openness results for uniform k stability assume that a projective variety together with a polarization is uniformly k stable if the polarization is canonical or anti canonical then the projective variety is uniformly k stable with respects to any polarization sufficiently close to the original polarization

compact moduli spaces of del pezzo surfaces and kahler einstein metrics we prove that the gromov hausdorff compactification of the moduli space of kahler einstein del pezzo surfaces in each degree agrees with certain algebro geometric compactification in particular this recovers tian s theorem on the existence of kahler einstein metrics on smooth del pezzo surfaces and classifies all the degenerations of such metrics the proof is based on a combination of both algebraic and differential geometric techniques

stable maps in higher dimensions we formulate a notion of stability for maps between polarised varieties which generalises kontsevich s definition when the domain is a curve and tian donaldson s definition of k stability when the target is a point we give some examples such as kodaira embeddings and fibrations we prove the existence of a projective moduli space of canonically polarised stable maps generalising the kontsevich alexeev moduli space of stable maps in dimensions one and two we also state an analogue of the yau tian donaldson conjecture in this setting relating stability of maps to the existence of certain canonical kahler metrics

sum of multiple zeta values of fixed weight depth and i height let r be a positive integer an explicit formula of the generating function of the sums of multiple zeta values of fixed weights depths and 1 heights 2 heights r heights is given in terms of generalized hypergeometric functions

special values of multiple polylogarithms historically the polylogarithm has attracted specialists and non specialists alike with its lovely evaluations much the same can be said for euler sums or multiple harmonic sums which within the past decade have arisen in combinatorics knot theory and high energy physics more recently we have been forced to consider multidimensional extensions encompassing the classical polylogarithm euler sums and the riemann zeta function here we provide a general framework within which previously isolated results can now be properly understood applying the theory developed herein we prove several previously conjectured evaluations including an intriguing conjecture of don zagier

linear differential equations and multiple zeta values iii zeta 3 we consider the hypergeometric equation 1 t t t g x3g 0 whose unique analytic solution 1 t x 1 o t near t 0 for t 1 becomes a generating function for multiple zeta values 1 1 x f3 x 1 3 x3 3 3 x6 we apply the so called wkb method to study solutions of the hypergeometric equation for large x and we calculate corresponding stokes matrices we prove that the function f3 x near x is also expressed via wkb type functions which subject to some stokes phenomenon this implies that f3 x satisfies a sixth order linear differential equation with irregular singularity at infinity

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

the l p robin problem for laplace equations in lipschitz and semi convex domains abstract let n 3 and be a bounded lipschitz domain in r n assume that p 2 and the function b l is non negative where denotes the boundary of denote by the outward unit normal to in this article the authors give two necessary and sufficient conditions for the unique solvability of the robin problem for the laplace equation u 0 in with boundary data u b u f l p respectively in terms of a weak reverse holder inequality with exponent p or the unique solvability of the robin problem with boundary data in some weighted l 2 space as applications the authors obtain the unique solvability of the robin problem for the laplace equation in the bounded semi convex domain with boundary data in weighted l p for any given p 1

scalar curvature and stability of toric varieties we define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a kahler metric of constant scalar curvature the main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces we prove that under the stability hypothesis the mabuchi functional is bounded below on invariant metrics and that minimising sequences have a certain convergence property in the reverse direction we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature the proofs use a general framework developed by guillemin and abreu in which invariant kahler metrics correspond to convex functions on the moment polytope of a toric variety this paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a kahler metric of constant scalar curvature the pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of stability in the sense of geometric invariant theory this expectation is probably now an item of folklore going back to suggestions put forward by yau in the case of kahlereinstein metrics and the many results of tian and others in this case reinforced by a detailed formal picture which makes clear the analogy with the well established relation between the stability of vector bundles and yang mills connections 5 here we begin the investigation of

on the proper moduli spaces of smoothable kahler einstein fano varieties in this paper we investigate the geometry of the orbit space of the closure of the subscheme parameterizing smooth kahler einstein fano manifolds inside an appropriate hilbert scheme in particular we prove that being k semistable is a zariski open condition and we establish the uniqueness of the gromov hausdorff limit for a punctured flat family of kahler einstein fano manifolds based on these we construct a proper scheme parameterizing the s equivalent classes of q gorenstein smoothable k semistable q fano varieties and we verify various necessary properties to guarantee that it is a good moduli space

