twistor spaces of hypercomplex manifolds are balanced abstract a hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations if in addition there exists a metric on the manifold which is hermitian with respect to the three structures and such that the corresponding hermitian forms are closed the manifold is said to be hyperkahler in the paper non hermitian yang mills connections 13 kaledin and verbitsky proved that the twistor space of a hyperkahler manifold admits a balanced metric these were first studied in the article on the existence of special metrics in complex geometry 17 by michelsohn in the present article we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced

the hp finite element method for problems in mechanics with boundary layers abstract we consider the numerical approximation of boundary layer phenomena occurring in many singularly perturbed problems in mechanics such as plate and shell problems we present guidelines for the effective resolution of such layers in the context of existing commercial p and hp finite element fe version codes we show that if high order spectral elements are available then just two elements are sufficient to approximate these layers at a near exponential rate independently of the problem parameters thickness or reynolds number we present hp mesh design principles for situations where both corner singularities and boundary layers are present

some exact solutions of the ideal mhd equations through symmetry reduction method abstract we use the symmetry reduction method based on lie group theory to obtain some exact solutions the so called invariant solutions of the ideal magnetohydrodynamic equations in 3 1 dimensions in particular these equations are invariant under a galilean similitude lie algebra for which the classification by conjugacy classes of r dimensional subalgebras 1 r 4 was already known we restrict our study to the three dimensional galilean similitude subalgebras that give us systems composed of ordinary differential equations here some examples of these solutions are presented with a brief physical interpretation

on the geometry of sasakian einstein 5 manifolds on simply connected five manifolds sasakian einstein metrics coincide with riemannian metrics admitting real killing spinors which are of great interest as models of near horizon geometry for three brane solutions in superstring theory 24 we expand on the recent work of demailly and kollar 14 and johnson and kollar 20 who give methods for constructing kahler einstein metrics on log del pezzo surfaces by a previous result of the first two authors 9 circle v bundles over log del pezzo surfaces with kahler einstein metrics have sasakian einstein metrics on the total space of the bundle here these simply connected 5 manifolds arise as links of isolated hypersurface singularities which by the well known work of smale 36 together with 11 must be diffeomorphic to s 5 l s 2 s 3 more precisely using methods from mori theory in algebraic geometry we prove the existence of 14 inequivalent sasakian einstein structures on s 2 s 3 and infinite families of such structures on l s 2 s 3 with 2 l 7 we also discuss the moduli problem for these sasakian einstein structures

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

non moishezon twistor spaces of 4 mathbf cp 2 with non trivial automorphism group we show that a twistor space of a self dual metric on 4cp 2 with u 1 isometry is not moishezon iff there is a c orbit biholomorphic to a smooth elliptic curve where the c action is the complexification of the u 1 action on the twistor space it follows that the u 1 isometry has a two sphere whose isotropy group is z 2 we also prove the existence of such twistor spaces in a strong form to show that a problem of campana and kreusler is affirmative even though a twistor space is required to have a non trivial automorphism group

on the construction of some complete metrics with exceptional holonomy on construit explicitement trois metriques completes distinctes d holonomie egale a g 2 une metrique complete d holonomie egale a spin 7 et diverses metriques incompletes d holonomie exceptionnelle

exact solutions of the multidimensional classical φ6 field equations obtained by symmetry reduction the often used 6 model of classical critical phenomena is studied in 3 1 dimensional minkowski and euclidean spaces the euler lagrange equations describing the kinetics of the scalar order parameter are in this case nonlinear klein gordon equations the method of symmetry reduction is systematically applied to derive all the solutions invariant under subgroups with generic orbits of codimension 1 whenever the obtained ordinary differential equations have the painleve property they can be transformed to one of two standard forms these are then solved in terms of elliptic functions or elementary ones this results in a large number of new exact solutions particularly interesting solutions are found in the immediate vicinity of the tricritical point our treatment of the 6 theory is complete only for the four dimensional spaces m 3 1 and e 4 but many of the results are given for the more general cases of m n 1 and e n 1

twistor examples of algebraic dimension zero threefolds our purpose is to present a fairly large class of algebraic dimension zero threefolds namely the twistor spaces of certain self dual 4 manifolds

on twistor spaces of anti self dual hermitian surfaces we consider a complex surface m with anti self dual hermitian metric h and study the holomorphic properties of its twistor space z we show that the naturally defined divisor line bundle x is isomorphic to the 1 2 power of the canonical bundle of z if and only if there is a kahler metric of zero scalar curvature in the conformal class of h this has strong consequences on the geometry of m which were also found by c boyer 3 using completely different methods we also prove the existence of a very close relation between holomorphic vector fields on m and z in the case that m is compact and kahler

projective uniformization extremal chern classes and quaternionic shimura curves projective structures on compact real manifolds are classical objects in real differential geometry complex manifolds with a holomorphic projective structure on the other hand form a special class as soon as the dimension is greater than one in the kahler einstein case mathbb p m tori and ball quotients are essentially the only examples they can be described purely in terms of chern class conditions we give a complete classification of all projective manifolds carrying a projective structure the only additional examples are modular abelian families over quaternionic shimura curves they can also be described purely in terms of chern class conditions

reduction and exact solutions of the navier stokes equations the authors construct a complete set of g 1 3 inequivalent ansatze of codimension 1 for the navier stokes ns field which reduce the ns equations to systems of ordinary differential equations ode having solved these odes they thereby obtain solutions of the ns equations formulae of group multiplication of solutions are given several non lie ansatze are discussed

invariant spinors and reduced dirac equations under subgroups of the euclidean group in four dimensional euclidean space the most general dirac spinors globally invariant under the representatives of the conjugacy classes of the connected part to the identity of the euclidean group in four dimensions are determined the dirac equation is reduced under each subgroup representative and a relation to the reduction by symmetry of a linear system of the self dual yang mills equations is given

conditional symmetries and riemann invariants for inhomogeneous hydrodynamic type systems a new approach to the solution of quasilinear nonelliptic first order systems of inhomogeneous partial differential equations in many dimensions is presented it is based on a version of the conditional symmetry and riemann invariant methods we discuss in detail the necessary and sufficient conditions for the existence of rank 2 and rank 3 solutions expressible in terms of riemann invariants we perform the analysis using the cayley hamilton theorem for a certain algebraic system associated with the initial system the problem of finding such solutions has been reduced to expanding a set of trace conditions on wave vectors and their profiles which are expressible in terms of riemann invariants a couple of theorems useful for the construction of such solutions are given these theoretical considerations are illustrated by the example of inhomogeneous equations of fluid dynamics which describe motion of an ideal fluid subjected to gravitational and coriolis forces several new rank 2 solutions are obtained some physical interpretation of these results is given

reductions by isometries of the self dual yang mills equations in four dimensional euclidean space a classification of the subgroups of the four dimensional euclidean group is given the anti self dual yang mills equations in four dimensional euclidean space are then reduced for each class representative and for any gauge group when a strict invariance of the gauge fields is required or for the gauge group so 3 if the gauge fields remain invariant up to gauge transformations amongst the residual systems many constitute sets of first order ordinary differential equations sharing the painleve property

stable hp mixed finite elements based on the hellinger reissner principle in the hellinger reissner formulation for linear elasticity both the displacement u and the stress are taken as unknowns giving rise to a saddle point problem we present new pairings of quadrilateral trunk finite element spaces for this method and prove stability and optimality in terms of both h and p the effect of mesh shape regularity on the stability constant is explicitly tracked our results provide a theoretical basis for recent numerical experiments in the context of a mixed p formulation for viscoelasticity that showed these spaces worked well computationally

compact self dual manifolds with positive scalar curvature on etudie l existence d une metrique auto duale a courbure scalaire positive sur une variete compacte simplement connexe qui n est pas conformement equivalente a ces metriques standard sur s 4 ou cp 2

symmetry reduction and simple supersymmetric models the reduction by the subgroups of the conformal group space time of the wess zumino model and the four dimensional simple n 1 supersymmetric yang mills theories is considered it is shown for both models that the translations are the only subgroups of the poincare group that leave a supersymmetric residual system on the reduced space if we require compatibility of the reduction with all poincare supersymmetry transformations

so 4 reduction of the su n yang mills dirac equations the equations in compactified minkowski space that describe su n yang mills fields minimally coupled to dirac spinor fields transforming under the lowest dimensional representation s of su n are reduced by the action of an so 4 subgroup of the conformal group of space time the reduced systems can be interpreted as hamiltonian systems with symmetry constrained by a condition on the associated moment maps

conformal invariance conditions for spinor fields with gauge freedom the global invariance conditions are given for the j j spinor fields with gauge freedom invariant under subgroups of c 3 1 we formulate two propositions concerning the nonexistence of nontrivial fields with and without compact gauge freedom and apply them to the maximal noncompact subgroups of c 3 1 we also determine explicitly the most general o 4 and o 2 o 4 invariant j j spinor fields with compact gauge freedom

