Celebratio Mathematica

F. Lalonde and D. McDuff: “The geometry of symplectic energy,” Ann. Math. (2) 141 : 2 (March 1995), pp. 349–371. MR 1324138 Zbl 0829.53025 article

F. Lalonde and D. McDuff: “Local non-squeezing theorems and stability,” Geom. Funct. Anal. 5 : 2 (March 1995), pp. 364–386. Dedicated to Misha Gromov on the occasion of his 50th birthday. MR 1334871 Zbl 0837.58014 article

Mikhael Leonidovich Gromov	Related
François Lalonde	Related

F. Lalonde and D. McDuff: “Erratum for ‘Hofer’s \( L^{\infty} \)-geometry: Energy and stability of Hamiltonian flows, II’,” Invent. Math. 123 : 3 (1996), pp. 613. Erratum for an article published in Invent. Math. 122:1 (1996). article

F. Lalonde and D. McDuff: “Erratum for ‘Hofer’s \( L^{\infty} \)-geometry: Energy and stability of Hamiltonian flows, I’,” Invent. Math. 123 : 3 (1996), pp. 613. Erratum for an article published in Invent. Math. 122:1 (1996). MR 1383964 article

F. Lalonde and D. McDuff: “The classification of ruled symplectic 4-manifolds,” Math. Res. Lett. 3 : 6 (1996), pp. 769–778. MR 1426534 Zbl 0874.57019 article

F. Lalonde and D. McDuff: “\( J \)-curves and the classification of rational and ruled symplectic 4-manifolds,” pp. 3–42 in Contact and symplectic geometry (Cambridge, UK, July–December 1994). Edited by C. B. Thomas. Publications of the Newton Institute 8. Cambridge University Press, 1996. MR 1432456 Zbl 0867.53028 incollection

Charles B. Thomas	Related
François Lalonde	Related

F. Lalonde and D. McDuff: “Hofer’s \( L^{\infty} \)-geometry: Energy and stability of Hamiltonian flows, I,” Invent. Math. 122 : 1 (1996), pp. 1–33. Errata were published in Invent. Math. 123:3 (1996). MR 1354953 Zbl 0844.58020 article

Consider the group \( \mathrm{Ham}^c(M) \) of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold \( (M,\omega) \) with the Hofer \( L^{\infty} \)-norm. A path in \( \mathrm{Ham}^c(M) \) will be called a geodesic if all sufficiently short pieces of it are local minima for the Hofer length functional \( \mathscr{L} \). In this paper, we give a necessary condition for a path \( \gamma \) to be a geodesic. We also develop a necessary condition for a geodesic to be stable, that is, a local minimum for \( \mathscr{L} \). This condition is related to the existence of periodic orbits for the linearization of the path, and so extends Ustilovsky’s work on the second variation formula. Using it, we construct a symplectomorphism of \( S^2 \) which cannot be reached from the identity by a shortest path. In later papers in this series, we will use holomorphic methods to prove the sufficiency of the condition given here for the characterisation of geodesics as well as the sufficiency of the condition for the stability of geodesics. We will also investigate conditions under which geodesics are absolutely length-minimizing.

F. Lalonde and D. McDuff: “Hofer’s \( L^{\infty} \)-geometry: Energy and stability of Hamiltonian flows, II,” Invent. Math. 122 : 1 (1996), pp. 35–69. An erratum was published in Invent. Math. 123:3 (1996). Zbl 0844.58021 article