the neumann problem of laplace s equation in semiconvex domains abstract let n 2 and be a bounded semiconvex domain in r n in this paper we show that the neumann problem for laplace s equation in with boundary data in l p is uniquely solvable for p 1 which improves the well known result for the neumann problem

one problem of the navier type for the stokes system in planar domains we study the problem u f u f u g u g in u gu g h h on for a bounded simply connected lipschitz domain in the plane for f 0f 0 g 0g 0 g lp g lp h lq h lq we study a solution in the sense of a nontangential limit for f ws 1 q r2 f ws 1 q r2 g ws q g ws q g wt 1 p p g wt 1 p p h ws 1 q q h ws 1 q q with t s 1t s 1 we prove the existence of a unique solution u wt p r2 ws q u wt p r2 ws q for f bs 1q r r2 g bsq r g bt 1 pp h bs 1 qq r with t s 1t s 1 we prove the existence of a unique solution u btp r2 bsq r for f ck 1 r2 g ck h ck h ck g ck 1 g ck 1 we prove the existence of a unique solution u ck 1 r2 ck

the neumann problem and helmholtz decomposition in convex domains we show that the neumann problem for laplace s equation in a convex domain with boundary data in lp is uniquely solvable for 1 p as a consequence we obtain the helmholtz decomposition of vector fields in lp rd

the lp boundary value problems on lipschitz domains abstract let be a bounded lipschitz domain in r n we develop a new approach to the invertibility on l p of the layer potentials associated with elliptic equations and systems in as a consequence for n 4 and 2 n 1 n 1 e p 2 where e 0 depends on we obtain the solvability of the l p neumann type boundary value problems for second order elliptic systems the analogous results for the biharmonic equation are also established

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

necessary and sufficient conditions for the solvability of the l p dirichlet problem on lipschitz domains we study the homogeneous elliptic systems of order 2 ell with real constant coefficients on lipschitz domains in mathbb r n n ge 4 for any fixed p 2 we show that a reverse holder condition with exponent p is necessary and sufficient for the solvability of the dirichlet problem with boundary data in l p we also obtain a simple sufficient condition as a consequence we establish the solvability of the l p dirichlet problem for n ge 4 and 2 epsilon p frac 2 n 1 n 3 epsilon the range of p is known to be sharp if ell ge 2 and 4 le n le 2 ell 1 for the polyharmonic equation the sharp range of p is also found in the case n 6 7 if ell 2 and n 2 ell 2 if ell ge 3

special test configuration and k stability of fano varieties for any at projective family x l c such that the generic bre x is a klt q fano variety and ljx q kx we use the techniques from the minimal model program mmp to modify the total family the end product is a family such that every ber is a klt q fano variety moreover we can prove that the donaldson futaki invariants of the appearing models decrease when the family is a test conguration of a xed fano variety x kx this implies tian s conjecture given x a fano manifold to test its k semi poly stability we only need to test on the special test congurations

relative k polystability of projective bundles over a curve let p e be the projectivization of a holomorphic vector bundle e over a compact complex curve c we characterize the existence of an extremal k ahler metric on the ruled manifold p e in terms of relative k polystability and the fact that e decomposes as a direct sum of stable bundles

kahler einstein metrics with positive scalar curvature in this paper we prove that the existence of kahler einstein metrics implies the stability of the underlying kahler manifold in a suitable sense in particular this disproves a long standing conjecture that a compact kahler manifold admits kahler einstein metrics if it has positive first chern class and no nontrivial holomorphic vector fields we will also establish an analytic criterion for the existence of kahler einstein metrics our arguments also yield that the analytic criterion is satisfied on stable kahler manifolds provided that the partial c 0 estimate posed in t6 is true

the neumann problem in lp on lipschitz and convex domains abstract given p 2 and a bounded lipschitz domain we obtain a necessary and sufficient condition for the solvability of the neumann problem for laplace s equation u 0 in with boundary data in l p as a result we show that the l p neumann problem on convex domains in r d is solvable for 1 p if d 2 for 1 p 4 if d 3 and for 1 p 3 e if d 4

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy