lipschitz operator ideals and the approximation property abstract we establish the basics of the theory of lipschitz operator ideals with the aim of recovering several classes of lipschitz maps related to absolute summability that have been introduced in the literature in the last years as an application we extend the notion and main results on the approximation property for banach spaces to the case of metric spaces

supports of derivations free factorizations and ranks of fixed subgroups in free groups for f a free group of finite rank it is shown that the fixed subgroup of any set b of endomorphisms of f has rank rank f generalizing a recent result of dicks and ventura the proof involves the combinatorics of derivations of groups some related questions are examined

test configurations large deviations and geodesic rays on toric varieties this article contains a detailed study in the case of a toric variety of the geodesic rays t defined by phong and sturm corresponding to test configurations t in the sense of donaldson we show that the bergman approximations k t z of phong and sturm converge in c1 to the geodesic ray t and that the geodesic ray itself is c1 1 and no better in particular the kahler metrics t 0 i t associated to the geodesic ray of potentials are discontinuous across certain hypersurfaces and are degenerate on certain open sets a novelty in the analysis is the connection between bergman metrics bergman kernels and the theory of large deviations we construct a sequence of measures kz on the polytope of the toric variety show that they satisfy a large deviations principle and relate the rate function to the geodesic ray

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

regularity of the geodesic equation in the space of sasakian metrics abstract in this paper we study a geodesic equation in the space of sasakian metrics h the equation leads to the dirichlet problem of a complex monge ampere type equation on the kahler cone this equation differs from the standard complex monge ampere equation in a significant way with gradient terms involved in the 1 1 symmetric tensor of the operator we establish appropriate regularity estimates for this complex monge ampere type equation as geometric application we show that the space of sasakian metrics h is a metric space and the constant transversal scalar curvature metric realizes the global minimum of k energy if the first basic chern class c 1 b 0 we also prove that the constant transversal scalar curvature metric is unique in each basic kahler class if the first basic chern class is either strictly negative or zero

the calabi metric for the space of kahler metrics given any closed kahler manifold we define following an idea by calabi bull am math soc 60 167 168 1954 a riemannian metric on the space of kahler metrics regarded as an infinite dimensional manifold we prove several geometrical features of the resulting space some of which we think were already known to calabi in particular the space is a portion of an infinite dimensional sphere and explicit unique smooth solutions for the cauchy and the dirichlet problems for the geodesic equation are given

convexity of the k energy on the space of kahler metrics and uniqueness of extremal metrics we establish the convexity of mabuchi s k energy functional along weak geodesics in the space of kahler potentials on a compact kahler manifold thus confirming a conjecture of chen and give some applications in kahler geometry including a proof of the uniqueness of constant scalar curvature metrics or more generally extremal metrics modulo automorphisms the key ingredient is a new local positivity property of weak solutions to the homogeneous monge ampere equation on a product domain whose proof uses plurisubharmonic variation of bergman kernels

long time existence of minimizing movement solutions of calabi flow abstract we recast the calabi flow in degiorgi s language of minimizing movements we establish the long time existence of minimizing movements for k energy with arbitrary initial condition furthermore we establish some a priori regularity results for these solutions and that sufficiently regular minimizing movements are smooth solutions to calabi flow

relative k stability for kahler manifolds we study the existence of extremal kahler metrics on kahler manifolds after introducing a notion of relative k stability for kahler manifolds we prove that kahler manifolds admitting extremal kahler metrics are relatively k stable along the way we prove a general l p lower bound on the calabi functional involving test configurations and their associated numerical invariants answering a question of donaldson when the kahler manifold is projective our definition of relative k stability is stronger than the usual definition given by szekelyhidi in particular our result strengthens the known results in the projective case even for constant scalar curvature kahler metrics and rules out a well known counterexample to the naive version of the yau tian donaldson conjecture in this setting

