the cauchy problem for the homogeneous monge ampere equation ii legendre transform abstract we continue our study of the cauchy problem for the homogeneous real and complex monge ampere equation hrma hcma in the prequel y a rubinstein and s zelditch 27 a quantum mechanical approach for solving the hcma was developed and was shown to coincide with the well known legendre transform approach in the case of the hrma in this article that uses tools of convex analysis and can be read independently we prove that the candidate solution produced by these methods ceases to solve the hrma even in a weak sense as soon as it ceases to be differentiable at the same time we show that it does solve the equation on its dense regular locus and we derive an explicit a priori upper bound on its monge ampere mass the technique involves studying regularity of legendre transforms of families of non convex functions

k energy on polarized compactifications of lie groups abstract in this paper we study mabuchi s k energy on a compactification m of a reductive lie group g which is a complexification of its maximal compact subgroup k we give a criterion for the properness of k energy on the space of k k invariant kahler potentials in particular it turns to give an alternative proof of delcroix s theorem for the existence of kahler einstein metrics in case of fano manifolds m we also study the existence of minimizers of k energy for general kahler classes of m

remarks on the extension of random contractions for x1 x2 d where d is a nonempty subset of x the problem of the existence of an extension of any contraction mapping is an interesting one this problem was studied for euclidean spaces by kirzbraun 7 and extended to hilbert spaces by valentine 15 16 recently myjak and zygadlewicz 11 beg and shahzad 3 and shahzad 14 gave stochastic versions of the kirzbraun valentine extension theorem let be a measurable space with a algebra of subsets of denote by i and f the sets of in nite and nite sequences of positive integers respectively let g be a family of sets and k f g be a map for each r ri i 1 i and n n we shall denote r n by r1 rn then r i n 1 k r n is said to be obtained from g by the souslin family if every set obtained from g in this way is also in g then g is called a souslin family if is an outer measure on the is a souslin family in particular if is a complete measurable space then is a souslin family see 17 let 2x be the family of all subsets of x and c x the family of all nonempty closed subsets of x a multifunction f 2x is said to be measurable resp weakly measurable if for every closed resp open subset a of x the set f 1 a f a 6 note that if x is a metric space measurability implies weak

complex symplectomorphisms and pseudo kahler islands in the quantization of toric manifolds let p be a delzant polytope we show that the quantization of the corresponding toric manifold x p in toric kahler polarizations and in the toric real polarization are related by analytic continuation of hamiltonian flows evaluated at time t sqrt 1 s we relate the quantization of x p in two different toric kahler polarizations by taking the time sqrt 1 s hamiltonian flow of strongly convex functions on the moment polytope p by taking s to infinity we obtain the quantization of x p in the singular real toric polarization recall that x p has an open dense subset which is biholomorphic to mathbb c n the quantization of x p in a toric kahler polarization can also be described by applying the complexified hamiltonian flow of the abreu guillemin symplectic potential g at time t sqrt 1 to an appropriate finite dimensional subspace of quantum states in the quantization of t mathbb t n in the vertical polarization by taking other imaginary times t k sqrt 1 k in mathbb r we describe toric kahler metrics with cone singularities along the toric divisors in x p for convex hamiltonian functions and sufficiently negative imaginary part of the complex time we obtain degenerate kahler structures which are negative definite in some regions of x p we show that the pointwise and l 2 norms of quantum states are asymptotically vanishing on negative definite regions

toric surfaces k stability and calabi flow let x be a toric surface and u be a normalized symplectic potential on the corresponding polygon p suppose that the riemannian curvature is bounded by a constant c 1 and int partial p u d sigma 0 depending only on c 1 c 2 and p such that donaldson s m condition holds for u as an application we show that if x p is analytic relative k stable then the modified calabi flow converges to an extremal metric exponentially fast by assuming that the calabi flow exists for all time and the riemannian curvature is uniformly bounded along the calabi flow

semi classical weights and equivariant spectral theory abstract we prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action these inverse spectral results involve the asymptotic equivariant spectrum which is the spectrum itself together with very large weights of the torus action on eigenspaces more precisely we show that the asymptotic equivariant spectrum of the laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold we also show that the asymptotic equivariant spectrum of a t n invariant schrodinger operator on r n determines its potential in some suitably convex cases in addition we prove that the asymptotic equivariant spectrum of an s 1 invariant metric on s 2 determines the metric itself in many cases finally we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces as a crucial ingredient in these inverse results we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi classical differential operators invariant under a torus action