double solid twistor spaces the case of arbitrary signature in a recent paper 9 we constructed a series of new moishezon twistor spaces which are a kind of variant of the famous lebrun twistor spaces in this paper we explicitly give projective models of another series of moishezon twistor spaces on ncp2 for arbitrary n 3 which can be regarded as a generalization of the twistor spaces of double solid type on 3cp2 studied by kreusler kurke poon and the author similarly to the twistor spaces of double solid type on 3cp2 projective models of the present twistor spaces have a natural structure of double covering of a cp2 bundle over cp1 we explicitly give a defining polynomial of the branch divisor of the double covering whose restriction to fibers is degree four if n 4 these are new twistor spaces to the best of the author s knowledge we also compute the dimension of the moduli space of these twistor spaces differently from 9 the present investigation is based on analysis of pluri half anticanonical systems of the twistor spaces

nonzero scalar curvature generalizations of the ale hyperkahler metrics a quaternionic quotient construction of families of self dual einstein metrics with positive scalar curvature on compact riemannian orbifolds v manifolds is described the metrics are the positive scalar curvature quaternionic analogs of the asymptotically locally euclidean ale gravitational instantons constructed by kronheimer

on non relativistic conformal symmetries and invariant tensor fields the largest symmetry group of the schrodinger equation the so called schrodinger group is analysed in connection with invariant 2forms 1 forms and 0 2 symmetric tensors through the beckers harnad perroud winternitz global method 1978 developed in the relativistic conformal context invariant fields and potentials are obtained and discussed in the physical context of schrodinger electromagnetism through the corresponding infinitesimal method specific attention is paid to magnetic monopole dynamical symmetries these results are obtained in correspondence with the subalgebra classification determined by using the patera winternitz zassenhaus algorithm 1975 the accent is put on the maximal subalgebras of the schrodinger algebra

new g _2 holonomy cones and exotic nearly kahler structures on s 6 and s 3 times s 3 there is a rich theory of so called strict nearly kaehler manifolds almost hermitian manifolds generalising the famous almost complex structure on the 6 sphere induced by octonionic multiplication nearly kaehler 6 manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional lie group g 2 the metric cone over a riemannian 6 manifold m has holonomy contained in g 2 if and only if m is a nearly kaehler 6 manifold a central problem in the field has been the absence of any complete inhomogeneous examples we prove the existence of the first complete inhomogeneous nearly kaehler 6 manifolds by proving the existence of at least one cohomogeneity one nearly kaehler structure on the 6 sphere and on the product of a pair of 3 spheres we conjecture that these are the only simply connected inhomogeneous cohomogeneity one nearly kaehler structures in six dimensions

symmetry reduction for the kadomtsev petviashvili equation using a loop algebra the kadomtsev petviashvili kp equation ut 3uux 2 1 4 uxxx x 3 uyy 4 0 allows an infinite dimensional lie group of symmetries i e a group transforming solutions amongst each other the lie algebra of this symmetry group depends on three arbitrary functions of time t and is shown to be related to a subalgebra of the loop algebra a 1 4 low dimensional subalgebras of the symmetry algebra are identified specifically all those of dimension n 3 and also a physically important six dimensional lie algebra containing translations dilations galilei transformations and quasirotations new solutions of the kp equation are obtained by symmetry reduction using the one dimensional subalgebras of the symmetry algebra these solutions contain up to three arbitrary functions of t

geometric and algebraic reduction for singular momentum maps this paper concerns the reduction of singular constraint sets of symplectic manifolds it develops a geometric reduction procedure as well as continues the work of sniatycki and patrick on reduction a la dirac the relationships among the dirac geometric and sniatycki weinstein algebraic reduction procedures are studied primary emphasis is placed on the case where the constraints are given by the vanishing of a singular momentum map associated to the hamiltonian action of a compact lie group specifically an explicit local normal form is given for the momentum map which is used to show that these various reductions are all well defined and a necessary and sufficient condition for them to agree is derived related conditions are investigated and shown to be sufficient in the case of torus actions numerous examples are computed illustrating the results some discussion and examples are given for the noncompact case

the twistor sections on the wolf spaces let m be a compact quaternion symmetric space a wolf space and v m an irreducible homogeneous vector bundle on m with its canonical connection whose rank is less than or equal to the dimension of m we classify the zero loci of the transversal twistor sections with a reality condition there exists a bijection between such zero loci and the real representations of simple compact connected lie groups with non trivial principal isotropy subgroups which are neither tori nor discrete groups next we obtain an embedding of the wolf space into a real grassmannian manifold using twistor sections which turns out to be a minimal embedding finally we focus our attention on the norm squared s 2 of a twistor section s we identify the subset s m where this function attains the maximum value under a suitable hypothesis such sets are classified and determine totally geodesic submanifolds of the wolf spaces moreover s 2 is a morse function in the sense of bott and its critical manifolds consist of the zero locus and s m

