Disk Bundle Research Articles

Three-Dimensional Quantitative Analysis of Internal Limiting Membrane Peeling Related Structural Changes in Retinal Detachment Repair

PurposeTo assess macular microstructural changes associated with internal limiting membrane peeling (ILMP) using three-dimensional optical coherence tomography (3D-OCT) in primary macula-off rhegmatogenous retinal detachment (RRD) repairs with vitrectomy and silicone oil (SO) tamponade. DesignRetrospective, consecutive, interventional case series. MethodsSetting: Institutional practice. Patient Population: Patients who received primary RRD repair by a single experienced surgeon between January 2017 and December 2021. Main Outcome Measures: In the qualitative comparative analysis, the presence of macular changes among patients who underwent primary RRD repair with (21 eyes) or without ILMP (20 eyes) were observed. Subsequently, a detailed quantitative analysis of ILMP-related microstructural changes in 56 eyes using both 3D and 2D-OCT images were performed. ResultsIn the qualitative comparative analysis, macular microstructural changes were observed in 95% of ILMP eyes and 5% of non-ILMP eyes (p < 0.001). In the quantitative analysis, four major macular microstructural changes were detected: dimple (75%), dissociated nerve fiber layer (DONFL) (55%), ILM peeling edge thinning (IPET) (64%), and temporal macular groove (TMG) (23%). Dimples (n = 251, average 4.5 ± 5.8 per eye) could be further classified into type I (confined to the inner plexiform layer [IPL]; 73%) and type II (beyond IPL, 27%). The average depth of the deepest dimples was 58 ± 18 μm. The extent of IPET was 6.0 ± 3.7 clock hours. The average length of TMG was 1.8 ± 0.4 mm. Comparing to unoperated fellow eyes, the eyes after ILMP showed decreased inner temporal over nasal retinal thickness ratio (0.86 ± 0.07 versus 0.96 ± 0.03, p < 0.001), shorter disc-fovea distance (4.61 ± 0.32 µm versus 4.78 ± 0.37 µm, p = 0.041), and wider retinal vein trajectories (c’ = 2.48 ± 0.84 versus 3.39 ± 1.61, p = 0.002). ConclusionsMacular microstructural changes are common after ILMP in RRD repair, encompassing both focal changes (dimples, DONFL) and zonal changes (IPET, TMG). DONFL and dimples may be part of a continuum of findings stemming from the same mechanism. IPET and TMG are the results of macular tissue shift due to contracture of the optic disc and neurovascular bundle.

Hofer–Zehnder capacity of disc tangent bundles of projective spaces

AbstractWe compute the Hofer–Zehnder capacity of disc tangent bundles of the complex and real projective spaces of any dimension. The disc bundle is taken with respect to the Fubini–Study resp. round metric, but we can obtain explicit bounds for any other metric. In the case of the complex projective space, we also compute the Hofer–Zehnder capacity for the magnetically twisted case, where the twist is proportional to the Fubini–Study form. For arbitrary twists, we can still give explicit upper bounds.

Kähler-Einstein metrics and obstruction flatness of circle bundles

Obstruction flatness of a strongly pseudoconvex hypersurface Σ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of Σ, complete up to Σ, has a potential −log⁡u such that u is C∞-smooth up to Σ. In general, u has only a finite degree of smoothness up to Σ. In this paper, we study obstruction flatness of hypersurfaces Σ that arise as unit circle bundles S(L) of negative Hermitian line bundles (L,h) over Kähler manifolds (M,g). We prove that if (M,g) has constant Ricci eigenvalues, then S(L) is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and (M,g) is complete, then we show that the corresponding disk bundle admits a complete Kähler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of S(L) when (M,g) is a Kähler surface (dim⁡M=2) with constant scalar curvature.

Symplectic cohomology and a conjecture of Viterbo

We identify a new class of closed smooth manifolds for which there exists a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in a unit cotangent disk bundle. This settles a well-known conjecture of Viterbo from 2007 as the special case of \(T^n,\) which has been completely open for \(n>1\). Our methods are different and more intrinsic than those of the previous work of the author first settling the case \(n=1\). The new class of manifolds is defined in topological terms involving the Chas–Sullivan algebra and the BV-operator on the homology of the free loop space. It contains spheres and is closed under products. We discuss generalizations and various applications, to \(C^0\) symplectic topology in particular.

