the exterior algebra λkrn and area minimization abstract the structure of the exterior algebra k r n is studied in low dimensions and consequences are drawn for k dimensional area minimizing surfaces in r n for a general form o 2 r 4 section 3 gives explicit formulas for the comass o and the face of the grassmannian exposed by o section 4 classifies the faces of the grassmannian of the 3 planes in r 6 and hence the associated geometries of area minimizing surfaces there are four types section 5 establishes an equality involving the comass norm in low dimension and draws implications on when the cartesian product of area minimizing surfaces is area minimizing new examples of area minimizing integral currents with singularities follow

on hamiltonian stationary lagrangian spheres in non einstein kahler surfaces hamiltonian stationary lagrangian spheres in kahler einstein surfaces are minimal we prove that in the family of non einstein kahler surfaces given by the product 1 2 of two complete orientable riemannian surfaces of different constant gauss curvatures there is only a non minimal hamiltonian stationary lagrangian sphere this example defined when the surfaces 1 and 2 are spheres is unstable

estimates on the spectral interval of validity of the anti maximum principle abstract the anti maximum principle for the homogeneous dirichlet problem to p u u p 2 u f x with positive f l states the existence of a critical value f 1 such that any solution of this problem with 1 f is strictly negative in this paper we give a variational upper bound for f and study its properties as an important supplementary result we investigate the branch of ground state solutions of the considered boundary value problem in 1 2

on the singular structure of three dimensional area minimizing surfaces a sufficient condition is given for the union of two three dimensional planes through the origin in r to be area minimizing the condition is in terms of the three angles 0 7 their singular sets have hausdorff dimension at most m 1 larger singular sets occur in higher codimensions for example complex analytic varieties which are automatically area minimizing provide real m dimensional surfaces with 7 2 dimensional singular sets recent work of almgren 1 seems to show that the dimension of a singular set never exceeds m 2 little is known about the structure of the singular set only for two dimensional area minimizing surfaces is even the first order structure of singularities understood 5 cf also 6 this paper studies the first order structure of singularities in three dimensional area minimizing surfaces at each point in the surface there is a tangent cone not known to be unique itself area minimizing which records first order behavior if a tangent cone is a plane with multiplicity one the surface is locally a smooth manifold since the next simplest cone is a sum i e union of planes we address the following basic question question when is the sum of two oriented 3 planes through the origin in r area minimizing the geometric relationship between two oriented 3 planes in r is characterized by three angles 0 y y2 w 2 y2 y3 s m y2 see lemma 1 and remarks we prove a sufficient condition in terms of these three angles theorem 7 y3 s y y2 the sum of the planes is area minimizing we conjecture that the converse holds as well except for the trivial case of the

the separation of a coupled system of differential equations in quantum mechanics the authors discuss various aspects of the methodology of the separation operation of a coupled system of differential equations in quantum mechanics namely the darboux transformation the t and n matrices approach for the case of symmetric and non symmetric coupling of the equations the analysis is also extended to systems with more than two equations with in each case specific examples illustrating their implementation in practical situations

the angle criterion we complete the proof of the angle conjecture on when a pair of orientedm planes is area minimizing the nonzero sum oriented union p1 p2 is area minimizing if and only if the characterizing angles between them satisfy the inequality beta m leqq beta 1 beta m 1

eventually and asymptotically positive semigroups on banach lattices abstract we develop a theory of eventually positive c 0 semigroups on banach lattices that is of semigroups for which for every positive initial value the solution of the corresponding cauchy problem becomes positive for large times we give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators complementing existing results on spaces of continuous functions this enables us to treat a range of new examples including the square of the laplacian with dirichlet boundary conditions the bi laplacian on l p spaces the dirichlet to neumann operator on l 2 and the laplacian with non local boundary conditions on l 2 within the one unified theory we also introduce and analyse a weaker notion of eventual positivity which we call asymptotic positivity where trajectories associated with positive initial data converge to the positive cone in the banach lattice as t this allows us to discuss further examples which do not fall within the above mentioned framework among them a network flow with non positive mass transition and a certain delay differential equation