7 dimensional compact riemannian manifolds with killing spinors using a link between einstein sasakian structures and killing spinors we prove a general construction principle of odd dimensional riemannian manifolds with real killing spinors in dimensionn 7 we classify all compact riemannian manifolds with two or three killing spinors finally we classify nonflat 7 dimensional riemannian manifolds with parallel spinor fields

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

teichmuller spaces and torelli theorems for hyperkahler manifolds kreck and yang su recently gave counterexamples to a version of the torelli theorem for hyperkahlerian manifolds as stated by verbitsky we extract the correct statement and give a short proof of it we also revisit a few of its consequences some of which are given new shorter proofs and ask some questions

superinvariant chiral and vector fields this work is concerned with the characterization of field supermultiplets on four dimensional minkowski space that are stable under the action of subgroups of the superconformal group su 2 2 1 the most general scalar and vector superfields whose lie derivative with respect to a fermionic tangent vector vanishes are determined invariance under subgroups of su 2 2 1 with more than one odd generator is also discussed

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

weak transversality and partially invariant solutions new exact solutions are obtained for several nonlinear physical equations namely the navier stokes and euler systems an isentropic compressible fluid system and a vector nonlinear schrodinger equation the solution method makes use of the symmetry group of the system in situations when the standard lie method of symmetry reduction is not applicable

symmetries and supersymmetries of the quantum harmonic oscillator the supersymmetric version of the one dimensional harmonic oscillator is studied by taking into account its conformal properties the largest superalgebra of symmetries and supersymmetries is derived as osp 2 2 square operator sh 1 the semidirect sum of osp 2 2 and the heisenberg superalgebra through a one to one correspondence between the nonrelativistic free case and the harmonic oscillator description the authors deduce the expected supersymmetries of the schrodinger equation the above structure appears as the largest spectrum generating superalgebra of the harmonic oscillator and its representation within an energy basis is given the physical three dimensional case is also considered when the maximal set of super symmetries is required and this case is compared with recent work

compact self dual hermitian surfaces in this paper we obtain a classification up to conformal equivalence of the compact self dual hermitian surfaces as an application we prove that every compact hermitian surface of pointwise constant holomorphic sectional curvature with respect to either the riemannian or the hermitian connection is kahler

symmetry reduction of variational bicomplexes and the principle of symmetric criticality consider a system of differential equations a 0 which is invariant under a lie group g of point transformations acting on the space e of independent and dependent variables by a method due to lie the g invariant solutions of these differential equations are found by solving a reduced system of differential equations a 0 on the space e of invariants of g in this paper we explore the relationship between the g invariant conservation laws and variational principles for the system of equations a 0 and the conservation laws and variational principles for the reduced equations a 0 this problem translates into one of constructing a certain cochain map qx between the g invariant variational bicomplex for the infinite jet space on e and the free variational bicomplex for e we prove that such a cochain map exists locally if and only if the relative lie algebra cohomology condition h f tooo 0 is satisfied where q is the orbit dimension of g r the lie algebra of vector fields on e which generate the infinitesimal action of g and toc o the linear isotropy subalgebra of t at eo as a simple consequence we prove that the vanishing of hq t tooo is the only local obstruction to palais principle of symmetric criticality 1 introduction when a system of partial differential equations is invari ant under a lie group g of point transformations one can construct a reduced system of partial or ordinary differential equations whose local solutions cor respond precisely to the local g invariant solutions of the original system of equations this simple and effective construction pioneered by lie himself is still one of the most powerful and extensively used techniques for finding exact solutions of many nonlinear partial differential equations 4 15 23 in this paper we explore the relationship between the g invariant conservation laws and variational principles for a system of partial differential equations and the con servation laws and variational principles of the reduced system of equations for the g invariant solutions since conservation laws and variational principles for differential equations are naturally represented by certain cohomology classes in the variational bicomplex associated to the given system of partial differential equations 1 22 our problem translates into that of constructing a cochain map between certain variational bicomplexes remarkably and contrary to the usual spirit of symmetry reduction these cochain maps do not always exist even lo cally we are able to completely classify the local obstructions to the existence of such cochain maps