Viterbo conjecture for Zoll symmetric spaces

We prove a conjecture of Viterbo from 2007 on the existence of a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in unit cotangent disk bundles, for bases given by compact rank one symmetric spaces \(S^n, {\mathbb {R}}P^n, {\mathbb {C}}P^n, {\mathbb {H}}P^n,\) \(n\ge 1.\) We discuss generalizations and give applications, in particular to \(C^0\) symplectic topology. Our key method consists in a quantitative deformation argument for Floer persistence modules that allows to excise a divisor.

Weakly 1‐completeness of holomorphic fiber bundles over compact Kähler manifolds

Diederich and Ohsawa (Publ. Res. Inst. Math. Sci. 21 (1985), no. 4, 819–833) proved that every disc bundle over a compact Kähler manifold is weakly 1-complete. In this paper, under certain conditions, we generalize this result to the case of fiber bundles over compact Kähler manifolds whose fibers are bounded symmetric domains. In particular, if the representation related to the fiber bundle is reductive, then it has a plurisubharmonic exhaustion function. If the bundle is obtained by the diagonal action on the product of bounded symmetric domains, we are able to show that it is hyperconvex.

Functional Amblyopia in Segmental Optic Nerve Hypoplasia

A 5-year-old boy had a visual acuity of 20/80 in the right eye (OD) and 20/20 in the left eye (OS). Examination showed anisometropia (+1.00 OD and plano OS) without strabismus. Retinal examination demonstrated segmental hypoplasia of the temporal optic disc with absence of the corresponding retinal nerve fiber layer (Fig A-B). Despite a structural defect involving the temporal disc and papillomacular bundle, combined treatment with glasses providing his full cycloplegic refraction and part-time occlusion OS resulted in visual improvement to 20/20 OD.

Skein lasagna modules for 2-handlebodies

Abstract Morrison, Walker, and Wedrich used the blob complex to construct a generalization of Khovanov–Rozansky homology to links in the boundary of a 4-manifold. The degree zero part of their theory, called the skein lasagna module, admits an elementary definition in terms of certain diagrams in the 4-manifold. We give a description of the skein lasagna module for 4-manifolds without 1- and 3-handles, and present some explicit calculations for disk bundles over S 2 {S^{2}} .

Symplectic fillings and cobordisms of lens spaces

We complete the classification of symplectic fillings of tight contact structures on lens spaces. In particular, we show that any symplectic filling X X of a virtually overtwisted contact structure on L ( p , q ) L(p,q) has another symplectic structure that fills the universally tight contact structure on L ( p , q ) L(p,q) . Moreover, we show that the Stein filling of L ( p , q ) L(p,q) with maximal second homology is given by the plumbing of disk bundles. We also consider the question of constructing symplectic cobordisms between lens spaces and report some partial results.

The quantitative nature of reduced Floer theory

We study the reduced symplectic cohomology of disk subbundles in negative symplectic line bundles. We show that this cohomology theory “sees” the spectrum of a quantum action on quantum cohomology. Precisely, quantum cohomology decomposes into generalized eigenspaces of the action of the first Chern class by quantum cup product. The reduced symplectic cohomology of a disk bundle of radius R sees all eigenspaces whose eigenvalues have size less than R, up to rescaling by a fixed constant. Similarly, we show that the reduced symplectic cohomology of an annulus subbundle between radii R1 and R2 captures all eigenspaces whose eigenvalues have size between R1 and R2, up to a rescaling. We show how local closed-string mirror symmetry statements follow from these computations

A couple of real hyperbolic disc bundles over surfaces

Applying the techniques developed in [1], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov–Lawson–Thurston conjecture [6], such bundles M \to S should satisfy the inequality |eM/\chi S|\leq 1 , where eM stands for the Euler number of the bundle and \chi S , for the Euler characteristic of the surface. In this paper, we construct new examples that provide a maximal value of |eM/\chi S|=\frac {3}{5} among all known examples. The former maximum, belonging to Feng Luo [10], was |eM/\chi S|=\frac {1}{2} .

Metrisability of projective surfaces and pseudo-holomorphic curves

We show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo-holomorphic curves. A projective structure {mathfrak {p}} and a volume form sigma on an oriented surface M equip the total space of a certain disk bundle Zrightarrow M with a pair (J_{{mathfrak {p}}},{mathfrak {J}}_{{mathfrak {p}},sigma }) of almost complex structures. A conformal structure on M corresponds to a section of Zrightarrow M and {mathfrak {p}} is metrisable by the metric g if and only if [g] : M rightarrow Z is a pseudo-holomorphic curve with respect to J_{{mathfrak {p}}} and {mathfrak {J}}_{{mathfrak {p}},dA_g}.

Convex plumbings in closed hyperbolic 4-manifolds

We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold.

