energy functionals and canonical kahler metrics yau y2 has conjectured that a fano manifold admits a kahler einstein metric if and only if it is stable in the sense of geometric invariant theory there has been much progress on this conjecture by tian t1 t2 donaldson do1 do2 and others the mabuchi energy functional plays a central role in these ideas we study the ek functionals introduced by x x chen and g tian ct1 which generalize the mabuchi energy we show that if a fano manifold admits a kahler einstein metric then the functional e1 is bounded from below and modulo holomorphic vector fields is proper this answers affirmatively a question raised by chen c2 in fact we show that e1 is proper if and only if there exists a kahler einstein metric giving a new analytic criterion for the existence of this canonical metric with possible implications for the study of stability we also show that on a fano kahler einstein manifold all of the functionals ek are bounded below on the space of metrics with nonnegative ricci curvature

blowing up kahler manifolds with constant scalar curvature ii in this paper we prove the existence of kahler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold that already carries a constant scalar curvature kahler metric in the case where the manifold has nontrivial holomorphic vector fields with zeros we give necessary conditions on the number and locations of the blow up points for the blow up to carry constant scalar curvature kahler metrics

scalar curvature moment maps and the deligne pairing in this paper we improve the bound for the moment map derivative proved by donaldson in his recent proof of the hilbert mumford stability of complex manifolds with constant scalar curvature the proof depends on the identification of donaldson s symplectic form with the curvature of a certain deligne pairing

moment maps scalar curvature and quantization of kahler manifolds building on donaldson s work on constant scalar curvature metrics we study the space of regular kahler metrics e i e those for which deformation quantization has been defined by cahen gutt and rawnsley after giving in sects 2 and 3 a review of donaldson s moment map approach we study the essential uniqueness of balanced basis i e of coherent states in a more general setting theorem 2 5 we then study the space e in sect 4 and we show in sect 5 how all the tools needed can be defined also in the case of non compact manifolds

metric dependence and asymptotic minimization of the expected number of critical points of random holomorphic sections we prove the main conjecture from douglas shiffman and zelditch 2006 concerning the metric dependence and asymptotic minimization of the expected number ncrit n hof critical points of random holomorphic sections of the nth tensor power of a positive line bundle the first non topological term in the asymptotic expansion of ncrit n h is the calabi functional multiplied by the constant 2 m which depends only on the dimension of the manifold we prove that 2 m is strictly positive in all dimensions showing that the expansion is non topological for all m and that the calabi extremal metric when it exists asymptotically minimizes ncrit n h

l3 solutions of the liquid crystals system abstract in this article we study the simplified system of the original ericksen leslie equations for the flow of liquid crystals it is shown that the l 3 solutions of the cauchy problem for the three dimensional incompressible liquid crystal system are smooth

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

regularity of area minimizing currents iii blow up this is the last of a series of three papers in which we give a new shorter proof of a slightly improved version of almgren s partial regularity of area minimizing currents in riemannian manifolds here we perform a blow up analysis deducing the regularity of area minimizing currents from that of dir minimizing multiple valued functions

minimal two spheres in kahler einstein fano manifolds abstract in this paper we show the existence of stable symplectic non holomorphic two spheres in kahler manifolds of positive constant scalar curvature of real dimension four and in kahler einstein fano manifolds of real dimension six some of the techniques used involve deformation theory of algebraic cycles

integral kahler invariants and the bergman kernel asymptotics for line bundles abstract on a compact kahler manifold one can define global invariants by integrating local invariants of the metric assume that a global invariant thus obtained depends only on the kahler class then we show that the integrand can be decomposed into a chern polynomial the integrand of a chern number and divergences of one forms which do not contribute to the integral we apply this decomposition formula to describe the asymptotic expansion of the bergman kernel for positive line bundles and to show that the cr q curvature on a sasakian manifold is a divergence

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

the abreu equation with degenerated boundary conditions abstract the abreu equation is a fully nonlinear 4th order partial differential equation that arises from the study of the extremal metrics on toric manifolds we study the dirichlet problem of the abreu equation with degenerated boundary conditions the solutions provide the kahler metrics of constant scalar curvature on the complex torus

