Disk Bundle Research Articles

On the slice-ribbon conjecture for Montesinos knots

We establish the slice-ribbon conjecture for a family P \mathscr {P} of Montesinos knots by means of Donaldson’s theorem on the intersection forms of definite 4 4 -manifolds. The 4 4 -manifolds that we consider are obtained by plumbing disc bundles over S 2 S^2 according to a star-shaped negative-weighted graph with 3 3 legs such that: i) the central vertex has weight less than or equal to − 3 - 3 ; ii) − total weight − 3 # vertices > − 1 -\,\mbox {total weight} - 3\, \# \mbox {vertices} >-1 . The Seifert spaces which bound these 4 4 -dimensional plumbing manifolds are the double covers of S 3 S^3 branched along the Montesinos knots in the family P \mathscr {P} .

M-theory, the signature theorem, and geometric invariants

The equations of motion and the Bianchi identity of the C-field in M-theory are encoded in terms of the signature operator. We then reformulate the topological part of the action in M-theory using the signature, which leads to connections to the geometry of the underlying manifold, including positive scalar curvature. This results in a variation on the miraculous cancellation formula of Alvarez-Gaum\'e and Witten in 12 dimensions and leads naturally to the Kreck-Stolz $s$-invariant in 11 dimensions. Hence M-theory detects the diffeomorphism type of 11-dimensional (and seven-dimensional) manifolds and in the restriction to parallelizable manifolds classifies topological 11 spheres. Furthermore, requiring the phase of the partition function to be anomaly-free imposes restrictions on allowed values of the $s$-invariant. Relating to string theory in ten dimensions amounts to viewing the bounding theory as a disk bundle, for which we study the corresponding phase in this formulation.

Rotationally symmetric pseudo-Kähler-Einstein metrics

In this paper, we construct some rotationally symmetric pseudo-Kahler-Einstein metrics on various holomorphic line bundles over projective spaces and their disc bundles. We also discuss the phase change phenomenon when one suitably changes parameters.

Rotationally symmetric pseudo-Kähler metrics of constant scalar curvatures

We study the ordinary differential equations related to rotationally symmetric pseudo-Kahler metrics of constant scalar curvatures. We present various solutions on various holomorphic line bundles over projective spaces and their disc bundles, and discuss the phase change phenomenon when one suitably changes initial values.

A generalization of the Wang sequence

The classical Wang sequence (for cohomology of fiber bundles over a sphere) is extended to a more generalized setting, given by gluing together two disc bundles over manifolds along their boundaries.

Complex Hyperbolic Structures on Disc Bundles over Surfaces

We study complex hyperbolic disc bundles over closed orientable surfaces that arise from discrete and faithful representations H_n->PU(2,1), where H_n is the fundamental group of the orbifold S^2(2,...,2) and thus contains a surface group as a subgroup of index 2 or 4. The results obtained provide the first complex hyperbolic disc bundles M->{\Sigma} that: admit both real and complex hyperbolic structures; satisfy the equality 2(\chi+e)=3\tau; satisfy the inequality \chi/2<e; and induce discrete and faithful representations \pi_1\Sigma->PU(2,1) with fractional Toledo invariant; where {\chi} is the Euler characteristic of \Sigma, e denotes the Euler number of M, and {\tau} stands for the Toledo invariant of M. To get a satisfactory explanation of the equality 2(\chi+e)=3\tau, we conjecture that there exists a holomorphic section in all our examples. In order to reduce the amount of calculations, we systematically explore coordinate-free methods.

Collapsing manifolds with boundary

This paper studies manifolds-with-boundary collapsing in the Gromov– Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The first group of results provide a fiber bundle structure to the manifolds-with-boundary. One of the main theorems establishes a disc bundle structure for any manifold-with-boundary having two-sided bounds on sectional curvature and second fundamental form, and a lower bound on intrinsic injectivity radius, which is sufficiently close in the Gromov–Hausdorff topology to a closed manifold. Another result is a rough version of Toponogov’s Splitting Theorem. The second group of results identify Gromov–Hausdorff limits of certain sequences of manifolds with non-convex boundaries as Alexandrov spaces of curvature bounded below.