bifurcation currents in holomorphic dynamics on p k potential theory has been introduced in one dimensional rational dynamics by brolin and tortrat 4 29 but does not play a central role there in higher dimension however as the classical tools are not any longer efficient pluri potential theory has revealed itself to be essential the fundamental works of hubbard papadopol fornaess sibony bedfordsmillie briend duval see 26 for precise references enlight the remarkable efficency of pluri potential theory in holomorphic dynamics on pk or ck it is therefore tempting to study the parameter spaces in a similar way more precisely one would like to relate the bifurcations of an holomorphic family f x of endomorphisms of pk to the powers of a certain current on the parameter space x let us recall that in dimension k 1 a bifurcation is said to occur at some point 0 x if the julia set of f does not move continuously around 0 the famous work of mane sadsullivan 16 which is based on the lemma and the fatou cremer sullivan classification relates the bifurcations with the instability of the critical orbits it also establishes that the bifurcations concentrate on the complement of an open dense subset of x for the quadratic family z2 x c the bifurcation locus is precisely the boundary of the mandelbrot set

the gursky streets equations gursky streets introduced a formal riemannian metric on the space of conformal metrics in a fixed conformal class of a compact riemannian four manifold in the context of the sigma 2 yamabe problem the geodesic equation of gursky streets metric is a fully nonlinear degenerate elliptic equation using this geometric structure and the geodesic equation gursky streets proved an important result in conformal geometry that the solution of the sigma 2 yamabe problem is unique the existence of such a solution was known more than a decade ago a key ingredient is the convexity of chang yang s mathcal f functional along the smooth geodesic however gursky streets have only proved uniform c 0 1 regularity for a perturbed equation and it turns out that the uniform c 1 1 regularity is very delicate without such a uniform c 1 1 regularity gursky streets arguments of the uniqueness theorem are very complicated in this paper we establish the uniform c 1 1 regularity of the gursky streets equation in the course of deriving regularity we also obtain very interesting and new convexity regarding matrices in gamma 2 as an application we can establish strictly the convexity of mathcal f functional along c 1 1 geodesic this gives a straightforward proof of the uniqueness of solutions of sigma 2 yamabe problem

a brunn minkowski type inequality for fano manifolds and some uniqueness theorems in kahler geometry for phi a metric on the anticanonical bundle k x of a fano manifold x we consider the volume of x begin aligned int x e phi end aligned in earlier papers we have proved that the logarithm of the volume is concave along geodesics in the space of positively curved metrics on k x our main result here is that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on x even with very low regularity assumptions on the geodesic as a consequence we get a simplified proof of the bando mabuchi uniqueness theorem for kahler einstein metrics a generalization of this theorem to twisted kahler einstein metrics and some classes of manifolds that satisfy weaker hypotheses than being fano is also given we moreover discuss a generalization of the main result to other bundles than k x and finally use the same method to give a new proof of the theorem of tian and zhu on uniqueness of kahler ricci solitons

remarks on the geodesic einstein metrics of a relative ample line bundle in this paper we introduce the associated geodesic einstein flow for a relative ample line bundle l over the total space mathcal x of a holomorphic fibration and obtain a few properties of that flow in particular we prove that the pair mathcal x l is nonlinear semistable if the associated donaldson type functional is bounded from below and the geodesic einstein flow has long time existence property we also define the associated s classes and c classes for mathcal x l and obtain two inequalities between them when l admits a geodesic einstein metric finally in the appendix of this paper we prove that a relative ample line bundle is geodesic einstein if and only if an associated infinite rank bundle is hermitian einstein

on the smoothness of convex envelopes we examine differentiability properties of the convex envelope conv e of a given function e rn cc cc in terms of properties of e it is shown that c1 as well as optimal c1 a regularity results 0 a 1 can be obtained under general conditions

geodesics in the space of singular kahler potentials motivated by the results of b berndtsson in this memoir we use the new estimates developed by w he to extend a theorem of the second author on the existence of weak c 1 1 geodesics between two smooth non degenerate kahler potentials to the case where the metrics on the end points may have singularities on some analytic set and may be positive semi definite