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

array shape calibration using sources in unknown locations a maximum likelihood approach sensor location uncertainty can severely degrade the performance of direction finding systems an iterative maximum likelihood method for simultaneously estimating directions of arrival doa and sensor locations is developed to alleviate this problem the case of nondisjoint sources i e sources observed in the same frequency cell and at the same time is emphasized the algorithm converges to the global maximum of the likelihood function if the initial conditions are sufficiently good numerical examples are presented illustrating the performance of the proposed technique a distinctive feature of the algorithm is its ability to locate the sensors accurately without deploying calibration sources at known locations

critical points and supersymmetric vacua ii asymptotics and extremal metrics motivated by the vacuum selection problem of string m theory we study a new geometric invariant of a positive hermitian line bundle l h m over a compact kahler manifold the expected distribution k crit z of critical points dlog s z h 0 of a gaussian random holomorphic section s h 0 m l with respect to h it is a measure on m whose total mass is the average number n crit h of critical points of a random holomorphic section we are interested in the metric dependence of n crit h especially metrics h which minimize n crit h we concentrate on the asymptotic minimization problem for the sequence of tensor powers l n h n m of the line bundle and their critical point densities k critn z we prove that k critn z has a complete asymptotic expansion in n whose coefficients are curvature invariants of h the first two terms in the expansion of n crit hn are topological invariants of l m the third term is a topological invariant plus a constantm 2 depending only on the dimension m of m times the calabi functional r m 2 dv olh whereis the scalar curvature of the curvature form of h we give an integral formula form 2 and show by a computer assisted calculation thatm 2 0 for m 3 hence that n crit hn is asymptotically minimized by the calabi extremal metric when one exists we conjecture thatm 2 0 in all dimensions i e that the calabi extremal metric is always the asymptotic minimizer

weighted projective embeddings stability of orbifolds and constant scalar curvature kahler metrics we embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of kodaira embedding dividing by the non reductive automorphisms of weighted projective space then formally gives a moduli space of orbifolds we show how to express this as a reductive quotient and so a git problem thus defining a notion of stability for orbifolds we then prove an orbifold version of donaldson s theorem the existence of an orbifold kahler metric of constant scalar curvature implies k semistability by extending the notion of slope stability to orbifolds we therefore get an explicit obstruction to the existence of constant scalar curvature orbifold kahler metrics we describe the manifold applications of this orbifold result and show how many previously known results troyanov ghigi kollar rollin singer the adscft sasaki einstein obstructions of gauntlett martelli sparks yau fit into this framework

on the lower order terms of the asymptotic expansion of tian yau zelditch we compute the first three coefficients of the asymptotic expansion of tian yau zelditch we also prove that in general the fcth coefficient is a polynomial of the curvature and its derivative of weight k

blowing up and desingularizing constant scalar curvature kahler manifolds this paper is concerned with the existence of constant scalar curvature kahler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature kahler metrics we also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature kahler metrics

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

global existence of solutions of the liquid crystal flow for the oseen frank model in r2 abstract in the first part of this paper we establish the global existence of solutions of the liquid crystal gradient flow for the well known oseen frank model the liquid crystal flow is a prototype of equations from the ericksen leslie system in the hydrodynamic theory and generalizes the heat flow for harmonic maps into the 2 sphere the ericksen leslie system is a system of the navier stokes equations coupled with the liquid crystal flow in the second part of this paper we also prove the global existence of solutions of the ericksen leslie system for a general oseen frank model in r 2

relative chow stability and extremal metrics abstract we prove that the existence of extremal metrics implies asymptotically relative chow stability an application of this is the uniqueness up to automorphisms of extremal metrics in any polarization

