Arbitrary Compact Manifold Research Articles

A new proof of regularity of weak solutions of the H -surface equation

Abstract. We give a new proof of a theorem of Bethuel, asserting that arbitrary weak solutions $u\in W^{1,2}({\mathbb B},{\mathb R}^3)$ of the H-surface system $\Delta u = 2H(u) u_{x1}\wedge u_{x2}$ are locally Holder continuous provided that H is a bounded Lipschitz function. Contrary to Bethuel's, our proof completely omits Lorentz spaces. Estimates below natural exponents of integrability are used instead. (The same method yields a new proof of Helein's theorem on regularity of harmonic maps from surfaces into arbitrary compact Riemannian manifolds.) We also prove that weak solutions with continuous trace are continuous up to the boundary, and give an extension of these results to the equation of hypersurfaces of prescribed mean curvature in ${\mathbb R}^{n+1}$ , this time assuming in addition that $|\nabla H(y)|$ decays at infinity like $|y|^{-1}$ .

On the Topology of Vacuum Spacetimes

We prove that there are no restrictions on the spatial topology of asymptotically flat solutions of the vacuum Einstein equations in (n+1)-dimensions. We do this by gluing a solution of the vacuum constraint equations on an arbitrary compact manifold to an asymptotically Euclidean solution of the constraints on R^n. For any compact manifold which does not admit a metric of positive scalar curvature, this provides for the existence of asymptotically flat vacuum spacetimes with no maximal slices. Our main theorem is a special case of a more general gluing construction for nondegenerate solutions of the vacuum constraint equations which have some restrictions on the mean curvature, but for which the mean curvature is not necessarily constant. This generalizes the construction [IMP] (gr-qc/0109045), which is restricted to constant mean curvature data.

On the geometry of the random representations for viscous fluids and a remarkable pure noise representation

Extending our previous work we present implicit representations for the Navier-Stokes equations (NS) for an incompressible fluid in a smooth compact manifold without boundary as well as for the kinematic dynamo equation (KDE, for short) of magnetohydrodynamics. We derive these representations from stochastic differential geometry, unifying gauge theoretical structures and the stochastic analysis on manifolds (the Ito-Elworthy formula for differential forms). From the diffeomorphism property of the random flow given by the scalar Lagrangian representations for the viscous and magnetized fluids, we derive the representations for NS and KDE, using the generalized Hamilton and Ricci random flows (for arbitrary compact manifolds without boundary), and the gradient diffusion processes (for isometric immersions on Euclidean space of these manifolds). Continuing with this method, we prove that NS and KDE in any dimension other than 1 can be represented as purely (geometrical) noise processes, with diffusion tensor depending on the fluid's velocity, and we represent the solutions of NS and KDE in terms of these processes. We discuss the relations between these representations and the problem of infinite-time existence of solutions of NS and KDE. We finally discuss the relations between this approach with the low dimensional chaotic dynamics describing the asymptotic regime of the solutions of NS.

Positive solutions for a fourth order equation invariant under isometries

Let ( M , g ) (M,g) be a smooth compact Riemannian manifold of dimension n ≥ 5 n\geq 5 . We consider the problem ( ⋆ ) Δ g 2 u + α Δ g u + a u = f u n + 4 n − 4 , \begin{equation*} \tag {$\star $}\Delta _g^2 u+\alpha \Delta _g u+au=f u^{\frac {n+4}{n-4}}, \end{equation*} where Δ g = − d i v g ( ∇ ) \Delta _g=-div_g(\nabla ) , α \alpha , a ∈ R a\in \mathbb {R} , u u , f ∈ C ∞ ( M ) f\in C^{\infty }(M) . We require u u to be positive and invariant under isometries. We prove existence results for ( ⋆ ) (\star ) on arbitrary compact manifolds. This includes the case of the geometric Paneitz-Branson operator on the sphere.

TEMPERED OPERATORS AND THE HEAT KERNEL AND COMPLEX POWERS OF ELLIPTIC PSEUDODIFFERENTIAL OPERATORS

The resolvent (A – λ)−1 of an elliptic b-pseudodifferential operator on a compact manifold with corners (of arbitrary codimension) is shown to lie in a calculus of operators, tempered in the parameter λ in a special way. We show that the Laplace and Mellin transforms, with respect to λ, of these tempered operators can be defined and that they have Schwartz kernels which can be described geometrically. As a corollary, we obtain the structures of the kernels of the heat operator and complex powers of b-pseudodifferential operators, as the heat operator and complex powers are the Laplace and Mellin transforms, respectively, of the resolvent. The heat operator and complex powers are then used to generalize the index formula of Atiyah, Patodi, and Singer for Dirac operators on manifolds with boundary to Fredholm b-pseudodifferential operators on arbitrary compact manifolds with corners.

