invariant and homogeneous bundles on g γ abstract let be a cocompact lattice in a connected complex lie group g given an invariant holomorphic vector bundle e on g we show that there is a trivial holomorphic subbundle f e such that any holomorphic section of e factors through holomorphic sections of f given two homomorphisms 1 and 2 from to a complex linear algebraic lie group h with relatively compact image we prove that any holomorphic isomorphism between the associated holomorphic principal h bundles e h 1 and e h 2 is automatically g equivariant

balanced metrics on non kahler calabi yau threefolds we construct balanced metrics on the family of non kahler calabi yau threefolds that are obtained by smoothing after contracting 1 1 rational curves on a kahler calabi yau threefold as an application we construct balanced metrics on complex manifolds diffeomorphic to the connected sum of k ge 2 copies of s 3 s 3

splitting comodules over hopf algebras and application to representation theory of quantum groups of type a0 0 abstract in this work we study some properties of comodules over hopf algebras possessing integrals co frobenius hopf algebras in particular we give a necessary and sufficient condition for a simple comodule to be injective we apply the result obtained to the classification of representations of quantum groups of type a 0 0

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2

co frobenius coalgebras abstract we investigate left and right co frobenius coalgebras and give equivalent characterizations which prove statements dual to the characterizations of frobenius algebras we prove that a coalgebra is left and right co frobenius if and only if c rat c c as right c modules and also that this is equivalent to the fact that the functors hom k k and hom c c from m c to m c are isomorphic this allows a definition of a left right symmetric concept of co frobenius coalgebras that is perfectly dual to the one of frobenius algebras and coincides to the existing notion left and right co frobenius coalgebra

colby fuller duality between coalgebras w x in 8 k morita introduced a useful notion of duality between categories of modules usually called morita duality he proved that every duality is given by contravariant hom functors defined by a bimodule which is an injective cogenerator for both categories of modules on the other hand the equivalences between categories of comodules over a coalgebra w x were characterized by m takeuchi 12 in this paper we study the dualities between categories of comodules a notion of duality for general grothendieck categories that seems to extend morita duality satisfactorily w x was introduced by r r colby and k r fuller in 1 it has been recently w x investigated by j l gomez pardo and p a guil asensio 3 4 6 section 1 is devoted to obtaining a complete characterization of colby fuller dualities between coalgebras a coalgebra c over a field k is right semiperfect if the category m c of right c comodules has enough projectives if c and d are coalgebras over a field k then either c and d are left and right semiperfect or there is no colby fuller duality between the category of right c comodules and the category of left d comodules theorem 1 11 this together with theorem 1 6 2 shows that there is a

the fu yau equation with negative slope parameter the fu yau equation is an equation introduced by fu and yau as a generalization to arbitrary dimensions of an ansatz for the strominger system as in the strominger system it depends on a slope parameter alpha prime the equation was solved in dimension 2 by fu and yau in two successive papers for alpha prime 0 and for alpha prime 0 in the present paper we solve the fu yau equation in arbitrary dimension for alpha prime 0 to our knowledge these are the first non trivial solutions of the fu yau equation in any dimension strictly greater than 2

a direct method for minimizing the yang mills functional over 4 manifolds a direct method is employed to minimize the yang mills functional over a 4 dimensional manifold the limiting connection is shown to be yang mills but in a possibly new bundle we show that a topological invariant of the bundle is preserved by the minimizing process this implies the existence of an absolute minimum of the yang mills functional in a wide class of bundles

infinitesimal moduli for the strominger system and killing spinors in generalized geometry we construct the space of infinitesimal variations for the strominger system and an obstruction space to integrability using elliptic operator theory we initiate the study of the geometry of the moduli space describing the infinitesimal structure of a natural foliation on this space the associated leaves are related to generalized geometry and correspond to moduli spaces of solutions of suitable killing spinor equations on a courant algebroid as an application we propose a unifying framework for metrics with holonomy mathrm su 3 and solutions of the strominger system