complete embedded minimal n dimensional submanifolds in ℂn a classical problem in the theory of minimal submanifolds of euclidean spaces is to understand whether a minimal submanifold exists with a prescribed behavior at infinity or to determine from the asymptotes the geometry of the whole submanifold beyond the intrinsic interest of these questions they are also of crucial importance when studying the possible singularities of minimal submanifolds in general riemannian manifolds for minimal surfaces i e two dimensional submanifolds the standard tool to solve these problems is given by the weierstrass representation formula which relates the geometry of the minimal surface to complex analytic properties of holomorphic 1 forms on riemann surfaces recently gluing techniques have been developed that provide an abundant number of new examples of minimal hypersurfaces in euclidean space for higher dimensional minimal submanifolds such a link clearly disappears and no complex analysis can be put into play although gluing techniques have been extensively used in the study of minimal hypersurfaces they have not been adapted to handle higher codimensional submanifolds the aim of this paper is to use a gluing technique for minimal submanifolds to make a step toward understanding these questions in higher codimension more precisely we will restrict ourselves to the case of real n dimensional submanifolds of cn there are two main reasons for doing so the first one comes from the fact that when trying to desingularize the intersection for example of a pair of n planes that give the desired asymptotic behavior one needs a model of a minimal submanifold with this behavior at infinity to rescale and to glue into the pair of planes where a neighborhood of the intersection is removed this local model needs to be sufficiently simple to allow a detailed study of the linearized mean curvature operator in our situation this is provided by a generalization of an area minimizing submanifold found by lawlor in 12 while in more general cases such an example is not known the second reason is that among minimal n submanifolds ofcn there is a

darboux transformations for the schrodinger equation in three dimensions the author applies darboux transformations to the schrodinger equation in three dimensions and derives the general form of the potentials and potential differences which can be treated by these transformations in particular the author establishes the existence of a new class of potentials whose solutions are related to each other by these transformations

coupled channel scattering and separation of coupled differential equations by generalized darboux transformations general methods for separation of coupled differential equations are applied in coupled channel scattering theory with appropriate asymptotic conditions it is shown that the problems with thresholds cannot be separated finally it is pointed out that separation of the coupled channel s wave problem without thresholds can be realized by the use of generalized darboux transformations

size minimizing rectifiable currents 1 introduct ion 333 2 on the existence and regularity of size minimizers 336 3 isoperimetric inequalities and size bounds 345 4 bernstein theorems 346 references 348

factorization of systems of differential equations it is shown that the classical infeld hull factorization method can be extended to coupled systems of second order equations a complete solution of the factorization equations in two dimensions is given and a partial enumeration of factorizable systems is made

uniform anti maximum principles abstract consider a second or higher order elliptic partial differential equation a u u f on an open bounded domain of r n with homogeneous boundary conditions b u 0 if there exists a simple eigenvalue for which the corresponding eigenfunction is positive and satisfies appropriate boundary estimates then an anti maximum principle holds for positive f lp with p large enough there exists f 0 such that for 1 1 f the solution is negative and for 1 f 1 the solution is positive we give conditions such that this sign reversing property is uniform there is 0 such that for all positive f the solution u is negative for 1 1 and positive for 1 1 two classes of higher order boundary value problems that satisfy these conditions will be given

calibrations modulo v abstract the theory of calibrations is extended to show that certain surfaces are area minimizing modulo v for example a complex algebraic variety in c n of degree d is area minimizing modulo v for all v 2 d

the unit mass ball of three vectors in r6 abstract we study the unit ball in 3 r 6 we give a test for distinguishing the types of mass decompositions establish an algorithm for finding mass decompositions and provide computational formulae for the mass norm

separation of coupled systems of differential equations by darboux transformations the author shows that certain classes of coupled schrodinger equations in one dimension which appear in various physical applications can be uncoupled by the application of an appropriate darboux transformation

