conformal deformation of a riemannian metric to a scalar flat metric with constant mean curvature on the boundary one of the most celebrated theorems in mathematics is the riemann mapping theorem it says that an open simply connected proper subset of the plane is conformally diffeomorphic to the disk in higher dimensions very few regions are conformally diffeomorphic to the ball however we can still ask whether a domain is conformally diffeomorphic to a manifold that resembles the ball in two ways namely it has zero scalar curvature and its boundary has constant mean curvature in this paper we generalize the riemann mapping theorem to higher dimensions in that sense

estimates for sums of eigenvalues of the laplacian the aim of this paper is to give bounds for the eigenvalues of the laplacian on a domain in euclidean space and on a compact riemannian manifold first we consider the eigenvalue problem for the laplacian on a bounded domain in euclidean space under dirichlet and neumann boundary conditions our method for obtaining an upper bound for sums of eigenvalues under dirichlet boundary conditions is closely related to the method used earlier j funct anal 106 1992 353 357 for the task of getting an upper bound for sums of eigenvalues under neumann boundary conditions on the other hand we modify the method used by p li and s t yau comm math phys 88 1983 309 318 for obtaining a lower bound for sums of eigenvalues under dirichlet boundary conditions in order to get a lower bound for sums of eigenvalues under neumann boundary conditions under the assumption that the domain under consideration is lipschitz equivalent to a ball finally we derive estimates for sums of squares of eigenvalues on a compact riemannian manifold without boundary

chow groups of projective varieties of very small degree see 12 5 and the references given there these facts together with various conjectures on the cohomology and chow groups of algebraic varieties suggest that the chow groups of x might satisfy chl x q chl pk q q for l 1 compare with remark 5 6 and corollary 5 7 this is explicitly formulated by v srinivas and k paranjape in 16 conjecture 1 8 the chain of reasoning goes roughly as follows suppose x is smooth one expects a good filtration 0 f j 1 f j f 0 ch x x q whose graded pieces f f l 1 are controlled by h2j l x x see 10 according to grothendieck s generalized conjecture 8 the groups h x should be generated by the image under the gysin morphism of the homology of a codimension subset together with the classes coming from p applying this to the diagonal in x x should then force the triviality of the chow groups in the desired range for zero cycles the conjecture follows from roitman s theorem see 17 and 18

all complete intersection varieties are fano visitors abstract we prove that the derived category of a smooth complete intersection variety is equivalent to a full subcategory of the derived category of a smooth projective fano variety this enables us to define a new invariant of smooth projective varieties and raise many interesting questions

lieb thirring inequality with semiclassical constant and gradient error term abstract in 1975 lieb and thirring derived a semiclassical lower bound on the kinetic energy for fermions which agrees with the thomas fermi approximation up to a constant factor whenever the optimal constant in their bound coincides with the semiclassical one is a long standing open question we prove an improved bound with the semiclassical constant and a gradient error term which is of lower order

a perturbation result for prescribing mean curvature in this article we consider a boundary value problem associated with the conformal deformation of metrics we are interested in the case in which a noncompact group of conformal transformation acts on the equation so that the kazdanwarner condition gives rise to obstruction as in the nirenberg problem the simplest situation is the following let bn 1 be the unit ball in rn 1 with euclidean metric g0 its boundary will be denoted by s n bn 1 let h be a smooth function on s n we study the problem of finding a conformal metric gu u4 n 1 g0 whose scalar curvature vanishes in bn 1 and on the boundary bn 1 the mean curvature is given by h this problem is equivalent to solving the following boundary value problem

the prescribed boundary mean curvature problem on b4 abstract in this paper we perform a fine blow up analysis for a boundary value elliptic equation involving the critical trace sobolev exponent related to the conformal deformation of the metrics on the standard ball namely the problem of prescribing the boundary mean curvature from this analysis some a priori estimates in low dimension are obtained with these estimates we prove the existence of at least one solution when an index counting formula associated to the prescribed mean curvature is different from zero

higher fano manifolds and rational surfaces let x be a fano manifold of pseudo index 3 such that c1 x 2c2 x is nef irreducibility of some spaces of rational curves on x implies a general point of x is contained in a rational surface