equivalence theorems and hopf galois extensions in this article we apply equivalence theorems for categories of modules d to the category m h of right d y a hopf modules to get a charactera ization of hopf galois extensions indeed let p q g mod r and t s end p and assume that p ber longs to the category s q of modules subgenerated by q in theorem r r w 2 3 by modifying a theorem formulated by dal pio and orsatti 5 x theorem 2 6 we study the situation where for every m belonging to the category s q the map r

on the dimension of the space of integrals on coalgebras abstract we study the injective envelopes of the simple right c comodules and their duals where c is a coalgebra this is used to give a short proof and to extend a result of iovanov on the dimension of the space of integrals on coalgebras we show that if c is right co frobenius then the dimension of the space of left m integrals on c is dim m for any left c comodule m of finite support and the dimension of the space of right n integrals on c is dim n for any right c comodule n of finite support some examples of integrals are computed for incidence coalgebras

geometry of compact complex homogeneous spaces with vanishing first chern class abstract we prove that any compact complex homogeneous space with vanishing first chern class after an appropriate deformation of the complex structure admits a homogeneous calabi yau with torsion structure provided that it also has an invariant volume form a description of such spaces among the homogeneous c spaces is given as well as many examples and a classification in the 3 dimensional case we calculate the cohomology ring of some of the examples and show that in dimension 14 there are infinitely many simply connected spaces admitting such a structure with the same hodge numbers and torsional chern classes we provide also an example solving the strominger s equations in heterotic string theory

extremal bundles on calabi yau threefolds we study constructions of stable holomorphic vector bundles on calabi yau threefolds especially those with exact anomaly cancellation which we call extremal by going through the known databases we find that such examples are rare in general and can be ruled out for the spectral cover construction for all elliptic threefolds we then introduce a general hartshorne serre construction and use it to find extremal bundles of general ranks and study their stability as well as computing their chern numbers based on both existing and our new constructions we revisit the dry conjecture for the existence of stable sheaves on calabi threefolds and provide theoretical and numerical evidence for its correctness our construction can be easily generalized to bundles with no extremal conditions imposed

torsion free generalized connections and heterotic supergravity this work revisits the notions of connection and curvature in generalized geometry with emphasis on torsion free generalized connections on a transitive courant algebroid as an application we provide a mathematical derivation of the equations of motion of heterotic supergravity in terms of the ricci tensor of a generalized metric inspired by the work of coimbra strickland constable and waldram

finiteness conditions co frobenius hopf algebras and quantum groups abstract some finiteness conditions for infinite dimensional coalgebras particularly right or left semiperfect coalgebras or co frobenius hopf algebras are studied as well examples of co frobenius hopf algebras are constructed via a hopf algebra structure on an ore extension of a group algebra and it is shown that bicrossproducts of co frobenius hopf algebras are themselves co frobenius

multi variable extended mean values on considere des fonctions dont les valeurs sont des valeurs moyennes d une suite x x 0 x n de nombre reels pour une bonne paire f g de fonctions on definit une valeur moyenne e f g x de chaque suite x a valeurs dans cette intervalle

on two conjectures of faith abstract we prove that a profinite algebra whose left right cyclic modules are torsionless is finite dimensional and qf we give a relative version of the notion of left right pf ring for pseudocompact algebras and prove it is left right symmetric and dual to the notion of quasi co frobenius coalgebras we also prove two ring theoretic conjectures of faith in the setting and supplementary hypothesis of profinite algebras any profinite semiartinian selfinjective algebra is finite dimensional and qf and any fgf profinite algebra is finite dimensional qf

invariant solutions to the strominger system on complex lie groups and their quotients using canonical 1 parameter family of hermitian connections on the tangent bundle we provide invariant solutions to the strominger system on certain complex lie groups and their quotients both flat and non flat cases are discussed in detail this paper answers a question proposed by andreas and garcia fernandez in comm math phys 332 3 1381 1383 2014