examples of unoriented area minimizing surfaces a comprehensive study is made of constructions of area minimizing flat chains modulo two many have singularities for instance any bounded area mini mizing submanifold of rn occurs as the singular set of some area minimizing flat chain modulo two in some rn introduction given a k l dimensional boundary in r one can seek to minimize area in some class of surfaces we want to take a large class that includes nonorientable as well as orientable surfaces without restrictions on the singularities or topological type such a class appears in geometric measure theory as the flat chains modulo two the existence and regularity of area minimizing flat chains modulo two are already well established cf 7 4 2 26 9 unfortunately examples have been rare this paper presents several methods for constructing area minimizing flat chains modulo two the first five theorems include the following five examples 1 a mobius strip 2 the graph over a convex region of any function satisfying the minimal surface equation

maximum and antimaximum principles for some nonlocal diffusion operators abstract in this work we consider the maximum and antimaximum principles for the nonlocal dirichlet problem j u u u h r n j x y u y d y u x u x h x 0 in a bounded domain with u x 0 in r n the kernel j in the convolution is assumed to be a continuous compactly supported nonnegative function with unit integral we prove that for 1 the solution verifies u 0 in if h l 2 h 0 while for 1 and close to 1 the solution verifies u 0 in provided h x x d x 0 h l this last assumption is also shown to be optimal the neumann version of the problem is also analyzed

theory of coupled differential equations the problem of decoupling a system of linear coupled differential equations without changing the order of the equations will be considered starting from a theorem on the separation of the coupled equations the author continues the investigations in the general case where the conditions set up by this theorem are not satisfied in deriving a method of separation of the equations in the strong coupling case an example on a system of two coupled schrodinger equations will be taken up as an illustration of this method

generalisation of the theory of coupled differential equations the author generalises a method derived from the theorem on the separation of coupled equations which has been stated previously to solve the case of two equations it will be shown that this generalisation is always possible with a sequence of special transformations in terms of scaling parameters the case of a system of three coupled differential equations is considered in some detail including a discussion on the search and use of these parameters

isospectral susy potentials the authors introduce a new linear transformation in order to construct isospectral potentials within the framework of supersymmetric quantum mechanics under this transformation the adjoint character of the ladder operator is destroyed but with preservation of the energy spectrum leading to some interesting consequences such as the concept of pseudo partnership of the two components connection with the darboux theorem as well as with some other recent methods of construction is analysed

self similar solutions and translating solitons for lagrangian mean curvature flow we construct many self similar and translating solitons for lagrangian mean curvature flow including self expanders and translating solitons with arbitrarily small oscillation on the lagrangian angle our translating solitons play the same role as cigar solitons in ricci flow and are important in studying the regularity of lagrangian mean curvature flow given two transverse lagrangian planes wn in cn with sum of characteristic angles less than a we show there exists a lagrangian self expander asymptotic to this pair of planes the maslov class of these self expanders is zero thus they can serve as local models for surgeries on lagrangian mean curvature flow families of self shrinkers and self expanders with different topologies are also constructed this paper generalizes the work of anciaux i joyce 12 lawlor 15 and lee and wang 17 18

special lagrangian cones with higher genus links for every odd natural number g 2d 1 we prove the existence of a countably infinite family of special lagrangian cones in mathbb c 3 over a closed riemann surface of genus g using a geometric pde gluing method

a theorem on the separation of a system of coupled differential equations considers the separation of a system of finite linear coupled differential equations the conditions which govern this separation in the case of a system of two coupled equations are discussed it is then shown how these results may be extended to the case of a system of a finite number of coupled equations for which a general theorem on the separability is formulated

a search for shape invariant solvable potentials the author investigates a simple method of constructing potentials which is related to the work of bhattacharjie and sudarshan 1962 and for which the schrodinger equation can be solved in terms of known special functions it turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions satisfying linear homogeneous second order differential equations can be solutions of the schrodinger equation with potentials of the form v x w2 x w x the author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape invariant potentials