Invariance of symplectic cohomology and twisted cotangent bundles over surfaces

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be nonexact and noncompactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of the BV-operator on symplectic cohomology.

Self-duality and associated parallel or cocalibrated G_2 structures

We find a remarkable family of $\mathrm{G}_2$ structures defined on certain principal $\mathrm{SO}(3)$-bundles $P_\pm\longrightarrow M$ associated with any given oriented Riemannian 4-manifold $M$. Such structures are always cocalibrated. The study starts with a recast of the Singer-Thorpe equations of 4-dimensional geometry. These are applied to the Bryant-Salamon cons\-truction of complete $\mathrm{G}_2$-holonomy metrics on the vector bundle of self- or anti-self-dual 2-forms on $M$. We then discover new examples of that special holonomy on disk bundles over ${\cal H}^4$ and ${\cal H}^2_{\mathbb{C}}$, respectively, the real and complex hyperbolic space. Only in the end we present the new $\mathrm{G}_2$ structures on principal bundles.

Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

In this paper, we establish the existence of periodic orbits belonging to any [Formula: see text]-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact [Formula: see text]-form [Formula: see text] admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any non-trivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran–Polterovich–Salamon capacity.

Finite TYCZ expansions and cscK metrics

Let (M,g) be a Kähler manifold whose associated Kähler form ω is integral and let (L,h)→(M,ω) be a quantization hermitian line bundle. In this paper we study those Kähler manifolds (M,g) admitting a finite TYCZ expansion, namely those for which the associated Kempf distortion function Tmg is of the form:Tmg(p)=fs(p)ms+fs−1(p)ms−1+⋯+fr(p)mr,fj∈C∞(M),s,r∈Z. We show that if the TYCZ expansion is finite then Tmg is indeed a polynomial in m of degree n, n=dimC⁡M, and the log-term of the Szegö kernel of the disc bundle D⊂L⁎ vanishes (where L⁎ is the dual bundle of L). Moreover, we provide a complete classification of the Kähler manifolds admitting finite TYCZ expansion either when M is a complex curve or when M is a complex surface with a cscK metric which admits a radial Kähler potential.

Plumbing constructions and the domain of outer communication for 5-dimensional stationary black holes

The topology of the domain of outer communication for 5-dimensional stationary bi-axisymmetric black holes is classified in terms of disc bundles over the 2-sphere and plumbing constructions. In particular we find an algorithmic bijective correspondence between the plumbing of disc bundles and the rod structure formalism for such spacetimes. Furthermore, we describe a canonical fill-in for the black hole region and cap for the asymptotic region. The resulting compactified domain of outer communication is then shown to be homeomorphic to S 4 S^4 , a connected sum of S 2 × S 2 S^2\times S^2 ’s, or a connected sum of complex projective planes C P 2 \mathbb {CP}^2 . Combined with recent existence results, it is shown that all such topological types are realized by vacuum solutions. In addition, our methods treat all possible types of asymptotic ends, including spacetimes which are asymptotically flat, asymptotically Kaluza-Klein, or asymptotically locally Euclidean.

Global linearization and fiber bundle structure of invariant manifolds

We study global properties of the global (center-)stable manifold of a normally attracting invariant manifold (NAIM), the special case of a normally hyperbolic invariant manifold (NHIM) with empty unstable bundle. We restrict our attention to continuous-time dynamical systems, or flows. We show that the global stable foliation of a NAIM has the structure of a topological disk bundle, and that similar statements hold for inflowing NAIMs and for general compact NHIMs. Furthermore, the global stable foliation has a Ck disk bundle structure if the local stable foliation is assumed Ck. We then show that the dynamics restricted to the stable manifold of a compact inflowing NAIM are globally topologically conjugate to the linearized transverse dynamics at the NAIM. Moreover, we give conditions ensuring the existence of a global Ck linearizing conjugacy. We also prove a Ck global linearization result for inflowing NAIMs; we believe that even the local version of this result is new, and may be useful in applications to slow-fast systems. We illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel normal form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form.

Chern–Gauss–Bonnet and Lefschetz duality from a currential point of view

We use the mapping cone for the relative deRham cohomology of a manifold with boundary in order to show that the Chern–Gauss–Bonnet Theorem for oriented Riemannian vector bundles over such manifolds is a manifestation of Lefschetz Duality in any of the two embodiments of the latter. We explain how Thom isomorphism fits into this picture, complementing thus the classical results about Thom forms with compact support. When the rank is odd, we construct, by using secondary transgression forms introduced here, a new closed pair of forms on the disk bundle associated to a vector bundle, pair which is Lefschetz dual to the zero section.