geodesics in the space of kahler cone metrics i in this paper we study the dirichlet problem of the geodesic equation in the space of k ahler cone metrics cal h cal b that is equivalent to a homogeneous complex monge amp ere equation whose boundary values consist of k ahler metrics with cone singularities our approach concerns the generalization of the function space defined by donaldson to the case of k ahler manifolds with boundary moreover we introduce a subspace cal h c of cal h cal b which we define by prescribing appropriate geometric conditions our main result is the existence uniqueness and regularity of c 1 1 cal b geodesics whose boundary values lie in cal h c moreover we prove that such geodesic is the limit of a sequence of c 2 alpha cal b approximate geodesics under the c 1 1 cal b norm as a geometric application we prove the metric space structure of cal h c

weak and quasi approximation properties in banach spaces in this paper we introduce weak versions the weak approximation property the bounded weak approximation property and the quasi approximation property of the approximation property and derive various characterizations of these properties and we show that if the dual of a banach space x has the weak approximation property respectively the bounded weak approximation property then x itself has the weak approximation property respectively the bounded weak approximation property also we observe that the bounded weak approximation property is closely related to the quasi approximation property

fixed subgroups of endomorphisms of free products abstract let g i 1 n g i and let be a symmetric endomorphism of g if is a monomorphism or if g is a finitely generated residually finite group then the fixed subgroup fix g g g g of has kurosh rank at most n

the space of poincare type kahler metrics on the complement of a divisor consider a divisor d with simple normal crossings in a compact kahler manifold x it has been known since the work by g tian and s t yau that if k d is ample there exists on x d a unique kahler einstein metric with cusp singularities along the divisor implying completeness and finite volume we show in this article that a kahler metric in an arbitrary class with constant scalar curvature and singularities analogous to that constructed by tian and yau is unique in this class when k d is ample this we do by generalizing chen s construction of approximate geodesics in the space of kahler metrics and proving an approximate version of the calabiyau theorem both independently of the ampleness of k d

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

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

a polynomial hutton theorem with applications abstract extending a classical linear result due to hutton to a nonlinear setting we prove that a continuous homogeneous polynomial between banach spaces can be approximated by finite rank polynomials if and only if its adjoint can be approximated by finite rank linear operators among other consequences we apply this result to generalize a classical result due to aron and schottenloher about the approximation property on spaces of polynomials and a recent result due to caliskan and rueda about the quasi approximation property on projective symmetric tensor products

linearization of bounded holomorphic mappings on banach spaces the main result in this paper is the following linearization theorem for each open set u in a complex banach space e there is a complex banach space goo u and a bounded holomorphic mapping gu u g u with the following universal property for each complex banach space f and each bounded holomorphic mapping f u f there is a unique continuous linear operator tf goo u f such that tf gu f the correspondence f tf is an isometric isomorphism between the space ho u f of all bounded holomorphic mappings from u into f and the space l g u f of all continuous linear operators from g u into f these properties characterize g u uniquely up to an isometric isomorphism the rest of the paper is devoted to the study of some aspects of the interplay between the spaces h o u f and l g u f this paper consists of five sections in 1 we establish our notation and terminology in 2 we prove the aforementioned linearization theorem in 3 we translate certain properties of a mapping f e h u f into properties of the corresponding operator tf e l g u f we show for instance that f has a relatively compact range if and only if tf is a compact operator in 4 we give a seminorm characterization of the unique locally convex topology t y on h u f such that the correspondence f tf is a topological isomorphism between the spaces h u f ty and l g u f t where tc denotes the compact open topology finally in 5 we use the preceding results to establish necessary and sufficient conditions for the spaces g u and h u to have the approximation property these are holomorphic analogues of classical results of a grothendieck 8 and complement results of r aron and m schottenloher 2 we show in particular that if u is a balanced bounded open set in a complex banach space e then g u has the approximation property if and only if e has the approximation property we also show that if u is an arbitrary open set in a complex banach space e then h u has the approximation property if and only if for each complex banach space f each mapping in h u f with a relatively compact range can be uniformly approximated on u by mappings in h u f with finite dimensional range since it is still unknown whether received by the editors april 10 1989 1980 mathematics subject classification 1985 revision primary 46g20 46e15 secondary 46e10 d 1991 american mathematical society 0002 9947 91 1 00 25 per page