In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group \( \mathrm{Ham}^c(M) \) of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction on \( M \). We then discuss conditions which guarantee that such a path minimizes the Hofer length. Our argument relies on a general geometric construction (the gluing of monodromies) and on an extension of Gromov’s non-squeezing theorem both to more general manifolds and to more general capacities. The manifolds we consider are quasi-cylinders, that is spaces homeomorphic to \( M\times D^2 \) which are symplectically ruled over \( D^2 \). When we work with the usual capacity (derived from embedded balls), we can prove the existence of paths which minimize the length among all homotopic paths, provided that \( M \) is semi-monotone. (This restriction occurs because of the well-known difficulty with the theory of \( J \)-holomorphic curves in arbitrary \( M \).) However, we can only prove the existence of length-minimizing paths (i.e. paths which minimize length amongst all paths, not only the homotopic ones) under even more restrictive conditions on \( M \), for example when \( M \) is exact and convex or of dimension 2. The new difficulty is caused by the possibility that there are non-trivial and very short loops in \( \mathrm{Ham}^c(M) \). When such length minimizing paths do exist, we can extend the Bialy–Polterovich calculation of the Hofer norm on a neighbourhood of the identity (\( C^1 \)-flatness).

Although it applies to a more restricted class of manifolds, the Hofer–Zehnder capacity seems to be better adapted to the problem at hand, giving sharper estimates in many situations. Also the capacity-area inequality for split cylinders extends more easily to quasi-cylinders in this case. As applications, we generalise Hofer’s estimate of the time for which an autonomous flow is length-minimizing to some manifolds other than \( \mathbf{R}^{2n} \), and derive new results such as the unboundedness of Hofer’s metric on some closed manifolds, and a linear rigidity result.

F. Lalonde and D. McDuff: “Positive paths in the linear symplectic group,” pp. 361–387 in The Arnold–Gelfand mathematical seminars. Edited by V. I. Arnold, I. M. Gelfand, M. Smirnov, and V. S. Retakh. Birkhäuser (Boston), 1997. MR 1429901 Zbl 0868.58031 incollection

A positive path in the linear symplectic group \( \mathrm{Sp}(2n) \) is a smooth path which is everywhere tangent to the positive cone. These paths are generated by negative definite (time-dependent) quadratic Hamiltonian functions on Euclidean space. A special case are autonomous positive paths, which are generated by time-independent Hamiltonians, and which all lie in the set \( \mathcal{U} \) of diagonalizable matrices with eigenvalues on the unit circle. However, as was shown by Krein, the eigenvalues of a general positive path can move off the unit circle. In this paper, we extend Krein’s theory: we investigate the general behavior of positive paths which do not encounter the eigenvalue 1, showing, for example, that any such path can be extended to have endpoint with all eigenvalues on the circle. We also show that in the case \( 2n = 4 \) there is a close relation between the index of a positive path and the regions of the symplectic group that such a path can cross. Our motivation for studying these paths came from a geometric squeezing problem [Lalonde and McDuff, 1995] in symplectic topology. However, they are also of interest in relation to the stability of periodic Hamiltonian systems [Gelfand and Lidskii 1958] and in the theory of geodesics in Riemannian geometry [Bott 1956].

Yuriĭ Mikhaĭlovich Smirnov	Related
Vladimir Solomonovich Retakh	Related
Vladimir Igorevich Arnold	Related
François Lalonde	Related
Israïl Moiseevich Gelfand	Related

D. McDuff: “Lectures on Gromov invariants for symplectic 4-manifolds,” pp. 175–210 in Gauge theory and symplectic geometry (Montreal, 3–14 July 1995). Edited by J. Hurtubise and F. Lalonde. NATO ASI Series C: Mathematical and Physical Sciences 488. Kluwer (Dordrecht), 1997. Based on notes taken by Wladyslav Lorek. MR 1461573 Zbl 0881.57037 incollection

Taubes’s recent spectacular work setting up a correspondence between \( J \)-holomorphic curves in symplectic 4-manifolds and solutions of the Seiberg–Witten equations counts \( J \)-holomorphic curves in a somewhat new way. The “standard” theory concerns itself with moduli spaces of connected curves, and gives rise to Gromov–Witten invariants: see, for example, [McDuff and Salamon 1994; Ruan and Tian 1995, 1996]. However, Taubes’s curves arise as zero sets of sections and so need not be connected. These notes are in the main expository. We first discuss the invariants as Taubes defined them, and then discuss some alternatives, showing, for example, a way of dealing with multiply-covered exceptional spheres. We also calculate some examples, in particular finding the Gromov invariant of the fiber class of an elliptic surface by counting \( J \)-holomorphic curves, rather than going via Seiberg–Witten theory.