gravitational instantons of type d k and a generalization of the gibbons hawking ansatz we describe a quaternionic based ansatz generalizing the gibbons hawking ansatz to a class of hyperkahler metrics with hidden symmetries we then apply it to obtain explicit expressions for gravitational instanton metrics of type d k

on the sum of the hermitian scalar curvatures of a compact hermitian manifold soit m une variete hermitienne compacte alors a u v 0 p m 0 m 0 b u v 0 p m 0 1 m 0 ou u et v sont les deux courbures scalaires de geometrie hermitienne et p m et q m sont les miemes plurigenres et le mieme plurigenre dual de m

group invariant solutions without transversality we present a generalization of lie s method for finding the group invariant solutions to a system of partial differential equations our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables the theoretical basis for our method is provided by a general existence theorem for the invariant sections both local and global of a bundle on which a finite dimensional lie group acts a simple and natural extension of our characterization of invariant sections leads to an intrinsic characterization of the reduced equations for the group invariant solutions for a system of differential equations the characterization of both the invariant sections and the reduced equations are summarized schematically by the kinematic and dynamic reduction diagrams and are illustrated by a number of examples from fluid mechanics harmonic maps and general relativity this work also provides the theoretical foundations for a further detailed study of the reduced equations for group invariant solutions

twistors kahler manifolds and bimeromorphic geometry i by considering deformations of the moishezon twistor spaces of cp2 cp2 constructed in 20 we showthat theblowupof c2 at n points in general position admits an asymptotically flat scalar flat kahler metric in each kahler class at least provided that the given points are nearly collinear department of mathematics suny at stony brook stony brook new york 11794 e mail address claude math sunysb edu this content downloaded from 157 55 39 153 on mon 19 sep 2016 04 58 00 utc all use subject to http about jstor org terms

on the construction of stable curvilinear p version elements for mixed formulations of elasticity and stokes flow summary the use of mixed finite element methods is well established in the numerical approximation of the problem of nearly incompressible elasticity and its limit stokes flow the question of stability over curved elements for such methods is of particular significance in the p version where since the element size remains fixed exact representation of the curved boundary by large elements is often used we identify a mixed element which we show to be optimally stable in both p and h refinement over curvilinear meshes we prove optimal p version up to o p epsilon and h version p 2 3 convergence for our element and illustrate its optimality through numerical experiments

conformal duality and compact complex surfaces on etudie les structures demi conformement plates sur une variete de riemann a 4 dimensions m g avec l hypothese supplementaire que m est complexe

on the geometry of lambda symmetries and pde reduction we give a geometrical characterization of prolongations of vector fields and hence of symmetries of odes this allows an extension to the case of pdes and systems of pdes in this context the central object is a horizontal 1 form and we speak of prolongations of vector fields and symmetries of pdes we show that these are as good as standard symmetries in providing symmetry reduction of pdes and systems and explicit invariant solutions

on the solutions of the cp1 model in 2 1 dimensions we use the methods of group theory to reduce the equations of motion of the cp1 model in 2 1 dimensions to sets of two coupled ordinary differential equations we decouple and solve many of these equations in terms of elementary functions elliptic functions and painleve transcendents some of the reduced equations do not have the painleve property the existence of a lax pair making the model integrable is hence very unlikely even though it possesses many properties of integrable systems such as stable numerical solitons

remarks on chern einstein hermitian metrics we study some basic properties and examples of hermitian metrics on complex manifolds whose traces of the curvature of the chern connection are proportional to the metric itself

so 4 reduction of the yang mills equations for the classical gauge groups we show that the reduction of the su n so n and sp n yang mills equations in compactified minkowski space by certain realizations of the so 4 subgroup of the conformal group leads to systems of ordinary differential equations the reduced systems are interpretable as hamiltonian systems with symmetry constrained so that the momentum map equals zero explicit solutions for these systems are given

singular angular momentum mappings we algebraically reduce the system consisting of a nonrelativistic particle moving in r with vanishing angular momentum after analyzing the conical structure of the constraint set 1 0 we use algebraic geometric techniques to explicitly construct the reduced poisson algebra of rotationally invariant observables this procedure enables us to completely identify the effects of the singularity in j 0 on the system we then group theoretically reduce the system and compare our results with those obtained algebraically

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

conformal o 3 2 symmetry of the two dimensional inverse square potential it is shown that the dynamical systems describing a point mass moving in a repulsive inverse square potential in a plane and a free relativistic massless particle are isomorphic to each other the obvious conformal invariance of the massless particle appears as a hidden dynamical symmetry of the inverse square potential