the splitting problem for coalgebras abstract let r be a ring and let t be a torsion preradical r is said to have the splitting property provided that for every left r module m the torsion submodule t m of m is a direct summand of m the characterization of rings with this property is a classical problem in particular the goldie and dickson torsion theories have been studied that for noncommutative rings remains open we consider the problem for the algebra c associated to a coalgebra c and the torsion preradical rat it is shown that if c has the splitting property with respect a rat then c is finite dimensional

means that minimize relative error and an associated integral equation abstract determinations are made of the means that minimize various relative errors in the sense that the harmonic mean of a and b minimizes the traditional relative error on a b the general problem for averaged relative error leads to a nonlinear integral equation for which we prove existence and uniqueness results as well as constructive solution procedures the inverse problem of finding the measure of relative error corresponding to a given mean is also analyzed in detail these studies shed light on both new and previously known inequalities for specific means

a flow of conformally balanced metrics with kahler fixed points while the anomaly flow was originally motivated by string theory its zero slope case is potentially of considerable interest in non kahler geometry as it is a flow of conformally balanced metrics whose stationary points are precisely kahler metrics we establish its convergence on kahler manifolds for suitable initial data we also discuss its relation to some current problems in complex geometry

qf functors and co monads abstract one reason for the universal interest in frobenius algebras is that their characterisation can be formulated in arbitrary categories a functor k a b between categories is frobenius if there exists a functor g b a which is at the same time a right and left adjoint of k a monad f on a is a frobenius monad provided the forgetful functor a f a is a frobenius functor where a f denotes the category of f modules with these notions an algebra a over a field k is a frobenius algebra if and only if a k is a frobenius monad on the category of k vector spaces the purpose of this paper is to find characterisations of quasi frobenius algebras by just referring to constructions available in any categories to achieve this we define qf functors between two categories by requiring conditions on pairings of functors which weaken the axioms for adjoint pairs of functors qf monads on a category a are those monads f for which the forgetful functor u f a f a is a qf functor applied to module categories or grothendieck categories our notions coincide with definitions first given k morita and others further applications show the relations of qf functors and qf monads with frobenius exact categories

balanced hermitian metrics from su 2 structures we study the intrinsic geometrical structure of hypersurfaces in six manifolds carrying a balanced hermitian su 3 structure which we call balanced su 2 structure we provide sufficient conditions in terms of suitable evolution equations which imply that a five manifold with such structure can be isometrically embedded as a hypersurface in a balanced hermitian su 3 manifold any five dimensional compact nilmanifold has an invariant balanced su 2 structure and we show how some of them can be evolved to give new explicit examples of balanced hermitian su 3 structures moreover for n 3 4 we present examples of compact solvmanifolds endowed with a balanced su n structure such that the corresponding bismut connection has holonomy equal to su n

series expansions of means abstract a mean m u v is defined to be a homogeneous symmetric function of two positive real variables satisfying min u v m u v max u v for all u and v setting m u v um 1 vu 1 um 1 1 t 0 t p 1 1 t 1 2 1 t p 2 1 p m p u v v p 1 u p 1 v u p 1 1 p stolarsky s mean m p u v u p v p u p 1 v p 1 lehmer s mean e r s u v r u s v s s u r v r 1 s r leach and sholander s mean and g r s u v u s v s u r v r 1 s r gini s mean the explicit power series coefficients and recurrence relations for these coefficients are found finally applications are shown by proving a theorem that generalizes one due to lehmer

instantons and geometric invariant theory we show that the yang mills instantons can be described in terms of certain holomorphic bundles on the projective plane the proof uses explicit matrix descriptions arising from monads and an analysis of the corresponding groups of symmetries

on estimates for the fu yau generalization of a strominger system we study an equation proposed by fu and yau as a natural n dimensional generalization of a strominger system that they solved in dimension 2 it is a complex hessian equation with right hand side depending on gradients building on the methods of fu and yau we obtain c0 c2 and c2 a priori estimates we also identify difficulties in extending the fu yau arguments for non degeneracy from dimension 2 to higher dimensions