complex legendre duality we introduce complex generalizations of the classical legendre transform operating on kahler metrics on a compact complex manifold these legendre transforms give explicit local isometric symmetries for the mabuchi metric on the space of kahler metrics around any real analytic kahler metric answering a question originating in semmes work

examples of retracts in free groups that are not the fixed subgroup of any automorphism let n be an arbitrary cardinal and let fn be a free group of rank n the flxed subgroup of an endomorphismof fn is the subgroup of elements in fn flxed by in this paper the relationship between the family of flxed subgroups of endomorphisms of fn and the family of flxed subgroups of automorphisms of fn will be studied we prove that these two families of subgroups do not coincide for n 3 by showing an inflnite sequence of explicit examples of retracts of fn and so flxed subgroups of endomorphisms of fn which are not flxed subgroups of any automorphism of fn

ricci flow and the metric completion of the space of kahler metrics we consider the space of kahler metrics as a riemannian submanifold of the space of riemannian metrics and study the associated submanifold geometry in particular we show that the intrinsic and extrinsic distance functions are equivalent we also determine the metric completion of the space of kahler metrics making contact with recent generalizations of the calabi yau theorem due to dinew and guedj zeriahi as an application we obtain a new analytic stability criterion for the existence of a kahler einstein metric on a fano manifold in terms of the ricci flow and the distance function we also prove that the kahler ricci flow converges as soon as it converges in the metric sense

weak strong continuity of multilinear mappings and the pelczynski pitt theorem abstract in generalizing a result of pelczynski the sequential weak to norm continuity ofn linear mappings between certain banach spaces including spaces of type and cotype will be studied in particular it is shown that everyn linear continuous mapping from lp1 lpninto lqis compact if and only if i 1n 1 pi

the group fixed by a family of endomorphisms of a surface group for a closed surface s with s 0 we show that the fixed subgroup of a family b of endomorphisms of 1 s has rankfixb rank 1 s in particular if b contains a non epimorphic endomorphism then rankfixb 12rank 1 s we also show that geometric subgroups of 1 s are inert and hence the fixed subgroup of a family of epimorphisms of 1 s is also inert

on equalizers of sections abstract let g and m be groups and a b g g m group theoretic sections of the natural projection g m g we use the almost stability theorem pro trees and new folding sequence techniques to show that if g is finitely generated then the equalizer of a and b is a free factor of g which answers a question of g m bergman

a donaldson type functional on a holomorphic finsler vector bundle in this paper we solve a problem of kobayashi posed in complex finsler vector bundles american mathematical society providence 1996 by introducing a donaldson type functional on the space f e of strongly pseudo convex complex finsler metrics on e a holomorphic vector bundle over a closed kahler manifold m this donaldson type functional is a generalization in the complex finsler geometry setting of the original donaldson functional and has finsler einstein metrics on e as its only critical points at which this functional attains the absolute minimum

partial coherent state transforms g t invariant kahler structures and geometric quantization of cotangent bundles of compact lie groups abstract in this paper we study the analytic continuation to complex time of the hamiltonian flow of certain g t invariant functions on the cotangent bundle of a compact connected lie group g with maximal torus t namely we will take the hamiltonian flows of one g g invariant function h and one g t invariant function f acting with these complex time hamiltonian flows on g g invariant kahler structures gives new g t invariant but not g g invariant kahler structures on t g we study the hilbert spaces h corresponding to the quantization of t g with respect to these non invariant kahler structures on the other hand by taking the vertical schrodinger polarization as a starting point the above g t invariant hamiltonian flows also generate families of mixed polarizations p 0 c im 0 each of these mixed polarizations is globally given by a direct sum of an integrable real distribution and of a complex distribution that defines a kahler structure on the leaves of a foliation of t g the geometric quantization of t g with respect to these mixed polarizations gives rise to unitary partial coherent state transforms corresponding to ksh maps as defined in 11 12