Wladyslav Lorek	Related
François Lalonde	Related
Jacques Hurtubise	Related

F. Lalonde, D. McDuff, and L. Polterovich: “On the flux conjectures,” pp. 69–85 in Geometry, topology, and dynamics (Montreal, 26–30 June 1995). Edited by F. Lalonde. CRM Proceedings Lecture Notes 15. American Mathematical Society (Providence, RI), 1998. MR 1619124 Zbl 0974.53062 ArXiv dg-ga/9706015 incollection

François Lalonde	Related
Leonid V. Polterovich	Related

F. Lalonde, D. McDuff, and L. Polterovich: “Topological rigidity of Hamiltonian loops and quantum homology,” Invent. Math. 135 : 2 (1999), pp. 369–385. MR 1666763 Zbl 0907.58004 article

This paper studies the question of when a loop \( \phi = \{\phi_t\}_{0\leq t\leq 1} \) in the group \( \mathrm{Symp}(M,\omega) \) of symplectomorphisms of a symplectic manifold \( (M,\omega) \) is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if \( \phi \) is Hamiltonian with respect to \( \omega \), and if \( \phi^{\prime} \) is a small perturbation of \( \phi \) that preserves another symplectic form \( \omega^{\prime} \), then \( \phi^{\prime} \) is Hamiltonian with respect to \( \omega^{\prime} \). This allows us to get some new information on the structure of the flux group, i.e. the image of \( \pi_1(\mathrm{Symp}(M,\omega)) \) under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general. The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of \( M \).

François Lalonde	Related
Leonid V. Polterovich	Related

F. Lalonde and D. McDuff: “Cohomological properties of ruled symplectic structures,” pp. 79–99 in Mirror symmetry IV: Proceedings of the conference on strings, duality, and geometry (Montreal, March 2000). Edited by E. D’Hoker, S.-T. Yau, and D. H. Phong. AMS/IP Studies in Advanced Mathematics 33. American Mathematical Society (Providence, RI), 2002. MR 1968218 Zbl 1090.53071 ArXiv math/0010277 incollection

François Lalonde	Related
Duong H. Phong	Related
Eric D'Hoker	Related
Shing-Tung Yau	Related

F. Lalonde and D. McDuff: “Symplectic structures on fiber bundles,” Topology 42 : 2 (March 2003), pp. 309–347. Errata were published in Topology 44:6 (2005). MR 1941438 Zbl 1032.53077 article

Let \( \pi:P\to B \) be a locally trivial fiber bundle over a connected \( \mathrm{CW} \) complex \( B \) with fiber equal to the closed symplectic manifold \( (M,\omega) \). Then \( \pi \) is said to be a symplectic fiber bundle if its structural group is the group of symplectomorphisms \( \mathrm{Symp}(M,\omega) \), and is called Hamiltonian if this group may be reduced to the group \( \mathrm{Ham}(M,\omega) \) of Hamiltonian symplectomorphisms. In this paper, building on prior work by Seidel and Lalonde, McDuff and Polterovich, we show that these bundles have interesting cohomological properties. In particular, for many bases \( B \) (for example when \( B \) is a sphere, a coadjoint orbit or a product of complex projective spaces) the rational cohomology of \( P \) is the tensor product of the cohomology of \( B \) with that of \( M \). As a consequence the natural action of the rational homology \( H_k(\mathrm{Ham}(M)) \) on \( H_*(M) \) is trivial for all \( M \) and all \( k > 0 \).

F. Lalonde and D. McDuff: “Errata to ‘Symplectic structures on fiber bundles’,” Topology 44 : 6 (2005), pp. 1301–1303. Errata for article published in Topology 42:2 (2003). MR 2168577 article

year	title	people

			clear

Dusa McDuff

Complete Bibliography

Works connected to François Lalonde

Filter the Bibliography List