Identification of Berezin-Toeplitz deformation quantization

We give a complete identification of the deformation quantization which was obtained from the Berezin- Toeplitz quantization on an arbitrary compact Kahler manifold. The deformation quantization with the opposite star-product proves to be a differential deformation quantization with separation of variables whose dassifying form is explicitly calculated. Its characteristic dass (which dassifies star-products up to equivalence) is obtained. The proof is based on the microlocal description of the Szego kernel of a strictly pseudoconvex domain given by Boutet de Monvel and Sjostrand.

Abel's theorem for divisors on an arbitrary compact complex manifold

We prove Abel's theorem for divisors on an arbitrary compact complex manifold by combining the Čech cohomology of sheaves, a logarithmic residue formula for 1-forms and de Rham's theory applied to open submanifolds.

COMBINATORIAL APPROXIMATION BY DEVANEY-CHAOTIC OR PERIODIC VOLUME PRESERVING HOMEOMORPHISMS

We modify a combinatorial idea of Peter Lax, to uniformly approximate any volume preserving homeomorphism of the closed unit cube In, n≥2, by another one which either: (i) rigidly permutes an infinite family of dyadic cubes of total volume one, or (ii) is chaotic in the sense of Devaney. Our methods apply as well to arbitrary compact manifolds endowed with a nonatomic, locally positive Borel measure.

Inégalités de Morse Holomorphes Singulières

We generalize Demailly’s holomorphic Morse inequalities to the case of a line bundle E equipped with a singular metric on an arbitrary compact complex manifold X. Our inequalities give an estimate of the cohomology groups with values in the tensor power E⊗k twisted by the corresponding sequence of multiplier ideal sheaves introduced by Nadel. The allowed singularities are of the following type: the metric is locally given by a weight exp(−φ) where $$\phi \sim \tfrac{c}{2}\log (\Sigma |f_j |^2 )$$ with holomorphic fj. As a consequence, we obtain a necessary and sufficient analytic condition, invariant by bimeromorphism, for a manifold X to be Moishezon. This characterization improves a result given by Ji and Shiffman. We finally recall and improve some results of Kollar in order to show that the corresponding sufficient conditions obtained by Siu and Demailly in the smooth case are not necessary.

The asymptotic expansion of a CR invariant and Grauert tubes

manifolds, the abstract models of real hypersurfaces in complex manifolds, are 2n + 1 dimensional manifolds M with a codimension one subbundle H of the tangent bundle, which carries a complex structure. The CR refers to Cauchy-Riemann because for M C C n+l, the subbundle H consists of induced Cauchy-Riemann operators. There is a wealth of geometry and analysis associated with these structures, especially when the manifolds are strictly pseudoconvex. For example, two strictly pseudoconvex domains are biholomorphically equivalent if and only if their boundaries are equivalent. A fundamental problem in geometry is to find computable invariants associated with the structures. The global invariant we will consider in this paper is the Chem-Simons type invariant/~ discovered by Bums and Epstein [B-E 1]. It is a real-valued global invariant of a compact 3dimensional strictly pseudo-convex manifold whose holomorphic tangent bundle is trivial. (Cheng and Lee independently found this invariant, and extend the definition of B-E invariant to a relative invariant on an arbitrary compact 3-dimensional manifold, cf. [C-L].) We will evaluate this g asymptotically on the boundary of small Grauert tubes. Before posing the question in a more precise form we will first say a few workds about Grauert tubes. Let X be a real analytic manifold. Then every coordinate patch U C R n can be thickened to obtain an open set r c C n. Since the coordinate changes of X are real analytic maps, by taking power series expansions and by shrinking C U to get convergence, they can be extended holomorphically to such enlarged domains and thus they can be used as holomorphic transition

Uniqueness for the harmonic map flow from surfaces to general targets

LetM be a two-dimensional compact Riemannian manifold with smooth (possibly empty) boundary,N an arbitrary compact manifold. Ifu andv are weak solutions of the harmonic map flow inH 1(Mx[0,T]; N) whose energy is non-increasing in time and having the same initial datau 0∈H1(M, N) (and same boundary values if ∂M≠O) thenu=v. Combined with a result of M. Struwe, this shows any suchu is smooth in the complement of a finite subset ofM×(0,T)c.