some discretizations of geometric evolution equations and the ricci iteration on the space of kahler metrics we propose the study of certain discretizations of geometric evolution equations as an approach to the study of the existence problem of some elliptic partial differential equations of a geometric nature as well as a means to obtain interesting dynamics on certain infinite dimensional spaces we illustrate the fruitfulness of this approach in the context of the ricci flow as well as another flow in kahler geometry we introduce and study dynamical systems related to the ricci operator on the space of kahler metrics that arise as discretizations of these flows we pose some problems regarding their dynamics we point out a number of applications to well studied objects in kahler and conformal geometry such as constant scalar curvature metrics kahler ricci solitons nadel type multiplier ideal sheaves balanced metrics the moser trudinger onofri inequality energy functionals and the geometry and structure of the space of kahler metrics e g we obtain a new sharp inequality strengthening the classical moser trudinger onofri inequality on the two sphere

embedding rings in completed graded rings 4 commutative algebras abstract if k is a commutative ring of finite characteristic and a a commutative associative k algebra with augmentation such that the intersection of the powers of the augmentation ideal is zero it is shown that a can be embedded in a completed graded commutative associative k algebra with degree 0 component just k the corresponding result is false in characteristic 0 though we do not know whether it can fail when k is a field the above finite characteristic result is also true with commutative associative algebra replaced by any variety of associative or nonassociative algebras defined by multilinear identities

embedding rings in completed graded rings 1 triangular embeddings abstract let k be a commutative ring with 1 r an associative k algebra without 1 and n a positive integer it is shown that the following conditions are equivalent a r can be embedded as a k algebra in the ring of strictly upper triangular n n matrices over an associative k algebra a b r has such an embedding with a further assumed commutative c r can be embedded as a k algebra in a graded associative k algebra c whose only nonzero components are c 1 c nn 1 the proof that the other conditions imply b is essentially a demonstration that a certain kind of k linear category always has a faithful functor into a commutative k algebra subsequent papers in this series determine sufficient conditions for a c above to hold in terms of nilpotence of r under various hypotheses

symmetries and other transformations of the complex monge ampere equation on definit des transformations tangentes en utilisant la structure symplectique holomorphe du fibre cotangent holomorphe

on p compact mappings and the p approximation property abstract the notion of p compact sets arises naturally from grothendieck s characterization of compact sets as those contained in the convex hull of a norm null sequence the definition due to sinha and karn 2002 leads to the concepts of p approximation property and p compact operators which form an ideal with its ideal norm p this paper examines the interaction between the p approximation property and certain space of holomorphic functions the p compact analytic functions in order to understand these functions we define a p compact radius of convergence which allows us to give a characterization of the functions in the class we show that p compact holomorphic functions behave more like nuclear than compact maps we use the product of schwartz to characterize the p approximation property of a banach space in terms of p compact homogeneous polynomials and in terms of p compact holomorphic functions with range on the space finally we show that p compact holomorphic functions fit into the framework of holomorphy types which allows us to inspect the p approximation property our approach also allows us to solve several questions posed by aron maestre and rueda 2010

geodesic einstein metrics and nonlinear stabilities in this paper we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic einstein metric on this line bundle which generalize the classical stabilities and hermitian einstein metrics of holomorphic vector bundles we introduce a donaldson type functional and show that this functional attains its absolute minimum at geodesic einstein metrics and we also discuss the relations between the existence of geodesic einstein metrics and the nonlinear stabilities of the line bundle as an application we will prove that a holomorphic vector bundle admits a finsler einstein metric if and only if it admits a hermitian einstein metric which answers a problem posed by s kobayashi