solutions of the strominger system via stable bundles on calabi yau threefolds we prove that a given calabi yau threefold with a stable holomorphic vector bundle can be perturbed to a solution of the strominger system provided that the second chern class of the vector bundle is equal to the second chern class of the tangent bundle if the calabi yau threefold has strict su 3 holonomy then the equations of motion derived from the heterotic string effective action are also satisfied by the solutions we obtain

vanishing theorems on l k strong kahler manifolds with torsion abstract we derive sufficient conditions for the vanishing of plurigenera p m j m 0 on compact l k strong l k 0 kahler manifolds with torsion in particular we show that the plurigenera of closed l k strong manifolds k n 1 for which hol s u n vanish where is the hermitian connection with skew symmetric torsion as a consequence all generalized k gauduchon manifolds for which hol s u n do not admit holomorphic n 0 forms furthermore we show that all conformally balanced l k strong kahler manifolds with torsion k n 1 are kahler we also give several examples of l k strong kahler and calabi yau manifolds with torsion

galois functors and entwining structures abstract galois comodules over a coring can be characterised by properties of the relative injective comodules they motivated the definition of galois functors over some comonad or monad on any category and in the first section of the present paper we investigate the role of the relative injectives projectives in this context then we generalise the notion of corings derived from an entwining of an algebra and a coalgebra to the entwining of a monad and a comonad hereby a key role is played by the notion of a grouplike natural transformation g i g generalising the grouplike elements in corings we apply the evolving theory to hopf monads on arbitrary categories and to opmonoidal monads with antipode on autonomous monoidal categories named hopf monads by bruguieres and virelizier which can be understood as an entwining of two related functors as well known for any set g the product g defines an endofunctor on the category of sets and this is a hopf monad if and only if g allows for a group structure in the final section the elements of this case are generalised to arbitrary categories with finite products leading to galois objects in the sense of chase and sweedler

frobenius monads and pseudomonoids six equivalent definitions of frobenius algebra in a monoidal category are provided in a monoidal bicategory a pseudoalgebra is frobenius if and only if it is star autonomous autonomous pseudoalgebras are also frobenius what it means for a morphism of a bicategory to be a projective equivalence is defined this concept is related to strongly separable frobenius algebras and weak monoidal morita equivalence wreath products of frobenius algebras are discussed

co frobenius hopf algebras integrals doi koppinen modules and injective objects abstract we investigate hopf algebras with non zero integral from a coalgebraic point of view categories of doi koppinen modules are studied in the special case where the defining coalgebra is left and right semiperfect and several pairs of adjoint functors are constructed as applications we give a very short proof for the uniqueness of the integrals and provide information about injective objects in the category of doi koppinen modules

means and taylor polynomials pour une fonction convexe derivable f sur 0 on note t a la tangente a la courbe de f en a f a puisque le point d intersection entre t a et t b note i t a t b i x t a t b i y t a t b verifie l inegalite min a b i x t a t b max a b il definit une moyenne et on note ainsi i x a h m a h cet article considere des generalisations et des modifications de ce type de moyenne a l aide de polynomes de taylor de degre impair ou d autres interpolants de hermite ainsi qu avec d autres points d intersection de tangentes on etudie des derivees de fonctions definies par ces moyennes

ricci flow killing spinors and t duality in generalized geometry abstract we introduce a notion of ricci flow in generalized geometry extending a previous definition by gualtieri on exact courant algebroids special stationary points of the flow are given by solutions to first order differential equations the killing spinor equations which encompass special holonomy metrics with solutions of the hull strominger system our main result investigates a method to produce new solutions of the ricci flow and the killing spinor equations for this we consider t duality between possibly topologically distinct torus bundles endowed with courant structures and demonstrate that solutions of the equations are exchanged under this symmetry as applications we give a mathematical explanation of the dilaton shift in string theory and prove that the hull strominger system is preserved by t duality