tropical counting from asymptotic analysis on maurer cartan equations let be a toric surface and let be its landau ginzburg lg mirror where is the hori vafa potential as shown in their preprint we apply asymptotic analysis to study the extended deformation theory of the lg model and prove that semi classical limits of fourier modes of a specific class of maurer cartan solutions naturally give rise to tropical disks in with maslov index 0 or 2 the latter of which produces a universal unfolding of for our construction reproduces gross perturbed potential adv math 224 2010 pp 169 245 which was proven to be the universal unfolding of written in canonical coordinates we also explain how the extended deformation theory can be used to reinterpret the jumping phenomenon of across walls of the scattering diagram formed by maslov index 0 tropical disks originally observed by gross in the same work in the case of

kahler metrics on toric orbifolds a theorem of e lerman and s tolman generalizing a result of t delzant states that compact symplectic toric orbifolds are classified by their moment polytopes together with a positive integer label attached to each of their facets in this paper we use this result and the existence of global actionangle coordinates to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold by means of smooth functions on the corresponding moment polytope this is equivalent to parametrizing all toric kahler metrics and generalizes an analogous result for toric manifolds a simple explicit description of interesting families of extremal kahler metrics arising from recent work of r bryant is given as an application of the approach in this paper the fact that in dimension four these metrics are self dual and conformally einstein is also discussed this gives rise in particular to a one parameter family of self dual einstein metrics connecting the well known eguchi hanson and taub nut metrics

ricci flat kahler metrics on crepant resolutions of kahler cones we prove that a crepant resolution y x of a ricci flat kahler cone x admits a complete ricci flat kahler metric asymptotic to the cone metric in every kahler class in h 2 c y mathbb r a kahler cone x bar g is a metric cone over a sasaki manifold s g i e x c s s times mathbb r 0 with bar g dr 2 r 2 g and x bar g is ricci flat precisely when s g einstein of positive scalar curvature this result contains as a subset the existence of ale ricci flat kahler metrics on crepant resolutions pi y rightarrow x mathbb c n gamma with gamma subset sl n mathbb c due to p kronheimer n 2 and d joyce n 2 we then consider the case when x c s is toric it is a result of a futaki h ono and g wang that any gorenstein toric kahler cone admits a ricci flat kahler cone metric it follows that if a toric kahler cone x c s admits a crepant resolution y x then y admits a tn invariant ricci flat kahler metric asymptotic to the cone metric x bar g in every kahler class in h 2 c y mathbb r a crepant resolution in this context is a simplicial fan refining the convex polyhedral cone defining x we then list some examples which are easy to construct using toric geometry

toric generalized kahler ricci solitons with hamiltonian 2 form we show that the generalized kahler ricci soliton equation on 4 dimensional toric kahler orbifolds reduces to odes assuming there is a hamiltonian 2 form this leads to an explicit resolution of this equation on labelled triangles and convex labelled quadrilaterals in particular we give the explicit expression of the kahler ricci solitons of weighted projective planes as well as new examples

t duality and homological mirror symmetry for toric varieties abstract let x be a complete toric variety the coherent constructible correspondence of fang et al 2011 14 equates p erf t x with a subcategory sh c c m r of constructible sheaves on a vector space m r the microlocalization equivalence of nadler and zaslow 2009 27 and nadler 2009 25 relates these sheaves to a subcategory fuk t m r of the fukaya category of the cotangent t m r when x is nonsingular taking the derived category yields an equivariant version of homological mirror symmetry dcoh t x dfuk t m r which is an equivalence of triangulated tensor categories the nonequivariant coherent constructible correspondence of treumann preprint 33 embeds p erf x into a subcategory s h c t r of constructible sheaves on a compact torus t r when x is nonsingular the composition of and microlocalization yields a version of homological mirror symmetry dcoh x dfuk t t r which is a full embedding of triangulated tensor categories when x is nonsingular and projective the composition is compatible with t duality in the following sense an equivariant ample line bundle l has a hermitian metric invariant under the real torus whose connection defines a family of flat line bundles over the real torus orbits this data produces a t dual lagrangian brane l on the universal cover t m r of the dual real torus fibration we prove l l in fuk t m r thus equivariant homological mirror symmetry is determined by t duality