complex monge ampere equations and totally real submanifolds abstract we study the dirichlet problem for complex monge ampere equations in hermitian manifolds with general non pseudoconvex boundary our main result theorem 1 1 extends the classical theorem of caffarelli kohn nirenberg and spruck in c n we also consider the equation on compact manifolds without boundary attempting to generalize yau s theorems in the kahler case as applications of the main result we study some connections between the homogeneous complex monge ampere hcma equation and totally real submanifolds and a special dirichlet problem for the hcma equation related to donaldson s conjecture on geodesics in the space of kahler metrics

the mabuchi geometry of finite energy classes abstract we introduce different finsler metrics on the space of smooth kahler potentials that will induce a natural geometry on various finite energy classes e x motivated by questions raised by r berman v guedj and y rubinstein we characterize the underlying topology of these spaces in terms of convergence in energy and give applications of our results to existence of kahler einstein metrics on fano manifolds

on automorphism fixed subgroups of a free group abstract let f be a finitely generated free group and let n denote its rank a subgroup h of f is said to be automorphism fixed or auto fixed for short if there exists a set s of automorphisms of f such that h is precisely the set of elements fixed by every element of s similarly h is 1 auto fixed if there exists a single automorphism of f whose set of fixed elements is precisely h we show that each auto fixed subgroup of f is a free factor of a 1 auto fixed subgroup of f we show also that if and only if n 3 then there exist free factors of 1 auto fixed subgroups of f which are not auto fixed subgroups of f a 1 auto fixed subgroup h of f has rank at most n by the bestvina handel theorem and if h has rank exactly n then h is said to be a maximum rank 1 auto fixed subgroup of f and similarly for auto fixed subgroups hence a maximum rank auto fixed subgroup of f is a maximum rank 1 auto fixed subgroup of f we further prove that if h is a maximum rank 1 auto fixed subgroup of f then the group of automorphisms of f which fix every element of h is free abelain of rank at most n 1 all of our results apply also to endomorphisms

kahler non collapsing eigenvalues and the calabi flow abstract we first prove a new compactness theorem of kahler metrics which confirms a prediction in 17 then we establish several eigenvalue estimates along the calabi flow combining the compactness theorem and these eigenvalue estimates we generalize the method developed for the kahler ricci flow in 22 to obtain several new small energy theorems of the calabi flow

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

the dirichlet and the weighted metrics for the space of kahler metrics in this work we study the intrinsic geometry of the space of kahler metrics under various riemannian metrics and the corresponding variational structures the first part is on the dirichlet metric a motivation for the study of this metric comes from chen and zheng j reine angew math 674 195 251 2013 there chen and the second author showed that the pseudo calabi flow is the gradient flow of the k energy when mathcal h is endowed precisely with the dirichlet metric the second part is on the family of weighted metrics whose distinguished element is the calabi metric studied in calamai math ann 353 373 402 2012 we investigate as well their geometric properties then we focus on the constant weight metric we use it to give an alternative proof of calabi s uniqueness of the kahler einstein metrics cf theorem 5 9 when c 1 le 0 we also introduce a new functional called g functional and we prove that its gradient flow has long time existence and also converges to the kahler einstein metric when c 1 le 0 cf theorem 5 10

the cauchy problem for the homogeneous monge ampere equation i toeplitz quantization the cauchy problem for the homogeneous real complex monge ampere equation hrma hcma arises from the initial value problem for geodesics in the space of kahler metrics equipped with the mabuchi metric this cauchy problem is believed to be ill posed and a basic problem is to characterize initial data of weak solutions which exist up to time t in this article we use a quantization method to construct a subsolution of the hcma on a general projective variety and we conjecture that it solves the equation for as long as the unique solution exists the subsolution called the quantum analytic continuation potential is obtained by i toeplitz quantizing the hamiltonian flow determined by the cauchy data ii analytically continuing the quantization and iii taking a certain logarithmic classical limit we then prove that in the case of torus invariant metrics where the hcma reduces to the hrma the quantum analytic continuation potential coincides with the well known legendre transform potential and hence solves the equation as long as it is smooth in the sequel the cauchy problem for the homogeneous monge ampere equation ii legendre transform adv math 228 2011 2989 3025 it is proved that the legendre transform potential ceases to solve the hcma once it ceases to be smooth the results here and in the sequels in particular characterize the initial data of smooth geodesic rays