frobenius structural matrix algebras abstract we discuss when the incidence coalgebra of a locally finite preordered set is right co frobenius as a consequence we obtain that a structural matrix algebra over a field k is frobenius if and only if it consists up to a permutation of rows and columns of diagonal blocks which are full matrix algebras over k

a construction of non kahler calabi yau manifolds and new solutions to the strominger system abstract we propose a new construction of compact non kahler calabi yau manifolds with balanced metrics and study the strominger system on them in particular we obtain explicit solutions to the strominger system with degeneracies on g t 4 where g is an immersed minimal surface of genus g 3 in flat t 3

monads and comonads on module categories let a be a ring and ma the category of right a modules it is well known in module theory that any a bimodule b is an a ring if and only if the functor ab ma ma is a monad or triple similarly an a bimodule c is an a coring provided the functor ac ma ma is a comonad or cotriple the related categories of modules or algebras of ab and comodules or coalgebras of ac are well studied in the literature on the other hand the right adjoint endofunctors homa b and homa c are a comonad and a monad respectively but the corresponding co module categories did not find much attention so far the category of homa b comodules is isomorphic to the category of b modules while the category of homa c modules called c contramodules by eilenberg and moore need not be equivalent to the category of c comodules the purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background this leads to a deeper understanding and characterisations of algebraic structures such as corings bialgebras and hopf algebras for example it turns out that the categories of c comodules and homa c modules are equivalent provided c is a coseparable coring furthermore we describe equivalences between categories of homa c modules and comodules over a coring d in terms of new galois properties of bicomodules finally we characterise hopf algebras h over a commutative ring r by properties of the functor homr h and the category of mixed homr h bimodules this generalises in particular the fact that a finite dimensional vector space h is a hopf algebra if and only if the dual space h is a hopf algebra

modules comodules and cotensor products over frobenius algebras abstract we characterize noncommutative frobenius algebras a in terms of the existence of a coproduct which is a map of left ae modules we show that the category of right left comodules over a relative to this coproduct is isomorphic to the category of right left modules this isomorphism enables a reformulation of the cotensor product of eilenberg and moore as a functor of modules rather than comodules we prove that the cotensor product m n of a right a module m and a left a module n is isomorphic to the vector space of homomorphisms from a particular left ae module d to n m viewed as a left ae module some properties of d are described finally we show that when a is a symmetric algebra the cotensor product m n and its derived functors are given by the hochschild cohomology over a of n m

vector bundles and f theory to understand in detail duality between heterotic string and f theory compactifications it is important to understand the construction of holomorphic g bundles over elliptic calabi yau manifolds for various groups g in this paper we develop techniques to describe these bundles and make several detailed comparisons between the heterotic string and f theory

linear compactness and morita duality for grothendieck categories the theory of morita duality for grothendieck categories arises from the desire of extending the classical results on morita duality to more genera1 contexts such as rings with local units 22 31 and graded rings 16 but no less interesting is the fact that it can be applied to the study of

constructing pointed hopf algebras by ore extensions abstract we present a general construction producing pointed co frobenius hopf algebras and give some classification results for the examples obtained

co frobenius hopf algebras and the coradical filtration we prove that a hopf algebra with a finite coradical filtration is co frobenius we also characterize co frobenius hopf algebras with coradical a hopf subalgebra let h be a hopf algebra whose coradical is a hopf algebra let gr h be the associated graded coalgebra and let r be the diagram of h c f 2 then the following are equivalent 1 h is co frobenius 2 gr h is co frobenius 3 r is finite dimensional 4 the coradical filtration of h is finite this theorem allows us to give systematically examples of co frobenius hopf algebras and opens the way to the classification of ample classes of such hopf algebras

the dickson subcategory splitting conjecture for pseudocompact algebras abstract let a be a pseudocompact or profinite algebra so a c where c is a coalgebra we show that the if the semiartinian part the dickson part of every a module m splits off in m then a is semiartinian giving thus a positive answer in the case of algebras arising as dual of coalgebras pseudocompact algebras to a well known conjecture of faith