a numerical criterion for uniruledness an n dimensional variety x over an algebraically closed field k is said to be uniruled if there exist an n 1 dimensional k variety w and a dominant rational map f p x w x x is called separably uniruled if f is a separable map i e k p x w is a finite separable extension of k x via f if x is uniniled then there exists a rational curve passing through a general k valued point of x and the converse is true by the countability of the components of the hilbert scheme provided that the ground field k is uncountable uniruled varieties are of importance in the classification of varieties it is conjectured that in characteristic 0 uniruledness is equivalent to k xc this article gives a numerical criterion for a projective variety to be uniruled

fractional hardy lieb thirring and related inequalities for interacting systems we prove analogues of the lieb thirring and hardy lieb thirring inequalities for many body quantum systems with fractional kinetic operators and homogeneous interaction potentials where no anti symmetry on the wave functions is assumed these many body inequalities imply interesting one body interpolation inequalities and we show that the corresponding one and many body inequalities are actually equivalent in certain cases

existence results for the fractional nirenberg problem abstract we consider the fractional nirenberg problem on the standard sphere s n with n 4 using the theory of critical points at infinity we establish an euler hopf type formula and obtain some existence results for curvature satisfying assumptions of bahri coron type

the prescribed boundary mean curvature problem on the standard n dimensional ball abstract in this paper we consider a boundary value problem associated with the conformal deformation of metrics on the unit ball b n n 3 using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem we prove the existence of a metric conformally equivalent to the euclidean metric with zero scalar curvature and prescribed mean curvature on the boundary

lengths of chains of minimal rational curves on fano manifolds abstract in this paper we consider a natural question how many minimal rational curves are needed to join two general points on a fano manifold x of picard number 1 in particular we study the minimal length of such chains in the cases where the dimension of x is at most 5 the coindex of x is at most 3 and x equips with a structure of a double cover as an application we give a better bound on the degree of fano 5 folds of picard number 1

on the minimal length of extremal rays for fano four folds the minimum of intersection numbers of the anti canonical divisor with rational curves on a fano manifold is called pseudo index it is expected that the intersection number of anti canonical divisor attains to the minimum on an extremal ray i e there exists an extremal rational curve whose intersection number with the anti canonical divisor equals the pseudo index in this note we prove this for smooth fano four folds having birational contractions

2 cycles on higher fano hypersurfaces let x d be a smooth hypersurface of degree d in mathbb p mathbb c n suppose that the fano variety mathrm f x d of lines of x d is smooth we prove that the griffiths group mathrm griff 1 mathrm f mathrm x d of mathrm f x d is trivial if the hypersurface mathrm x d is of 2 fano type as a result we give a positive answer to a question of professor voisin about the first griffiths groups of fano varieties in some cases base on this result we prove that mathrm ch 2 mathrm x d mathbb z for a smooth 3 fano hypersurface mathrm x d subseteq mathbb p n mathbb c with smooth fano variety of lines

polarized minimal families of rational curves and higher fano manifolds in this paper we investigate fano manifolds x whose chern characters rm ch k x satisfy some positivity conditions our approach is via the study of it polarized minimal families of rational curves h x l x through a general point x in x first we translate positivity properties of the chern characters of x into properties of the pair h x l x this allows us to classify polarized minimal families of rational curves associated to fano manifolds x satisfying rm ch 2 x geq 0 and rm ch 3 x geq 0 this classification enables us to find new examples of higher fano manifolds we also provide sufficient conditions for these manifolds to be covered by subvarieties isomorphic to bbb p 2 and bbb p 3

universal relations on stable map spaces in genus zero we introduce a factorization for the map between moduli spaces of stable maps which forgets one marked point this leads to a study of universal relations in the cohomology of stable map spaces in genus zero

kontsevich spaces of rational curves on fano hypersurfaces we investigate the spaces of rational curves on a general hypersurface in particular we show that for a general degree d hypersurface in pn with n d 2 the space m0 0 x e of degree e kontsevich stable maps from a rational curve to x is an irreducible local complete intersection stack of dimension e n d 1 n 4 this resolves all but one case of a conjecture of coskun harris and starr