differential geometry of real monge ampere foliations on etudie les feuilletages qui proviennent de la condition de monge ampere recquerant le hessien d une fonction definie sur une variete affine d avoir un rang inferieur au maximal

aspects of free groups if n tuples g h in a rank 2 free group satisfy the same existential formulas then there is an automorphism taking g to h fix a prime p then fk is up to isomorphism the only k generated group having all finite k generated p groups as homomorphic images the theory of non abelian free groups has no prime model

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

products of test elements of free factors of a free group abstract let g be a group an element g g is called a test element of g if for every endomorphism g g g g implies that is an automorphism let f x be a free group on a finite non empty set x and let x x 1 x 2 x r be a finite partition of x into r 2 non empty subsets for i 1 2 r let u i x i f x and let w z 1 z r be a word in the variables z 1 z r we give several sufficient conditions on u i 1 i r and w for w u 1 u r to be a test element of f x as an application of these results we give examples of test elements of a free group of rank greater than two that are not test elements in any pro p completion of the group

uniqueness and examples of compact toric sasaki einstein metrics in 11 it was proved that given a compact toric sasaki manifold with positive basic first chern class and trivial first chern class of the contact bundle one can find a deformed sasaki structure on which a sasaki einstein metric exists in the present paper we first prove the uniqueness of such einstein metrics on compact toric sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure and secondly remark that the result of 11 implies the existence of compatible einstein metrics on all compact sasaki manifolds obtained from the toric diagrams with any height or equivalently on all compact toric sasaki manifolds whose cones have flat canonical bundle we further show that there exists an infinite family of inequivalent toric sasaki einstein metrics on s 5 sharp k s 2 times s 3 for each positive integer k

primitive almost primitive test and δ primitive elements of free algebras with the nielsen schreier property abstract we study generalized primitive elements of free algebras of finite ranks with the nielsen schreier property and their automorphic orbits a primitive element of a free algebra is an element of some free generating set of this algebra almost primitive elements are not primitive elements which are primitive in any proper subalgebra primitive elements are elements whose partial derivatives generate the same one sided ideal of the universal multiplicative envelope algebra of a free algebra as the set of free generators generate we prove that an endomorphism preserving an automorphic orbit of a nonzero element of a free algebra of rank two is an automorphism an algorithm to determine test elements of free algebras of rank two is described a series of almost primitive elements is constructed and new examples of test elements are given we prove that if the rank n of the free lie algebra l is even n 2m then any primitive element of l is an automorphic image of the element w x1 x2 x2m 1 x2m there are no primitive elements of l if n is odd and the group of automorphisms of the algebra l acts transitively on the set of all primitive elements

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

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

calabi flow geodesic rays and uniqueness of constant scalar curvature kahler metrics we prove that constant scalar curvature kahler metric adjacent to a fixed kahler class is unique up to isomorphism the proof is based on the study of a fourth order evolution equation namely the calabi flow from a new geometric perspective and on the geometry of the space of kahler metrics

on the space of kahler potentials we consider the geodesic equation for the generalized kahler potential with only mixed second derivatives bounded we show that given two such generalized kahler potentials there is a unique geodesic segment such that for each point on the geodesic the generalized kahler potential has uniformly bounded mixed second derivatives in manifold directions this generalizes a fundamental theorem of chen 2000 on the space of kahler potentials 2014 wiley periodicals inc