ESTIMATES AND COMPUTATIONS IN RABINOWITZ–FLOER HOMOLOGY

The Rabinowitz–Floer homology of a Liouville domain W is the Floer homology of the Rabinowitz free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. This invariant has been introduced by K. Cieliebak and U. Frauenfelder and has already found several applications in symplectic topology and in Hamiltonian dynamics. Together with A. Oancea, the same authors have recently computed the Rabinowitz–Floer homology of the cotangent disk bundle D* M of a closed Riemannian manifold M, by means of an exact sequence relating the Rabinowitz–Floer homology of D* M with its symplectic homology and cohomology. The first aim of this paper is to present a chain level construction of this exact sequence. In fact, we show that this sequence is the long homology sequence induced by a short exact sequence of chain complexes, which involves the Morse chain complex and the Morse differential complex of the energy functional for closed geodesics on M. These chain maps are defined by considering spaces of solutions of the Rabinowitz–Floer equation on half-cylinders, with suitable boundary conditions which couple them with the negative gradient flow of the geodesic energy functional. The second aim is to generalize this construction to the case of a fiberwise uniformly convex compact subset W of T* M whose interior part contains a Lagrangian graph. Equivalently, W is the energy sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any energy level which is larger than the strict Mañé critical value of L. In this case, the energy functional for closed geodesics is replaced by the free period Lagrangian action functional associated to a suitable calibration of L. An important issue in our analysis is to extend the uniform estimates for the solutions of the Rabinowitz–Floer equation — both on cylinders and on half-cylinders — to Hamiltonians which have quadratic growth in the momenta. These uniform estimates are obtained by the Aleksandrov integral version of the maximum principle. In the case of half-cylinders, they are obtained by an Aleksandrov-type maximum principle with Neumann conditions on part of the boundary.

Generalized Huygens principle with pulsed-beam wavelets

Huygens' geometric construction explaining wave motion has a well-known problem with unphysical back-propagation due to the spherical nature of the secondary wavelets. We solve this by analytically continuing the surface of integration. If the surface is a sphere SR of radius R, this is done by complexifying R to α = R + ia. The resulting complex sphere Sα is shown to be equivalent to the real tangent disk bundle with base SR consisting of all disks with radius a tangent to SR. Huygens' secondary source points are thus replaced by disks, and his secondary wavelets by well-focused pulsed beams propagating outward. This solves the back-propagation problem. The generalized Huygens principle is a completeness relation for these pulsed-beam wavelets enabling a pulsed-beam representation of all radiation fields. Furthermore, this yields a natural and extremely efficient way to compute radiation fields numerically because all pulsed beams missing a given observer can be ignored with minimal error. Increasing the disk radius a sharpens the focus of the pulsed beams, which in turn raises the compression ratio of the representation.

Cohomogeneity one disk bundles with normal homogeneous collars

We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its rank has to be in $\{2,3,4,6,8\}$. Moreover, we give a complete classification of such bundles of rank 6 and 8, and a partial classification for rank 3.

Ultrasensitive Electrochemical DNA Assay Based on Counting of Single Magnetic Nanobeads by a Combination of DNA Amplification and Enzyme Amplification

An ultrasensitive electrochemical method for determination of DNA is developed based on counting of single magnetic nanobeads (MNBs) corresponding to single DNA sequences combined with a double amplification (DNA amplification and enzyme amplification). In this method, target DNA (t-DNA) is captured on a streptavidin-coated substrate via biotinylated capture DNA. Then, MNBs functionalized with first-probe DNAs (p1-DNA-MNBs) are conjugated to t-DNA sequences with a ratio of 1:1. Subsequently, the p1-DNA-MNBs are released from the substrate via dehybridization. The released p1-DNA-MNBs are labeled with alkaline phosphatase (AP) using biotinylated second-probe DNAs (p2-DNAs) and streptavidin-AP conjugates. The resultant AP-p2-DNA-p1-DNA-MNBs with enzyme substrate disodium phenyl phosphate (DPP) are continuously introduced through a capillary as the microsampler and microreactor at 40 degrees C. AP on the AP-p2-DNA-p1-DNA-MNBs converts a huge number of DPP into its product phenol, and phenol zones are produced around each moving AP-p2-DNA-p1-DNA-MNB. The phenol zones are continuously delivered to the capillary outlet and detected by a carbon fiber disk bundle electrode at 1.05 V. An elution curve with peaks is obtained. Each peak is corresponding to a phenol zone relative to single t-DNA sequence. The peaks on the elution curve are counted for quantification. The number of the peaks is proportional to the concentration of t-DNA in a range of 5.0 x 10(-16) to 1.0 x 10(-13) mol/L.