on the convergence of constrained parallel variable distribution algorithms we consider the parallel variable distribution pvd approach proposed by ferris and mangasarian siam j optim 4 1994 pp 815 832 for solving optimization problems the problem variables are distributed among p processors with each processor having the primary responsibility for updating its block of variables while allowing the remaining secondary variables to change in a restricted fashion along some easily computable directions for constrained nonlinear programs convergence in m c ferris and o l mangasarian siam j optim 4 1994 pp 815 832 was established in the special case of convex block separable constraints for general inseparable constraints it was suggested that a dual differentiable exact penalty function reformulation of the problem be used we propose to apply the pvd approach to problems with general convex constraints directly and show that the algorithm converges provided certain conditions are imposed on the change of secondary variables these conditions are both natural and practically implementable we also show that the original requirement of exact global solution of the parallel subproblems can be replaced by a less stringent sufficient descent condition the first rate of convergence result for the class of constrained pvd algorithms is also given

calabi yau metrics on canonical bundles of complex flag manifolds abstract in the present paper we provide a description of complete calabi yau metrics on the canonical bundle of generalized complex flag manifolds by means of lie theory we give an explicit description of complete ricci flat kahler metrics obtained through the calabi ansatz technique we use this approach to provide several explicit examples of noncompact complete calabi yau manifolds these examples include canonical bundles of non toric flag manifolds e g grassmann manifolds and full flag manifolds

lagrangian floer theory on compact toric manifolds i we introduced the notion of weakly unobstructed lagrangian submanifolds and constructed their potential function po purely in terms of a model data in fooo3 in this article we carry out explicit calculations involving po on toric manifolds and study the relationship between this class of lagrangian submanifolds with the earlier work of givental g1 which advocates that the quantum cohomology ring is isomorphic to the jacobian ring of a certain function called the landau ginzburg superpotential combining this study with the results from fooo3 we also apply the study to various examples to illustrate its implications to symplectic topology of lagrangian fibers of toric manifolds in particular we relate it to the hamiltonian displacement property of lagrangian fibers and to entov polterovich s symplectic quasi states

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

interior estimates for the n dimensional abreuʼs equation abstract we study the abreu s equation in n dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform k stability

hamiltonian 2 forms in kahler geometry ii global classification we present a classification of compact kahler manifolds admitting a hamiltonian 2 form which were classified locally in part i of this work this involves two components of independent interest the first is the notion of a rigid hamiltonian torus action this natural condition for torus actions on a kahler manifold was introduced locally in part i but such actions turn out to be remarkably well behaved globally leading to a fairly explicit classification up to a blow up compact kahler manifolds with a rigid hamiltonian torus action are bundles of toric kahler manifolds the second idea is a special case of toric geometry which we call orthotoric we prove that orthotoric kahler manifolds are diffeomorphic to complex projective space but we extend our analysis to orthotoric orbifolds where the geometry is much richer we thus obtain new examples of kahler einstein 4 orbifolds combining these two themes we prove that compact kahler manifolds with hamiltonian 2 forms are covered by blow downs of projective bundles over kahler products and we describe explicitly how the kahler metrics with a hamiltonian 2 form are parameterized we explain how this provides a context for constructing new examples of extremal kahler metrics in particular a subclass of such metrics which we call weakly bochner flat we also provide a self contained treatment of the theory of compact toric kahler manifolds since we need it and find the existing literature incomplete

numerical kahler einstein metric on the third del pezzo the third del pezzo surface admits a unique kahler einstein metric which is not known in closed form the manifold s toric structure reduces the einstein equation to a single monge ampere equation in two real dimensions we numerically solve this nonlinear pde using three different algorithms and describe the resulting metric the first two algorithms involve simulation of ricci flow in complex and symplectic coordinates respectively the third algorithm involves turning the pde into an optimization problem on a certain space of metrics which are symplectic analogues of the algebraic metrics used in numerical work on calabi yau manifolds our algorithms should be applicable to general toric manifolds using our metric we compute various geometric quantities of interest including laplacian eigenvalues and a harmonic 1 1 form the metric and 1 1 form can be used to construct a klebanov tseytlin like supergravity solution