hamiltonian structure of real monge ampere equations the variational principle for the real homogeneous monge ampere equation in two dimensions is shown to contain three arbitrary functions of four variables there exist two different specializations of this variational principle where the lagrangian is degenerate and furthermore contains an arbitrary function of two variables the hamiltonian formulation of these degenerate lagrangian systems requires the use of dirac s theory of constraints as in the case of most completely integrable systems the constraints are second class and dirac brackets directly yield the hamiltonian operators thus the real homogeneous monge ampere equation in two dimensions admits two classes of infinitely many hamiltonian operators namely a family of local as well as another family non local hamiltonian operators and symplectic 2 forms which depend on arbitrary functions of two variables the simplest non local hamiltonian operator corresponds to the kac moody algebra of vector fields and functions on the unit circle hamiltonian operators that belong to either class are compatible with each other but between classes there is only one compatible pair in the case of real monge ampere equations with constant right hand side this compatible pair is the only pair of hamiltonian operators that survives then the complete integrability of all these real monge ampere equations follows by magri s theorem some of the remarkable properties we have obtained for the hamiltonian structure of the real homogeneous monge ampere equation in two dimensions turn out to be generic to the real homogeneous monge ampere equation and the geodesic flow for the complex homogeneous monge ampere equation in arbitrary number of dimensions hence among all integrable nonlinear evolution equations in one space and one time dimension the real homogeneous monge ampere equation is distinguished as one that retains its character as an integrable system in multiple dimensions

local geodesics for plurisubharmonic functions we study geodesics for plurisubharmonic functions from the cegrell class mathcal f 1 on a bounded hyperconvex domain of mathbb c n and show that as in the case of metrics on kahler compact manifolds they linearize an energy functional as a consequence we get a uniqueness theorem for functions from mathcal f 1 in terms of total masses of certain mixed monge ampere currents geodesics of relative extremal functions are considered and a reverse brunn minkowski inequality is proved for capacities of multiplicative combinations of multi circled compact sets we also show that functions with strong singularities generally cannot be connected by sub geodesic arcs

the calabi homomorphism lagrangian paths and special lagrangians let mathcal o be an orbit of the group of hamiltonian symplectomorphisms acting on the space of lagrangian submanifolds of a symplectic manifold x omega we define a functional mathcal c mathcal o rightarrow mathbb r for each differential form beta of middle degree satisfying beta wedge omega 0 and an exactness condition if the exactness condition does not hold mathcal c is defined on the universal cover of mathcal o a particular instance of mathcal c recovers the calabi homomorphism if beta is the imaginary part of a holomorphic volume form the critical points of mathcal c are special lagrangian submanifolds we present evidence that mathcal c is related by mirror symmetry to a functional introduced by donaldson to study einstein hermitian metrics on holomorphic vector bundles in particular we show that mathcal c is convex on an open subspace mathcal o subset mathcal o as a prerequisite we define a riemannian metric on mathcal o and analyze its geodesics finally we discuss a generalization of the flux homomorphism to the space of lagrangian submanifolds and a lagrangian analog of the flux conjecture

curvatures of direct image sheaves of vector bundles and applications let mathcal p x to s be a proper kahler fibration and mathcal e to x a hermitian holomorphic vector bundle as motivated by the work of berndtsson curvature of vector bundles associated to holomorphic fibrations by using basic hodge theory we derive several general curvature formulas for the direct image mathcal p k x s otimes e for general hermitian holomorphic vector bundle mathcal e in a simple way a straightforward application is that if the family mathcal x to s is infinitesimally trivial and hermitian vector bundle mathcal e is nakano negative along the base mathcal s then the direct image mathcal p k x s otimes e is nakano negative we also use these curvature formulas to study the moduli space of projectively flat vector bundles with positive first chern classes and obtain that if the chern curvature of direct image p k x otimes e of a positive projectively flat family e h t t in mathbb d to x vanishes then the curvature forms of this family are connected by holomorphic automorphisms of the pair x e