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

generalized kahler geometry generalized kahler geometry is the natural analogue of kahler geometry in the context of generalized complex geometry just as we may require a complex structure to be compatible with a riemannian metric in a way which gives rise to a symplectic form we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure we prove that generalized kahler geometry is equivalent to the bi hermitian geometry on the target of a 2 dimensional sigma model with 2 2 supersymmetry we also prove the existence of natural holomorphic courant algebroids for each of the underlying complex structures and that these split into a sum of transverse holomorphic dirac structures finally we explore the analogy between pre quantum line bundles and gerbes in the context of generalized kahler geometry

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

geometric flows and strominger systems a geometric flow on 2 2 forms is introduced which preserves the balanced condition of metrics and whose stationary points satisfy the anomaly equation in strominger systems the existence of solutions for a short time is established using hamilton s version of the nash moser implicit function theorem

means generalized divided differences and intersections of osculating hyperplanes abstract we define means innvariables by taking the intersection point inrnofnosculating hyperplanes to a given curve inrn these planes are the natural extensions of the osculating plane inr3 more precisely letcbe a curve inrn and let 0

integrals for dual quasi hopf algebras applications abstract a classical result in the theory of hopf algebras concerns the uniqueness and existence of integrals for an arbitrary hopf algebra the integral space has dimension 1 and for a finite dimensional hopf algebra this dimension is exactly one we generalize these results to quasi hopf algebras and dual quasi hopf algebras in particular it will follow that the bijectivity of the antipode follows from the other axioms of a finite dimensional quasi hopf algebra we give a new version of the fundamental theorem for quasi hopf algebras we show that a dual quasi hopf algebra is co frobenius if and only if it has a non zero integral in this case the space of left or right integrals has dimension one

infinite dimensional serial algebras and their representations abstract we study arbitrary algebras all of whose indecomposable finite dimensional representations are uniserial or more generally comodules over serial coalgebras we show that the infinite representation theory for such algebras parallels and non trivially extends the theory of primary infinite abelian groups we obtain a kulikov and a prufer decomposability criterion and determine indecomposables in the category of locally finite modules over such an algebra as applications we obtain consequences on the infinite representation theory of the extended affine dynkin quiver a n and the infinite line and half line quivers and obtain jordan form results for operators on infinite dimensional spaces we also find a class of counterexamples to an infinite version of the pure semisimplicity conjecture there exist algebras all of whose locally finite left modules are direct sums of indecomposables but not all of whose right locally finite modules decompose as such on the way we also provide classifications of one sided serial co algebras and introduce and classify a class of quantum groups whose indecomposable representations are uniserial namely coserial pointed hopf algebras these are one parameter deformations of group algebras and are obtained as certain ore extensions of these

integrals quantum galois extensions and the affineness criterion for quantum yetter drinfel d modules abstract in this paper we shall generalize the notion of an integral on a hopf algebra introduced by sweedler by defining the more general concept of an integral of a threetuple h a c where h is a hopf algebra coacting on an algebra a and acting on a coalgebra c we prove that there exists a total integral c hom c a of h a c if and only if any representation of h a c is injective in a functorial way as a corepresentation of c in particular the quantum integrals associated to yetter drinfel d modules are defined let now a be an h bicomodule algebra h y d a the category of quantum yetter drinfel d modules and b a a s 1 a 1 a 1 a 0 1h a the subalgebra of coinvariants of the verma structure a h y d a we shall prove the following affineness criterion if there exists h hom h a a total quantum integral and the canonical map a b a h a a b b s 1 b 1 b 1 ab 0 is surjective i e a b is a quantum homogeneous space then the induction functor b a m b h y d a is an equivalence of categories the affineness criteria proven by cline parshall and scott and independently by oberst for affine algebraic groups schemes and schneider in the noncommutative case are recovered as special cases

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we