quantization and the hessian of mabuchi energy let l x be an ample bundle over a compact complex manifold fix a hermitian metric in l whose curvature defines a kahler metric on x the hessian of mabuchi energy is a fourth order elliptic operator d d on functions which arises in the study of scalar curvature we quantize d d by the hessian p k pk of balancing energy a function appearing in the study of balanced embeddings p k pk is defined on the space of hermitian endomorphisms of h0 x lk endowed with the l2 inner product we first prove that the leading order term in the asymptotic expansion of p k pk is d d we next show that if aut x l is discrete then the eigenvalues and eigenspaces of p k pk converge to those of d d we also prove convergence of the hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature kahler metric as consequences of our results we prove that an estimate of phong and sturm is sharp and give a negative answer to a question posed by donaldson we also discuss some possible applications to the study of calabi flow 2012

optimal synthesis of reconfigurable planar arrays with simplified architectures for monopulse radar applications a new approach to the power synthesis of fixed geometry reconfigurable planar arrays radiating sum and difference patterns is presented the proposed design technique allows to maximize the radiation performance field slope amplitude or even directivity of both beam patterns over assigned directions subject to completely arbitrary masks for sidelobe bounds whatever the elements disposal and the array boundary the overall problem is solved through a convex quadratic programming procedure moreover if centrosymmetric antenna layouts are adopted the synthesis is reduced to an even more powerful linear programming routine which preserves the solution uniqueness and optimality guaranteed by quadratic programming codes while dramatically decreasing their computational weight and hence enabling the design of much larger planar arrays the proposed approach also takes into account the need of reducing the complexity of the beam forming network bfn which is fulfilled by sharing part of the excitation amplitudes between the two radiation modalities a set of numerical examples is reported and discussed to show the versatility and effectiveness of the proposed approach

on smoothness of l3 solutions to the navier stokes equations up to boundary we show that l3 solutions to the three dimensional navier stokes equations near a flat part of the boundary are smooth

curvature of vector bundles associated to holomorphic fibrations let l be a semi positive line bundle over a kahler manifold x fibered over a complex manifold y assuming the fibers are compact and nonsingular we prove that the hermitian vector bundle e over y whose fibers over points y are the spaces of global sections over xy to l kx y endowed with the l2 metric is semi positive in the sense of nakano we also discuss various applications among them a partial result on a conjecture of griffiths on the positivity of ample bundles

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

global existence of weak solutions of the nematic liquid crystal flow in dimension three for any bounded smooth domain 3 or 3 we establish the global existence of a weak solution u d 0 3 s2 of the initial boundary value or the cauchy problem of the simplified ericksen leslie system llf modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary or cauchy data u0 d0 h h1 s2 with d0 s 2 the upper hemisphere furthermore u d satisfies the global energy inequality 2016 wiley periodicals inc

finite time singularity of the nematic liquid crystal flow in dimension three in this paper we consider the initial and boundary value problem of a simplified nematic liquid crystal flow in dimension three and construct two examples of finite time singularity the first example is constructed within the class of axisymmetric solutions while the second example is constructed for any generic initial data u 0 d 0 that has sufficiently small energy and d 0 has a nontrivial topology

on einstein hermitian 4 manifolds let m h be a compact 4 dimensional einstein manifold and suppose that h is hermitian with respect to some complex structure j on m then either m j h is kahler einstein or else up to rescaling and isometry it is one of the following two exceptions the page metric on mathbb cp 2 overline mathbb cp 2 or the einstein metric on mathbb cp 2 2 overline mathbb cp 2 discovered in on conformally kahler einstein manifolds j amer math soc 21 2008 pp 1137 1168

simultaneous nulling in sum and difference patterns by amplitude control a method is discussed of suppressing jammers simultaneously in the sum and the difference patterns of a linear antenna array by either using complex weights or controlling the current amplitudes only with special emphasis on the latter method for a given array size the total number of jammers that can be nulled is doubled if complex weights are used instead of real weights however the amplitude only technique has many attractions it is shown that because the resulting current amplitude distribution has a double symmetry the number of attenuators required for simultaneous nulling can be as low as a quarter of the total number of elements in the array this means a much faster computing speed as well as a considerable saving in hardware the method also allows the main beam of the antenna to be scanned by using phase shifters while nulls are placed in directions of known jammers in both the sum and difference patterns by using attenuators as a result the maximization of the signal to jammer ratio can be achieved with relative ease because the stronger a jammer is the easier it is to locate it and hence to suppress it

