Cotangent Bundle Research Articles

Correction to: Stein and Weinstein structures on disk cotangent bundles of surfaces

After our article entitled “Stein and Weinstein structures on disk cotangent bundles of surfaces” was accepted and appeared online, it was pointed out by Sylvain Courte that our proof of the existence of the symplectomorphisms in Theorem 1.1 is flawed. Our proof of the existence of the contactomorphisms, however, holds true in both parts of Theorem 1.1.

On different geometric approaches to the dynamics of finite‐dimensional mechanical systems

AbstractThe language of differential geometry allows a coordinate‐free representation of physical quantities. This led to the development of several geometric theories for the description of finite‐dimensional mechanical systems. These approaches differ in the mathematical concepts they invoke and in the classes of mechanical systems they can describe. This short note aims to give an overview on the following three popular approaches, all of which are limited to time‐independent mechanical systems. While in the first approach, the motion of the mechanical system is considered as a curve in the system's configuration manifold, in the latter two, the corresponding motions are interpreted as curves in the tangent or the cotangent bundle of the configuration manifold.

Compact Exact Lagrangian Intersections in Cotangent Bundles via Sheaf Quantization

We show that the cardinality of the transverse intersection of two compact exact Lagrangian submanifolds in a cotangent bundle is bounded from below by the dimension of the Hom space of sheaf quantizations of the Lagrangians in Tamarkin's category. Our sheaf-theoretic method can also deal with clean and degenerate Lagrangian intersections.

Poisson-Lie analogues of spin Sutherland models

We present generalizations of the well-known trigonometric spin Sutherland models, which were derived by Hamiltonian reduction of ‘free motion’ on cotangent bundles of compact simple Lie groups based on the conjugation action. Our models result by reducing the corresponding Heisenberg doubles with the aid of a Poisson-Lie analogue of the conjugation action. We describe the reduced symplectic structure and show that the ‘reduced main Hamiltonians’ reproduce the spin Sutherland model by keeping only their leading terms. The solutions of the equations of motion emerge from geodesics on the compact Lie group via the standard projection method and possess many first integrals. Similar hyperbolic spin Ruijsenaars–Schneider type models were obtained previously by L.-C. Li using a different method, based on coboundary dynamical Poisson groupoids, but their relation with spin Sutherland models was not discussed.

A note on coherent orientations for exact Lagrangian cobordisms

Let L \subset \R \times J^1(M) be a spin, exact Lagrangian cobordism in the symplectization of the 1-jet space of a smooth manifold M . Assume that L has cylindrical Legendrian ends \Lambda_\pm \subset J^1(M) . It is well known that the Legendrian contact homology of \Lambda_\pm can be defined with integer coefficients, via a signed count of pseudo-holomorphic disks in the cotangent bundle of M . It is also known that this count can be lifted to a mod 2 count of pseudo-holomorphic disks in the symplectization \R \times J^1(M) , and that L induces a morphism between the \Z_2 -valued DGA:s of the ends \Lambda_\pm in a functorial way. We prove that this hold with integer coefficients as well. The proofs are built on the technique of orienting the moduli spaces of pseudo-holomorphic disks using capping operators at the Reeb chords. We give an expression for how the DGA:s change if we change the capping operators.

RC-POSITIVITY, VANISHING THEOREMS AND RIGIDITY OF HOLOMORPHIC MAPS

AbstractLet $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle ${\mathcal{O}}_{T_{M}^{\ast }}(1)$ is not pseudo-effective and ${\mathcal{O}}_{T_{N}^{\ast }}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

Einstein metrics, projective structures and the [formula omitted] Toda equation

We establish an explicit correspondence between two-dimensional projective structures admitting a projective vector field, and a class of solutions to the SU(∞) Toda equation. We give several examples of new, explicit solutions of the Toda equation, and construct their mini-twistor spaces. Finally we discuss the projective-to-Einstein correspondence, which gives a neutral signature Einstein metric on a cotangent bundle T∗N of any projective structure (N,[∇]). We show that there is a canonical Einstein of metric on an R∗-bundle over T∗N, with a connection whose curvature is the pull-back of the natural symplectic structure from T∗N.

Bergman kernel on Riemann surfaces and Kähler metric on symmetric products

Let [Formula: see text] be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer [Formula: see text], we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle [Formula: see text], where [Formula: see text] is the holomorphic cotangent bundle of [Formula: see text]. Our first main result estimates the corresponding Bergman metric on [Formula: see text] in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of [Formula: see text] into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of [Formula: see text]. The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of [Formula: see text]. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of [Formula: see text] and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on [Formula: see text].

Dispersionless integrable hierarchies andGL(2, ℝ) geometry

AbstractParaconformal orGL(2, ℝ) geometry on ann-dimensional manifoldMis defined by a field of rational normal curves of degreen– 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source ofGL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context,GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutiveGL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-freeGL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesicGL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesicα-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.Our main result states that involutiveGL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n– 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

In simply connected cotangent bundles, exact Lagrangian cobordisms are h-cobordisms

Abstract Let Q be a simply connected manifold. We show that every exact Lagrangian cobordism between compact, exact Lagrangians in T * Q is an h-cobordism. This is a corollary of the Abouzaid–Kragh Theorem.

A Construction of Special Lagrangian Submanifolds by Generalized Perpendicular Symmetries

We show a method to construct a special Lagrangian submanifold L' from a given special Lagrangian submanifold L in a Calabi-Yau manifold with the use of generalized perpendicular symmetries. We use moment maps of the actions of Lie groups, which are not necessarily abelian. By our method, we construct some non-trivial examples in the cotangent bundles of the spheres which are non-flat Calabi-Yau manifolds equipped with the Stenzel metrics.