Floer homology in disk bundles and symplectically twisted geodesic flows

We show that if $K: P\to\mathbb{R}$ is an autonomous Hamiltonian on a symplectic manifold $(P,\Omega)$ which attains a Morse-Bott nondegenerate minimum of 0 along a symplectic submanifold $M$ and if $c_1(TP)$↾M vanishes in real cohomology, then the Hamiltonian flow of $K$ has contractible periodic orbits with bounded period on all sufficiently small energy levels. As a special case, if the geodesic flow on T*M is twisted by a symplectic magnetic field form, then the resulting flow has contractible periodic orbits on all low energy levels. These results were proven by Ginzburg and Gürel when $\Omega$↾M is spherically rational, and our proof builds on their work; the argument involves constructing and carefully analyzing at the chain level a version of filtered Floer homology in the symplectic normal disk bundle to $M$.

On the Displacement Rigidity of Levi Flat Hypersurfaces—The Case of Boundaries of Disc Bundles Over Compact Riemann Surfaces

Non-existence theorems for Levi flat hypersurfaces have found great interest in the literature. The question next to this that has to be asked is, when existing Levi flat hypersurfaces are at least rigid under deformations. Here, the case of boundaries of disc bundles over certain compact Riemann surfaces is considered.

Equivariant Morse relations

For a finite group G, Costenoble and Waner defined a cellular (co-)homology theory for G-spaces X, which is graded on virtual representations of the equivariant fundamental groupoid …G(X). Using this homology, we associate an infinite (Morse) series with an equivariant Morse function f defined on a closed Riemannian G-manifold M. Wasserman has shown that when the critical locus of f is a disjoint union of orbits, M has a canonical decomposition into disc bundles. We show that if this decomposition ‘corresponds’ to a virtual representation ∞ of …G(M), then the Morse relations are satisfied by the ‘∞th homology groups’. For semi-free G-actions, we characterise the Morse functions which naturally give rise to such representations ∞ of …G(M). We also show that corresponding to any equivariant Morse function on a Z2-manifold, it is always possible to define virtual representations ∞ so that the Morse relation is satisfied by the ‘∞th homology groups’. In particular, the Morse relation is satisfied by Bredon homology.

Relative Mapping Class Group of the Trivial and the Tangent Disk Bundles over the Sphere

Relative Mapping Class Group of the Trivial and the Tangent Disk Bundles over the Sphere

Symplectic rigidity for Anosov hypersurfaces

We show, using standard results in length spectrum rigidity and symplectic homology, that if the unit tangent bundles of two compact surfaces of negative curvature are exact symplectomorphic, then the underlying surfaces are isometric, and hence the disk bundles are symplectomorphic smoothly up to the boundaries. Motivated by the work of Eliashberg and Hofer on unseen symplectic boundaries, we try to extend the above result to other classes of domains in (co)tangent bundles of surfaces. The domains we can treat consist of those bounded by deformations of negatively curved unit cotangent bundles through boundaries with Anosov characteristic (Reeb) flows. We show that if two such are exact symplectomorphic, then the two domains are symplectomorphic by a diffeomorphism smooth up to the boundary. Furthermore, if two such surfaces have the same marked length spectrum then they are connected by a 1-parameter family of domains bounded by hypersurfaces with smoothly time preserving conjugate Reeb flows by a smooth family of Hamiltonian diffeomorphisms. A few counterexamples to easy generalizations are given.

Quasiconformally instable disc bundles with complex structures

We discuss deformations and the quasiconformal instability of the Kahler geometry of disc bundles that are locally modeled on symmetric rank-one manifolds. The Kahler geometry of these manifolds is associated with natural complex or hypercomplex structures of pinched negative sectional curvature and infinite volume. Their fundamental groups are isomorphic to discrete subgroups of PU(n, 1), PSp(n, 1), or F 4 −20 .

Photodynamic therapy in symptomatic parafoveal telangiectasia secondary to Osler−Rendu−Weber disease