potentially non klt locus and its applications we introduce the notion of potentially klt pairs for normal projective varieties with pseudoeffective anticanonical divisor the potentially non klt locus is a subset of x which is birationally transformed precisely into the non klt locus on a k x minimal model of x we prove basic properties of potentially non klt locus in comparison with those of classical non klt locus as applications we give a new characterization of varieties of fano type and we also improve results on the rational connectedness of uniruled varieties with pseudoeffective anticanonical divisor

fano n folds with nef tangent bundle and picard number greater than n 5 we prove that fano n folds with nef tangent bundle and picard number greater than n 5 are rational homogeneous manifolds

on manifolds whose tangent bundle is big and 1 ample a line bundle over a complex projective variety is called big and 1 ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1 dimensional fibers a vector bundle is called big and 1 ample if the relative hyperplane line bundle over its projectivisation is big and 1 ample the main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more whose tangent bundle is big and 1 ample is equal either to a projective space or to a smooth quadric since big and 1 ample bundles are almost ample the present result is yet another extension of the celebrated mori paper projective manifolds with ample tangent bundles ann of math 110 1979 593 606 the proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective in the appendix we re prove these results

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

conformal transformation of metrics on the n ball abstract in this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of r n n 3 under the assumption that the prescribed function is flat near its critical point we give precise estimates on the losses of the compactness and we provide new existence and multiplicity results of bahri coron type

problemes de neumann non lineaires sur les varietes riemanniennes nonlinear neumann problems on riemannian manifolds let m g be a c compact riemannian manifold of dimension n 2 whose boundary b is an n 1 dimensional submanifold and let m m b be the interior of m study of neumann problems of the form x 0 in m d dn g y 0 on b where for every t x y r m b t x and g t y are bounded by c 1 t a or c exp t a application to the determination of a conformal metric for which the scalar curvature of m and the mean curvature of b take prescribed values

on a nirenberg type problem involving the square root of the laplacian abstract through an euler hopf type formula we establish existence result to a nirenberg type problem involving the square root of the laplacian in sphere s 2

a positive density analogue of the lieb thirring inequality the lieb thirring inequalities give a bound on the negative eigenvalues of a schrodinger operator in terms of an l p norm of the potential these are dual to bounds on the h 1 norms of a system of orthonormal functions here we extend these bounds to analogous inequalities for perturbations of the fermi sea of noninteracting particles i e for perturbations of the continuous spectrum of the laplacian by local potentials

conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary abstract this paper considers the prescribed zero scalar curvature and mean curvature problem on the n dimensional euclidean ball for n 3 we consider the limits of solutions of the regularization obtained by decreasing the critical exponent we characterize those subcritical solutions which blow up at the least possible energy level determining the points at which they can concentrate and their morse indices we show that when n 3 this is the only blow up which can occur for solutions we use this in combination with the morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three dimensional euclidean ball in the higher dimensional case n 4 we give conditions on the function h to guarantee there is only one simple blow up point

compactness results in conformal deformations of riemannian metrics on manifolds with boundaries this paper is devoted to the study of a problem arising from a geometric context namely the conformal deformation of a riemannian metric to a scalar flat one having constant mean curvature on the boundary by means of blow up analysis techniques and the positive mass theorem we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the c2 norm and the total leray schauder degree of all solutions is equal to 1 then we deduce from this compactness result the existence of at least one solution to our problem

congruences for rational points on varieties over finite fields we prove the existence of rational points on singular varieties over finite fields arising as degenerations of smooth proper varieties with trivial chow group of 0 cycles we also obtain congruences for the number of rational points of singular varieties appearing as fibres of a proper family with smooth total and base space and such that the chow group of 0 cycles of the generic fibre is trivial in particular this leads to a vast generalization of the classical chevalley warning theorem the above results are obtained as special cases of our main theorem which can be viewed as a relative version of a theorem of h esnault on the number of rational points of smooth proper varieties over finite fields with trivial chow group of 0 cycles