 for n 4

proof of the riemannian penrose inequality with charge for multiple black holes we present a proof of the riemannian penrose inequality with charge in the context of asymptotically flat initial data sets for the einstein maxwell equations having possibly multiple black holes with no charged matter outside the horizon and satisfying the relevant dominant energy condition the proof is based on a generalization of hubert bray s conformal flow of metrics adapted to this setting

isometric embeddings into the minkowski space and new quasi local mass the definition of quasi local mass for a bounded space like region in space time is essential in several major unsettled problems in general relativity the quasi local mass is expected to be a type of flux integral on the boundary two surface sigma partial omega and should be independent of whichever space like region sigma bounds an important idea which is related to the hamiltonian formulation of general relativity is to consider a reference surface in a flat ambient space with the same first fundamental form and derive the quasi local mass from the difference of the extrinsic geometries this approach has been taken by brown york 4 5 and liu yau 16 17 see also related works 3 6 9 12 14 15 28 32 to define such notions using the isometric embedding theorem into the euclidean three space however there exist surfaces in the minkowski space whose quasilocal mass is strictly positive 19 it appears that the momentum information needs to be accounted for to reconcile the difference in order to fully capture this information we use isometric embeddings into the minkowski space as references in this article we first prove an existence and uniqueness theorem for such isometric embeddings we then solve the boundary value problem for jang s 13 equation as a procedure to recognize such a surface in the minkowski space in doing so we discover a new expression of quasi local mass for a large class of admissible surfaces see theorem a and remark 1 1 the new mass is positive when the ambient space time satisfies the dominant energy condition and vanishes on surfaces in the minkowski space it also has the nice asymptotic behavior at spatial infinity and null infinity some of these results were announced in 29

scalar curvature rigidity of geodesic balls in sn in this paper we prove a scalar curvature rigidity result for geodesic balls in s n this result contrasts sharply with the counterexamples to min oo s conjecture constructed in deformations of the hemisphere that increase scalar curvature invent math 185 2011 175 197

reduced open gromov witten invariants on k3 surfaces and multiple cover formula abstract in the paper we study the wall crossing phenomenon of reduced open gromov witten invariants on k3 surfaces with rigid special lagrangian boundary condition as a corollary we derived the multiple cover formula for the reduced open gromov witten invariants which is an open analog of the gopakumar vafa conjecture of genus zero

a polyhedron comparison theorem for 3 manifolds with positive scalar curvature the study of comparison theorems in geometry has a rich history in this paper we establish a comparison theorem for polyhedra in 3 manifolds with nonnegative scalar curvature answering affirmatively a dihedral rigidity conjecture by gromov for a large collections of polyhedra with interior non negative scalar curvature and mean convex faces we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding euclidean model unless it is isometric to a flat polyhedron

the gbc mass for asymptotically hyperbolic manifolds the paper consists of two parts in the first part by using the gauss bonnet curvature which is a natural generalization of the scalar curvature we introduce a higher order mass the gauss bonnet chern mass m mathbb h k for asymptotically hyperbolic manifolds and show that it is a geometric invariant moreover we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding penrose type inequality for this mass and weighted alexandrov fenchel inequalities in the hyperbolic space mathbb h n in the second part we establish these weighted alexandrov fenchel inequalities in mathbb h n for any horospherical convex hypersurface sigma begin aligned int sigma v sigma 2k 1 d mu ge c n 1 2k 1 omega n 1 left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned where sigma j is the j th mean curvature of sigma subset mathbb h n v cosh r and sigma is the area of sigma as an application we obtain an optimal penrose type inequality for the new mass defined in the first part begin aligned m k mathbb h ge frac 1 2 k left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned for asymptotically hyperbolic graphs with a horizon type boundary sigma provided that a dominant energy type condition tilde l k ge 0 holds both inequalities are optimal

the homotopy lie algebra of symplectomorphism groups of 3 fold blow ups of the projective plane by a result of kedra and pinsonnault we know that the topology of groups of symplectomorphisms of symplectic 4 manifolds is complicated in general however in all known very specific examples the rational cohomology rings of symplectomorphism groups are finitely generated in this paper we compute the rational homotopy lie algebra of symplectomorphism groups of the 3 point blow up of the projective plane with an arbitrary symplectic form and show that in some cases depending on the sizes of the blow ups it is infinite dimensional moreover we explain how the topology is generated by the toric structures one can put on the manifold our method involve the study of the space of almost complex structures compatible with the symplectic structure and it depends on the inflation technique of lalonde mcduff