quasi uniformity for the abstract neumann antimaximum principle and applications with a priori estimates abstract in this paper we prove a general result giving the maximum and the antimaximum principles in a unitary way for linear operators of the form l i provided that 0 is an eigenvalue of l with associated constant eigenfunctions to this purpose we introduce a new notion of quasi uniform maximum principle named k uniform maximum principle it holds for belonging to certain neighborhoods of 0 depending on the fixed positive multiplier k 0 which selects the good class of right hand sides our approach is based on a l l p estimate for some related problems as an application we prove some generalization and new results for elliptic problems and for time periodic parabolic problems under neumann boundary conditions

special lagrangian submanifolds with isolated conical singularities v survey and applications this is the last in a series of five papers studying compact special lagrangian submanifolds sl m folds x in almost calabi yau m folds m with singularities x1 x n locally modelled on special lagrangian cones c1 c n in c m with isolated singularities at 0 readers are advised to begin with this paper we survey the major results of the previous four papers giving brief explanations of the proofs we apply the results to describe the boundary of a moduli space of compact nonsingular sl m folds n in m we prove the existence of special lagrangian connected sums n1 nk of sl mfolds n1 n k in m we also study sl 3 folds with t 2 cone singularities proving results related to ideas of the author on invariants of calabi yau 3 folds and the syz conjecture let x be a compact sl m fold with isolated conical singularities xi and cones ci for i 1 n the first paper studied the regularity of x near its singular points and the the second the moduli space of deformations of x the third and fourth papers construct desingularizations of x realizing x as a limit of a family of compact nonsingular sl m folds n t in m for small t 0 let li be an asymptotically conical sl m fold in cm asymptotic to ci at infinity we make n t by gluing tli into x at xi for i 1 n

minimal submanifolds of ⁿ arising from degenerate 3 orbits on the grassmannian we give new examples of minimal submanifolds of e 2n characterised by having their gauss map s image lie in degenerate s0 3 orbits of gp2n 1 the grassmannian of p planes in e where the action on 2n 1 n gp 2n 1 is induced from the irreducible s0 3 action on r2n these submanifolds are all given explicitly in terms of holomorphic data and are linearly 2n l1 full in e

constructing special lagrangian m folds in c m by evolving quadrics this is the second in a series of papers constructing explicit examples of special lagrangian submanifolds sl m folds in c the first paper of the series 7 studied sl m folds with large symmetry groups and subsequent papers 8 9 10 11 construct examples of sl 3 folds in c using evolution equations symmetries ruled submanifolds and integrable systems the principal motivation for these papers is to lay the foundations for a study of the singularities of compact special lagrangian m folds in calabi yau m folds particularly in low dimensions such as m 3 special lagrangian m folds in c and especially special lagrangian cones should provide local models for singularities of sl m folds in calabi yau m folds understanding such singularities will be essential in making rigorous the explanation of mirror symmetry of calabi yau 3 folds x x proposed by strominger yau and zaslow 13 which involves dual fibrations of x x by special lagrangian 3 tori with some singular fibres it will also be important in resolving conjectures made by the author 6 which attempt to define an invariant of calabi yau 3 folds by counting special lagrangian homology 3 spheres the paper falls into three parts the first this section and 2 is introductory the second part 3 4 describes a general construction of special lagrangianm folds n in c depending on a set of evolution data p where p is an m 1 submanifold in r then n is the subset of c swept out by the image of p under a 1 parameter family of linear or affine maps t r n c which satisfy a first order nonlinear o d e in t examples of sets of evolution data will be given in 4 together with some progress towards a classification of such data the simplest interesting sets of evolution data occur when n m and p is a nondegenerate quadric in r in the third part 5 7 we apply the construction to these examples in this case t r must be a lagrangian plane in c for each t thus n is fibred by quadrics in lagrangian planes r in c the construction of 3 4 will also be used in the sequel to this paper 8 with different evolution data to construct families of sl 3 folds in c the construction has both a linear and an affine version in the linear version

supersymmetry factorisation of the schrodinger equation and a hamiltonian hierarchy the author presents a systematic procedure for constructing a hierarchy of non relativistic hamiltonians with the property that the adjacent members of the hierarchy are supersymmetric partners i e they share the same eigenvalue spectrum except for the missing ground state and the eigenvectors are simply related