Extremal maps in Sobolev type inequalities: Some remarks

In this work we make some observations on the existence of extremal maps for sharp L²-Riemannian Sobolev type inequalities as Nash and logarithmic Sobolev ones. Among other results, we prove also that there exist smooth compact Riemannian manifolds with scalar curvature changing signal on which there exist extremal maps.     

Infinitely many leaf-wise intersections on cotangent bundles

If the homology of the free loop space of a closed manifold B$B$ is infinite dimensional then generically there exist infinitely many leaf-wise intersection points for fiberwise star-shaped hypersurfaces in T^∗B$T^{\ast }B$. We illustrate this in the case of the restricted three body problem.     

Generalized symplectic Cayley transforms and a higher order formula for the Conley–Zehnder index of symplectic paths

Using a generalized notion of symplectic Cayley transform in the symplectic group, we introduce a sequence of integer valued invariants (higher order signatures) associated with a degeneracy instant of a smooth path of symplectomorphisms. In the real analytic case, we give a formula for the Conley–Zehnder index in terms of the higher order signatures.     

Multiplicity results for problems involving the Hardy-Sobolev operator via Morse theory

We establish some multiplicity results for a class of boundary value problems involving the Hardy-Sobolev operator using Morse theory.     

A Nash type solution for hemivariational inequality systems

In this paper, we prove an existence result for a general class of hemivariational inequality systems using the Ky Fan version of the KKM theorem Fan (1984) [10] or Tarafdar fixed points Tarafdar (1987) [11]. As application, we give an infinite-dimensional version for the existence result of Nash generalized derivative points introduced recently by Kristály (2010) [5]. We also give an application to a general hemivariational inequality system.     

Multiplicity and stability of closed geodesics on bumpy Finsler 3-spheres

We prove that for every Q-homological Finsler 3-sphere (M, F) with a bumpy and irreversible metric F, either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics. Huagui Duan: Partially supported by NNSF and RFDP of MOE of China. Yiming Long: Partially supported by the 973 Program of MOST, Yangzi River Professorship, NNSF, MCME, RFDP, LPMC of MOE of China, S. S. Chern Foundation, and Nankai University.     

Geometry of curves and surfaces through the contact map

We introduce a new approach to the study of affine equidistants and centre symmetry sets via a family of maps obtained by reflexion in the midpoints of chords of a submanifold of affine space. We apply this to surfaces in R³, previously studied by Giblin and Zakalyukin, and then apply the same ideas to surfaces in R⁴, elucidating some of the connexions between their geometry and the family of reflexion maps. We also point out some connexions with symplectic topology.     

Fundamental solution of a multidimensional difference equation with periodical and matrix coefficients

Summary The multidimensional (partial) difference equation with periodical coefficients is transformed into an equation for a vector sequence. Integral formulae for the vector fundamental solution are developed and some results about its asymptotic properties are explained. As an example, the results are used for a simple difference equation on a hexagonal grid.     

Nonlinear Liouville theorems for Grushin and Tricomi operators

The aim of this paper is to study necessary conditions for existence of weak solutions of the inequality

     

A Liouville-type result for quasi-linear elliptic equations on complete Riemannian manifolds

We extend a Liouville-type result of D. G. Aronson and H. F. Weinberger and E.N. Dancer and Y. Du concerning solutions to the equation Δ_pu=b(x)f(u) to the case of a class of singular elliptic operators on Riemannian manifolds, which include the ?-Laplacian and are the natural generalization to manifolds of the operators studied by J. Serrin and collaborators in Euclidean setting. In the process, we obtain an a priori lower bound for positive solutions of the equation in consideration, which complements an upper bound previously obtained by the authors in the same context.     

Orthogonal trajectories on Riemannian manifolds and applications to Plane Wave type spacetimes

We present a result on trajectories of a Lagrangian system joining two given submanifolds of a Riemannian manifold, under the action of an unbounded potential. As an application, we consider geodesics in a class of semi-Riemannian manifolds, the Plane Wave type spacetimes.     

Solutions to a gradient system with resonance at both zero and infinity

In this paper, we study the existence and multiplicity of nontrivial solutions for a gradient system with resonance at both zero and infinity via Morse theory.     

Existence and nonexistence of the global solution for the semilinear parabolic system on Riemannian manifold

In this paper, we study the global existence and nonexistence of solutions to the following semilinear parabolic system

     

Existence and regularity for generalised harmonic maps associated to a nonlocal polyconvex energy of Skyrme type

We prove existence and regularity of critical points of arbitrary degree for a generalised harmonic map problem, in which there is an additional nonlocal polyconvex term in the energy, heuristically of the same order as the Dirichlet term. The proof of regularity hinges upon a special nonlinear structure in the Euler–Lagrange equation similar to that possessed by the harmonic map equation. The functional is of a type appearing in certain models of the quantum Hall effect describing nonlocal Skyrmions.     

Transgression on hyperkähler manifolds and generalized higher torsion forms

We propose a generalization of the Hodge dd_c-lemma to the case of hyperk?hler manifolds. As an application we derive a global construction of the fourth order transgression of the Chern character forms of hyperholomorphic bundles over compact hyperk?hler manifolds. In Section 3 we consider the fourth order transgression for the infinite-dimensional bundle arising from local families of hyperk?hler manifolds. We propose a local construction of the fourth order transgression of the Chern character form. We derive an explicit expression for the arising hypertorsion differential form. Its zero-degree part may be expressed in terms of the Laplace operators defined on the fibres of the local family.     

Intrinsic third derivatives for Plateau's problem and the Morse inequalities for disc minimal surfaces in ℝ3

Summary A simply branched minimal surface in ³ cannot be a non-degenerate critical point of Dirichlet's energy since the Hessian always has a kernel. However such minimal surface can be non-degenerate in another sense introduced earlier by R. Böhme and the author. Such surfaces arise as the zeros of a vector field on the space of all disc surfaces spanning a fixed contour. In this paper we show that the winding number of this vector field about such a surface is ±2 ^p , wherep is the number of branch points. As a consequence we derive the Morse inequalities for disc minimal surfaces in ³, thereby completing the program initiated by Morse, Tompkins, and Courant. Finally, this result implies that certain contours in ⁴ arbitrarily close to the given contour must span at least 2 ^p disc minimal surfaces.     

A cohomology attached to a function

In this paper, we study a certain cohomology attached to a smooth function, which arose naturally in Poisson geometry. We explain how this cohomology depends on the function, and we prove that it satisfies both the excision and the Mayer-Vietoris axioms. For a regular function we show that the cohomology is related to the de Rham cohomology. Finally, we use it to give a new proof of a well-known result of A. Dimca [Compositio Math. 76 (1990) 19-47] in complex analytic geometry.     

A generalization of Higgs bundles to higher dimensional varieties

Let X be a smooth n-dimensional projective variety defined over and let L be a line bundle on X. In this paper we shall construct a moduli space parametrizing -cohomology L-twisted Higgs pairs, i.e., pairs where E is a vector bundle on X and . If we take , the canonical line bundle on X, the variety is canonically identified with the cotangent bundle of the smooth locus of the moduli space of stable vector bundles on X and, as such, it has a canonical symplectic structure. We prove that, in the general case, in correspondence to the choice of a non-zero section , one can define, in a natural way, a Poisson structure on . We also analyze the relations between this Poisson structure on and the canonical symplectic structure of the cotangent bundle to the smooth locus of the moduli space of parabolic bundles over X, with parabolic structure over the divisor D defined by the section s. These results generalize to the higher dimensional case similar results proved in [Bo1] in the case of curves. Received November 4, 1997; in final form May 28, 1998