the positive mass theorem for manifolds with distributional curvature we formulate and prove a positive mass theorem for n dimensional spin manifolds whose metrics have only the sobolev regularity c 0 cap w rm loc 1 n at this level of regularity the curvature of the metric is defined in the distributional sense only and we propose here a generalized notion of adm mass for such a metric our main theorem establishes that if the manifold is asymptotically flat and has non negative scalar curvature distribution then its generalized adm mass is well defined and non negative and vanishes only if the manifold is isometric to euclidian space prior applications of witten s spinor method by lee and parker and by bartnik required the much stronger regularity w rm loc 2 p with p n our proof is a generalization of witten s arguments in which we must treat the dirac operator and its associated lichnerowicz weitzenbock identity in the distributional sense and cope with certain averages of first order derivatives of the metric over annuli that approach infinity finally we observe that our arguments are not specific to scalar curvature and also allow us to establish a universal positive mass theorem

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

enumerative geometry of calabi yau 4 folds gromov witten theory is used to define an enumerative geometry of curves in calabi yau 4 folds the main technique is to find exact solutions to moving multiple cover integrals the resulting invariants are analogous to the bps counts of gopakumar and vafa for calabi yau 3 folds we conjecture the 4 fold invariants to be integers and expect a sheaf theoretic explanation

on scalar curvature rigidity of vacuum static spaces in this paper we extend the local scalar curvature rigidity result in brendle and marques j differ geom 88 379 394 2011 to a small domain on general vacuum static spaces which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper corvino commun math phys 214 137 189 2000 we obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces we also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains where the lapse functions are positive on vacuum static spaces with positive scalar curvature and show such domains are maximal which generalizes the work in hang and wang commun anal geom 14 91 106 2006

the topology of the space of symplectic balls in rational 4 manifolds we study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball b4 c r4 into 4 dimensional rational symplectic manifolds where c r2 is the capacity of the standard ball of radius r we compute the rational homotopy groups of that space when the 4 manifold has the form m s2 s2 0 0 where 0 is the area form on the sphere with total area 1 and belongs to the interval 1 2 we show that when is 1 this space retracts to the space of symplectic frames for any value of c however for any given 1 2 the rational homotopy type of that space changes as c crosses the critical parameter 1 which is the difference of areas between the two s2 factors we prove moreover that the full homotopy type of that space changes only at that value that is that the restriction map between these spaces is a homotopy equivalence as long as these values of c remain either below or above that critical value the same methods apply to all other values of and other rational 4 manifolds as well the methods rely on two different tools the study of the action of symplectic groups on the stratified space of almost complex structures developed by gromov abreu and mcduff and the analysis of the relations between the group corresponding to a manifold m the group corresponding to its blow up m and the space of symplectic embedded balls in m

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

curvature estimates for immersed hypersurfaces in riemannian manifolds we establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in riemannian manifold n n 1 bar g through regularity study of a degenerate fully nonlinear curvature equation in general riemannian manifold the estimate has a direct consequence for the weyl isometric embedding problem of mathbb s 2 g in 3 dimensional warped product space n 3 bar g we also discuss isometric embedding problem in spaces with horizon in general relativity like the anti de sitter schwarzschild manifolds and the reissner nordstrom manifolds

geometric inequalities for quasi local masses in this paper lower bounds are obtained for quasi local masses in terms of charge angular momentum and horizon area in particular we treat three quasi local masses based on a hamiltonian approach namely the brown york liu yau and wang yau masses the geometric inequalities are motivated by analogous results for the adm mass they may be interpreted as localized versions of these inequalities and are also closely tied to the conjectured bekenstein bounds for entropy of macroscopic bodies in addition we give a new proof of the positivity property for the wang yau mass which is used to remove the spin condition in higher dimensions furthermore we generalize a recent result of lu and miao to obtain a localized version of the penrose inequality for the static wang yau mass

positive scalar curvature with skeleton singularities we study positive scalar curvature on the regular part of riemannian manifolds with singular uniformly euclidean l infty metrics that consolidate gromov s scalar curvature polyhedral comparison theory and edge metrics that appear in the study of einstein manifolds we show that in all dimensions edge singularities with cone angles le 2 pi along codimension 2 submanifolds do not affect the yamabe type in three dimensions we prove the same for more general singular sets which are allowed to stratify along 1 skeletons exhibiting edge singularities angles le 2 pi and arbitrary l infty isolated point singularities we derive as an application of our techniques positive mass theorems for asymptotically flat manifolds with analogous singularities