liquid crystal flows in two dimensions this paper is concerned with a simplified hydrodynamic equation proposed by ericksen and leslie modeling the flow of nematic liquid crystals in dimension two we establish both interior and boundary regularity theorems for such a flow under smallness conditions as a consequence we establish the existence of global in time weak solutions on a bounded smooth domain in mathbb r 2 which are smooth everywhere with possible exceptions of finitely many singular times

duality of positive currents and plurisubharmonic functions in calibrated geometry recently the authors showed that there is a robust potential theory attached to any calibrated manifold x phi in particular on x there exist phi plurisubharmonic functions phi convex domains phi convex boundaries etc all inter related and having a number of good properties in this paper we show that in a strong sense the plurisubharmonic functions are the polar duals of the phi submanifolds or more generally the phi currents studied in the original paper on calibrations in particular we establish an analogue of duval sibony duality which characterizes points in the phi convex hull of a compact set k subset x in terms of phi positive green s currents on x and jensen measures on k we also characterize boundaries of phi currents entirely in terms of phi plurisubharmonic functions specific calibrations are used as examples throughout analogues of the hodge conjecture in calibrated geometry are considered

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

the singular set of 1 1 integral currents we prove that 2 dimensional integer multiplicity 2 dimensional rectifiable currents which are almost complex cycles in an almost complex manifold admitting locally a compatible positive symplectic form are smooth surfaces aside from isolated points and therefore are j holomorphic curves

subarray synthesized low side lobe sum and difference patterns with partial common weights methods of synthesizing low side lobe sum and difference patterns for linear arrays are described by subarray beam forming the sum and difference patterns exhibit a common factor leading to a simple monopulse estimator a procedure is proposed for the synthesis of the subarray based sum and difference patterns which approximate a set of reference patterns in least squares sense supervised nulling is incorporated as a means of suppression of strong interference to further reduce system complexity the constraint that parts of the beam forming weights are shared by the sum and difference channels is imposed the relationship between the maximum number of common weights allowable and the number of subarrays used is discussed numerical results are presented to demonstrate the efficacy of the proposed methods

uniqueness of tangent cones to positive p p integral cycles we prove that every positive p p integral cycle in an arbitrary almost complex manifold possesses at every point a unique tangent cone the argument relies on an algebraic blow up perturbed in order to face the analysis issues of this problem in the almost complex setting primary subjects 53c38 secondary subjects 49q05 49q20 35j99

design of sum and difference patterns with common nulls and low slls simultaneously in the presence of array errors sum sigma and difference delta beampatterns are commonly utilized to estimate the bearings of targets in the radar system however the nulls of delta pattern are not always aligned with those of sigma pattern and this phenomenon would become more severe in practical scenario since the array errors exist meanwhile the sidelobe level sll of the array would also increase due to the unavoidable errors the above described factors would dramatically decrease the performance of radar in the adaptive suppression of interference to increase the output signal to interference plus noise ratio of a radar system this paper presents an approach to design the low sll sigma delta patterns where the nulls of the delta pattern excluding the central null are always aligned with those of the sigma pattern in the presence of errors two schemes are addressed to obtain the covariance matrix for null alignment constraint which is exploited to force nulling in desired angles accordingly in the sequel the matrix tapering method is adopted to broaden the nulls and enhance the robustness for the possible errors with the proposed approach the low sll and nulls alignment can be achieved simultaneously numerical examples are provided and discussed to demonstrate the effectiveness of the proposed solution including the results based on a linearly spaced microstrip array antenna

on partial regularity results for the navier stokes equations looking at the regularity results of scheffer respectively caffarelli kohn and nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the navier stokes equations

a closed formula for the asymptotic expansion of the bergman kernel we prove a graph theoretic closed formula for coefficients in the tian yau zelditch asymptotic expansion of the bergman kernel the formula is expressed in terms of the characteristic polynomial of the directed graphs representing weyl invariants the proof relies on a combinatorial interpretation of a recursive formula due to m englis and a loi

existence of solutions for the ericksen leslie system we study the general ericksen leslie system which describes the flow of liquid crystal materials the dissipation property of the system is established and is used to prove the global existence of weak solutions we also study the existence of classical solutions and the asymptotic stability of the solutions