The integrability of symplectic twist maps without conjugate points

It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$-dimensional torus that is without conjugate points is $C^{0}$-integrable, that is $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs.

Invariant Nijenhuis Tensors and Integrable Geodesic Flows

We study invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ of a reductive Lie group $G$ from the point of view of integrability of a Hamiltonian system of differential equations with the $G$-invariant Hamiltonian function on the cotangent bundle $T^*(G/K)$. Such a tensor induces an invariant Poisson tensor $\Pi_1$ on $T^*(G/K)$, which is Poisson compatible with the canonical Poisson tensor $\Pi_{T^*(G/K)}$. This Poisson pair can be reduced to the space of $G$-invariant functions on $T^*(G/K)$ and produces a family of Poisson commuting $G$-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. As an application we prove Liouville integrability in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces $G/K$ of compact Lie groups $G$ for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of $G$ to two subgroups $G=G_1\cdot G_2$, where $G/G_i$ are symmetric spaces, $K=G_1\cap G_2$.

Lusternik–Schnirelmann theory and closed Reeb orbits

We develop a variant of Lusternik–Schnirelmann theory for the shift operator in equivariant Floer and symplectic homology. Our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated. As an application, we prove new multiplicity results for simple closed Reeb orbits on the standard contact sphere, the unit cotangent bundle to the sphere and some other contact manifolds. We also show that the lower Conley–Zehnder index enjoys a certain recurrence property and revisit and reprove from a different perspective a variant of the common jump theorem of Long and Zhu. This is the second, combinatorial ingredient in the proof of the multiplicity results.

Lagrangian Pairs of Pants

AbstractWe construct a Lagrangian submanifold, inside the cotangent bundle of a real torus, which we call a Lagrangian pair of pants. It is given as the graph of an exact one form on the real blowup of a Lagrangian coamoeba. Lagrangian pairs of pants are the main building blocks in a construction of smooth Lagrangian submanifolds of $( {\mathbb{C}}^*)^n$ that lift tropical subvarieties in $\mathbb R^n$. As an example we explain how to lift tropical curves in $ {\mathbb{R}}^2$ to Lagrangian submanifolds of $( {\mathbb{C}}^*)^2$. We also give several new examples of Lagrangian submanifolds inside toric varieties, some of which are monotone.

Fillings and fittings of unit cotangent bundles of odd-dimensional spheres

AbstractWe introduce the concept of fittings to symplectic fillings of the unit cotangent bundle of odd-dimensional spheres. Assuming symplectic asphericity we show that all fittings are diffeomorphic to the respective unit co-disc bundle.

Quantization commutes with singular reduction: Cotangent bundles of compact Lie groups

We analyze the ‘quantization commutes with reduction’ problem (first studied in physics by Dirac, and known in the mathematical literature also as the Guillemin–Sternberg Conjecture) for the conjugate action of a compact connected Lie group [Formula: see text] on its own cotangent bundle [Formula: see text]. This example is interesting because the momentum map is not proper and the ensuing symplectic (or Marsden–Weinstein quotient) [Formula: see text] is typically singular.In the spirit of (modern) geometric quantization, our quantization of [Formula: see text] (with its standard Kähler structure) is defined as the kernel of the Dolbeault–Dirac operator (or, equivalently, the spin[Formula: see text]–Dirac operator) twisted by the pre-quantum line bundle. We show that this quantization of [Formula: see text] reproduces the Hilbert space found earlier by Hall (2002) using geometric quantization based on a holomorphic polarization. We then define the quantization of the singular quotient [Formula: see text] as the kernel of the twisted Dolbeault–Dirac operator on the principal stratum, and show that quantization commutes with reduction in the sense that either way one obtains the same Hilbert space [Formula: see text].

Stein and Weinstein structures on disk cotangent bundles of surfaces

Gompf (Ann Math (2) 148(2):619–693, 1998) describes a Stein domain structure on the disk cotangent bundle of any closed surface S by a Legendrian handlebody diagram. We prove that Gompf’s Stein domain is symplectomorphic to the disk cotangent bundle equipped with its canonical symplectic structure and the boundary of this domain is contactomorphic to the unit cotangent bundle of S equipped with its canonical contact structure. As a corollary, we obtain a surgery diagram for the canonical contact structure on the unit cotangent bundle of S.

Grassmann–Grassmann conormal varieties, integrability, and plane partitions

We give a conjectural formula for sheaves supported on (irreducible) conormal varieties inside the cotangent bundle of the Grassmannian, such that their equivariant K-class is given by the partition function of an integrable loop model, and furthermore their K-theoretic pushforward to a point is a solution of the level 1 quantum Knizhnik–Zamolodchikov equation. We prove these results in the case that the Lagrangian is smooth (hence is the conormal bundle to a subGrassmannian). To compute the pushforward to a point, or equivalently to the affinization, we simultaneously degenerate the Lagrangian and sheaf (over the affinization); the sheaf degenerates to a direct sum of cyclic modules over the geometric components, which are in bijection with plane partitions, giving a geometric interpretation to the Razumov–Stroganov correspondence satisfied by the loop model.

Special Lagrangian Submanifolds and Cohomogeneity One Actions on the Complex Projective Space

We construct examples of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle over the complex projective space, whose Calabi-Yau structure was given by Stenzel. For each example, we describe the condition of special Lagrangian as an ordinary differential equation. Our method is based on a moment map technique and the classification of cohomogeneity one actions on the complex projective space classified by Takagi.