Periodic orbits for Hamiltonian systems in cotangent bundles

We prove the existence of at least cl ⁡ ( M ) \operatorname {cl}(M) periodic orbits for certain time-dependent Hamiltonian systems on the cotangent bundle of an arbitrary compact manifold M. These Hamiltonians are not necessarily convex but they satisfy a certain boundary condition given by a Riemannian metric on M. We discretize the variational problem by decomposing the time-1 map into a product of "symplectic twist maps". A second theorem deals with homotopically non-trivial orbits of negative curvature.

Cobordism of polarized t-structures

Polarized t-structures seem to be attractive topological objects. One readily sees how to define the notion of the cobordism between them. This cobordism group Qrr (in dimension 3) is our principal object of interest in this paper. When one looks for cobordism invariants from this perspective, the following idea comes to mind first: The family g, gives rise to a metric on M x R+, as explained in Cl]. The conditions l-3 imply that the sectional curvature of this metric is bounded by one, and the volume is finite. Filling M with an arbitrary compact manifold W, and extending the metric to the filling we obtain a complete, finite volume and bounded curvature metric on the interior of W which is equal to the Cheeger-Gromov metric in the neighborhood of infinity. One can show that the integrals over W of the Pontrjagin forms computed from the metric, reduced mod 2 are cobordism invariants of the polarized t-structure. In fact complete open manifolds of bounded curvature and finite volume give rise to an interesting cobordism theory, and the integrals of Pontrjagin forms supply us with cobordism invariants. It is unclear just how well do these integrals gauge the cobordism group. In the present paper we construct, in the restricted context of polarized t-structures, finer invariants, which in fact allow the computation of the 3-dimensional cobordism group .$“I modulo torsion. We then use an invariant related to the integral of the Pontrjagin form to deal with the torsion.

Rotation sets for homeomorphisms and homology

In this article we propose a definition of rotation sets for homeomorphisms of arbitrary compact manifolds. This approach is based on taking the suspended flow and using ideas of Schwartzmann on homology and winding cycles for flows. Our main application is to give a generalisation of a theorem of Llibre and MacKay for tori to the context of surfaces of higher genus.

The origin of the three generations in compactifications of the E 8×E 8 heterotic string

The index for a twisted spin complex with the SU(N) Z N bundle over an arbitrary six-dimensional compact manifold K is computed. Three generations of quarks/leptons with the gauge group SO(10) are shown to be universally realized in compactifications of the E 8×E 8 heterotic string from the field theory limit if the dimension of the first homology group of K is three or larger than three, i.e., dim H 1( K)⩾3.

The Gauss-Bonnet-Chern theorem and supersymmetry

Using the path integral representation of the Witten index for supersymmetric QM systems, we compute the boundary corrections to the Gauss-Bonnet theorem for an arbitrary compact manifold with boundary.

Charges from extra dimensions

A prescription is given for calculating gauge coupling constants in (4 + N)-dimensional models in which N dimensions form an arbitrary compact manifold: Each coupling is given by the ration of 2π(16πG) 1 2 to an appropriate root-mean-square circumference of the manifold. It is speculated that these couplings will be of order 1/√ n, where n is the number of matter fields whose one-loop quantum energy density is responsible for the compactification.

Certain Controllability Properties of Analytic Control Systems

In this article we consider analytic control systems defined on an analytic manifold M. In particular, we study the relationship between the accessibility property, i.e., attainable sets having nonempty interior, and controllability. This study is restricted to analytic control systems because they admit a very simple algebraic criterion for determining the accessibility property; this criterion is a generalization of the well-known rank condition for linear systems.The main difficulty in extending the accessibility property to yield controllability lies in the fact that the time coordinate is nondecreasing. Beside systems of the form $F(x,u) = - F(x, - u)$, right invariant control systems defined on a compact Lie group are controllable whenever they have the accessibility property. It seems reasonable that a similar result should hold for arbitrary compact manifolds.