a variational characterization of complex submanifolds in this note we generalize our results in arezzo and sun reine angew math doi 10 1515 crelle 2013 0097 2012 to integer p currents of any degree we prove that if the mass of a current as a functional of the ambient metric has a critical or stable point in some special directions then the current is complex this holds for any dimension and codimension we also study a natural functional on the space of currents representing a fixed homology class closely related to the first derivative of the mass in our new approach detecting the deviation of a surface from being holomorphic

positivity of direct image bundles and convexity on the space of kahler metrics we develop some results from 4 on the positivity of direct image bundles in the particular case of a trivial fibration over a one dimensional base we also apply the results to study variations of kahler metrics

on the convergence of a modified kahler ricci flow we study the convergence of a modified kahler ricci flow defined by zhou zhang we show that the modified kahler ricci flow converges to a singular metric when the limit class is degenerate this proves a conjecture of zhang

on canonical metrics on cartan hartogs domains the cartan hartogs domains are defined as a class of hartogs type domains over irreducible bounded symmetric domains the purpose of this paper is twofold firstly for a cartan hartogs domain omega b d 0 mu endowed with the canonical metric g mu we obtain an explicit formula for the bergman kernel of the weighted hilbert space mathcal h alpha of square integrable holomorphic functions on left omega b d 0 mu g mu right with the weight exp alpha varphi where varphi is a globally defined kahler potential for g mu for alpha 0 and furthermore we give an explicit expression of the rawnsley s varepsilon function expansion for left omega b d 0 mu g mu right secondly using the explicit expression of the rawnsley s varepsilon function expansion we show that the coefficient a 2 of the rawnsley s varepsilon function expansion for the cartan hartogs domain left omega b d 0 mu g mu right is constant on omega b d 0 mu if and only if left omega b d 0 mu g mu right is biholomorphically isometric to the complex hyperbolic space so we give an affirmative answer to a conjecture raised by m zedda

geodesic convexity of small neighborhood in the space of kahler potentials abstract we show that given k 4 0 j min 1 4 k 4 4 any point in the space of non degenerate smooth kahler potentials has a small neighborhood with respect to c k norm s t any two points in this neighborhood can be connected by a geodesic of at least c k j regularity

development and analysis of a two stage beamformer for multiple correlated interferers using rectangular array a two stage two dimensional 2 d beamformer is proposed for signal enhancement in an environment of multiple correlated interferers using a rectangular array in the first stage subarray beamformers are constructed which exhibit reliable interference cancellation using difference preprocessed and spatially smoothed data a secondary combining of all possible subarray beamformers is then performed to fully exploit the array aperture it is shown that compared to the conventional one stage optimum beamformer the two stage beamformer performs equally well and requires a much lower complexity of implementation when a suitable subarray size is chosen

projective embeddings and lagrangian fibrations of abelian varieties it is well known that every abelian variety can be embedded into projective spaces by theta functions and the basis of theta functions are determined by choosing a lagrangian fibration in this paper we prove that the restriction of natural lagrangian fibrations moment maps of projective spaces converge to that of the abelian variety in the gromov hausdorff topology this is in some sense a lagrangian fibration version of the convergence theorem of g tian 6 and s zelditch 7 for kahler metrics

non minimal scalar flat kahler surfaces and parabolic stability a new construction is presented of scalar flat kahler metrics on non minimal ruled surfaces the method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank 2 parabolically stable holomorphic bundles this rather general construction is shown also to give new examples of low genus in particular it is shown that mathbb cp 2 blown up at 10 suitably chosen points admits a scalar flat kahler metric this answers a question raised by claude lebrun in 1986 in connection with the classification of compact self dual 4 manifolds

radial balanced metrics on the unit ball of the kepler manifold abstract we show that there is no radial balanced metric on the unit ball of the kepler manifold with not too wild boundary behavior additionally we identify explicitly the weights corresponding to radial metrics with such boundary behavior which satisfy the balanced condition as far as germs at the boundary are concerned related results for poincare metrics are also established