Editor, O sler Rendu Weber disease is an autosomal-dominant hereditary haemorrhagic telangiectasia (HHT) leading to arteriovenous malformations in the skin, mucosa and viscera (Osler 1901). While mucosal complications may cause epistaxis and gastrointestinal bleeding, additional visceral involvement frequently concerns the lung, liver or brain. The diagnosis of HHT is based on three of four criteria: epistaxis, telangiectasia, visceral lesions and appropriate family history. Frequent ocular abnormalities include conjunctival telangiectasia causing bloody tears. Retinal malformation include visceral angiodysplasia such as arteriovenous fistula, angiectasia, phlebectasia and angioma (Vase & Vase 1979; Brant et al. 1989; Mahmoud et al. 2002). We performed photodynamic therapy (PDT) to treat choroidal neovascularization (CNV) that occurred secondary to parafoveal telangiectasia in a patient with HHT. A 76-year-old woman complained of decreased visual acuity (VA) of 20/60 with metamorphopsia in the left eye. Hereditary haemorrhagic telangiectasia had been diagnosed 10 years earlier due to telangiectasia located at her tongue, distal fingers and conjunctiva (Fig. 1A, B). Although epistaxis occurred frequently, no surgery was required. On fundus examination in both eyes, the retinal vessels appeared well perfused (Fig. 1C, D). However, the macula in the left eye showed a significant oedema with subretinal lipid exudates superiorly and temporally (Fig. 1D). Fluorescein angiography disclosed parafoveal telangiectasia in both eyes (Fig. 1E, F), whereas an additional occult CNV caused a severe leakage temporal to the macula in the left eye (Fig. 1F). Optical coherence tomography in the left eye showed a neuroretinal elevation of 480 mm. Our patient had no additional retinal disease including venous occlusion, hypertensive or diabetic retinopathy. After the patient had signed informed consent, one PDT session was performed. One year after PDT the metamorphopsia disappeared and the subject’s VA stabilized at 20/125. Some subtle exudates and retinal pigment epithelium (RPE) defects maintained temporal to the macula (Fig. 2A). Fluorescein angiography showed a mild atrophy of the choriocapillaris in the area of the PDT application; however, there was no persisting leakage or oedema (Fig. 2B). Optical coherence tomography disclosed no subretinal fluid and a retinal thickness of 240 mm. The tumour growth factor-beta (TGF-b) binding protein endoglin was recently identified as causing HHT with pulmonary arteriovenous malformations. While a corresponding candidate gene for endoglin is localized at 9q33–34 (McAllister et al. 1994), additional genetic studies have demonstrated the pivotal role of TGF-b signalling during angiogenesis. Knockout mice for endoglin, endothelial-specific deletion of the TGF-b-type 2 receptor (TbR2) or activin receptor-like-kinase-5 (ALK-5) failed to differentiate vascular smooth muscle cells as well as endothelial cells, inducing fragile and dilated blood vessels (Carvalho et al. 2004). The prevalence of ocular abnormalities in HHT is approximately 35% for extraocular lesions (conjunctival telangiectasia) and only 10% for retinal vascular malformations (Brant et al. 1989). Although this retrospective analysis observed intraocular findings, the authors questioned the coherence of HHT in selected cases. A great variety of retinal malformations may occur, including telangiectatic vessels at the optic disc or papillomacular bundle, and beading of retinal vessels. Retinal parafoveal telangiectasia, as presented in our HHT patient, has not yet been described in detail. Our HHT patient clearly presented typical findings of parafoveal telangiectasia in both eyes. Whether the idiopathic parafoveal telangiectasia was part of the underlying disease or just coincidental remains speculative as overall retinal involvement in HHT is very rare. However, the presence of telangiectasia in other mucosa and viscera structures supports this assumption. The unilateral symptomatic alteration was associated with severe leakage and oedema corresponding from an adjacent occult CNV in the left eye. Photodynamic therapy reduced the leakage and exudation significantly, maintaining stable VA for our patient. Novel treatment modalities such as intravitreal triamcinolone or antivascular endothelial-growth-factors (anti-VEGF) are currently under investigation in proliferative disease. Further research is necessary to determine the influence of endoglin in the pathogenesis of parafoveal telangiectasia in order to evaluate a possible treatment approach by anti-TGF-b in vascular disease such as HHT.

Symplectomorphism groups and isotropic skeletons

The symplectomorphism group of a 2-dimensional surface is homotopy equivalent to the orbit of a filling system of curves. We give a generalization of this statement to dimension 4. The filling system of curves is replaced by a decomposition of the symplectic 4-manifold (M, omega) into a disjoint union of an isotropic 2-complex L and a disc bundle over a symplectic surface Sigma which is Poincare dual to a multiple of the form omega. We show that then one can recover the homotopy type of the symplectomorphism group of M from the orbit of the pair (L, Sigma). This allows us to compute the homotopy type of certain spaces of Lagrangian submanifolds, for example the space of Lagrangian RP^2 in CP^2 isotopic to the standard one.

REAL SURFACES IN ELLIPTIC SURFACES

We study the structure of complex points on real surfaces, embedded into complex elliptic surfaces. We show, for example, that any compact connected surface has a totally real embedding into a blow-up of a K3 surface. We also exhibit smooth disc bundles over compact connected orientable surfaces that have a Stein structure as Stein domains inside elliptic surfaces.