Symplectic Geometry: A Festschrift in Honour of Claude Viterbo’s 60th Birthday

1157 Pages·2023·33.782 MB·English

by  Helmut Hofer, Alberto Abbondandolo, Urs Frauenfelder, Felix Schlenk

Helmut Hofer
Alberto Abbondandolo
Urs Frauenfelder
Felix Schlenk
Editors
Symplectic
Geometry
A Festschrift in Honour of Claude
Viterbo’s 60th Birthday

Helmut Hofer • Alberto Abbondandolo
Urs Frauenfelder • Felix Schlenk
Editors

Symplectic Geometry
A Festschrift in Honour of Claude Viterbo’s
60th Birthday
Previously published in Journal of Fixed Point Theory
and Applications Volume 24, issue 2, June 2022
Editors
Helmut Hofer Alberto Abbondandolo
Institute for Advanced Study Fakultät für Mathematik
Princeton, NJ, USA Ruhr-Universität Bochum

Bochum, Nordrhein-Westfalen, Germany
Urs Frauenfelder
Institut für Mathematik Felix Schlenk
Universität Augsburg Institut de Mathématiques
Augsburg, Bayern, Germany Université de Neuchâtel
Neuchâtel, Switzerland

ISBN 978-3-031-19110-7
Mathematics Subject Classification (2020): 53Dxx

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Contents
Volume 1
Dedication .......................................................................................................... 1
Helmut Hofer, Alberto Abbondandolo, Urs Frauenfelder,
and Felix Schlenk: Journal of Fixed Point Theory and Applications
2022, 2022:54 (11, July 2022) DOI: 10.1007/s11784-022-00969-2

Symplectically convex and symplectically star-shaped curves:
a variational problem ....................................................................................... 5
Peter Albers and Serge Tabachnikov: Journal of Fixed Point
Theory and Applications 2022, 2022:27 (06, April 2022)
DOI: 10.1007/s11784-022-00931-2
C0-robustness of topological entropy for geodesic flows ............................... 29
Marcelo R. R. Alves, Lucas Dahinden, Matthias Meiwes,
and Louis Merlin: Journal of Fixed Point Theory and Applications
2022, 2022:42 (29, April 2022) DOI: 10.1007/s11784-022-00959-4
B ifurcations of balanced configurations for the Newtonian n-body
problem in R4 .................................................................................................... 73
Luca Asselle, Marco Fenucci, and Alessandro Portaluri: Journal
of Fixed Point Theory and Applications 2022, 2022:22 (04, April 2022)
DOI: 10.1007/s11784-022-00932-1

Relative Hofer–Zehnder capacity and positive symplectic homology .......... 99
Gabriele Benedetti and Jungsoo Kang: Journal of Fixed Point
Theory and Applications 2022, 2022:44 (13, May 2022)
DOI: 10.1007/s11784-022-00963-8

An Arnold-type principle for non-smooth objects ......................................... 131
Lev Buhovsky, Vincent Humilière, and Sobhan Seyfaddini: Journal
of Fixed Point Theory and Applications 2022, 2022: 24 (04, April 2022)
DOI: 10.1007/s11784-022-00934-z

Quantitative h-principle in symplectic geometry ........................................... 153
Lev Buhovsky and Emmanuel Opshtein: Journal of Fixed
Point Theory and Applications 2022, 2022:38 (15, April 2022)
DOI:10.1007/s11784-022-00947-8
v
vi Contents
On symplectomorphisms and Hamiltonian Flows ....................................... 169
Franco Cardin: Journal of Fixed Point Theory and Applications 2022,
2022:33 (08, April 2022) DOI: 10.1007/s11784-022-00938-9

Lagrangian skeleta and plane curve singularities ........................................ 181
Roger Casals: Journal of Fixed Point Theory and Applications 2022,
2022:34 (08, April 2022) DOI: 10.1007/s11784-022-00939-8

Reeb chords of Lagrangian slices .................................................................. 225
Baptiste Chantraine: Journal of Fixed Point Theory and Applications
2022, 2022:26 (06, April 2022) DOI: 10.1007/s11784-022-00940-1

Basic facts and naive questions ...................................................................... 233
Marc Chaperon: Journal of Fixed Point Theory and Applications 2022,
2022:31 (08, April 2022) DOI: 10.1007/s11784-022-00941-0

Another look at the Hofer-Zehnder conjecture ........................................... 251
Erman Çineli, Viktor L. Ginzburg, and Başak Z. Gürel: Journal
of Fixed Point Theory and Applications 2022, 2022:53 (28, June 2022)
DOI: 10.1007/s11784-022-00937-w

Higher symplectic capacities and the stabilized embedding
problem for integral elllipsoids ...................................................................... 275
Daniel Cristofaro-Gardiner, Richard Hind, and Kyler Siegel: Journal
of Fixed Point Theory and Applications 2022, 2022:49 (10, June 2022)
DOI: 10.1007/s11784-022-00942-z

Closed geodesics on reversible Finsler 2-spheres ......................................... 301
Guido De Philippis, Michele Marini, Marco Mazzucchelli,
and Stefan Suhr: Journal of Fixed Point Theory and Applications 2022,
2022:19 (30, March 2022) DOI: 10.1007/s11784-022-00962-9

Families of Legendrians and Lagrangians with unbounded
spectral norm .................................................................................................. 365
Georgios Dimitroglou Rizell: Journal of Fixed Point Theory
and Applications 2022, 2022:43 (05, May 2022)
DOI: 10.1007/s11784-022-00964-7
Legendrian persistence modules and dynamics ........................................... 397
Michael Entov and Leonid Polterovich: Journal of Fixed Point
Theory and Applications 2022, 2022:30 (08, April 2022)
DOI: 10.1007/s11784-022-00944-x

A Lagrangian Klein bottle you can’t squeeze ............................................... 451
Jonathan David Evans: Journal of Fixed Point Theory and Applications
2022, 2022:47 (04, June 2022) DOI: 10.1007/s11784-022-00945-w

Construction of a linear K-system in Hamiltonian Floer theory ................ 461
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono: Journal
of Fixed Point Theory and Applications 2022, 2022:39 (18, April 2022)
DOI: 10.1007/s11784-022-00960-x
Contents vii
What does a vector field know about volume? ............................................. 571
Hansjörg Geiges: Journal of Fixed Point Theory and Applications 2022,
2022:23 (04, April 2022) DOI: 10.1007/s11784-022-00946-9
On the symplectic fillings of standard real projective spaces ..................... 597
Paolo Ghiggini and Klaus Niederkrüger-Eid: Journal of Fixed
Point Theory and Applications 2022, 2022:37 (12, April 2022)
DOI: 10.1007/s11784-022-00943-y

Volume 2
On curves with the Poritsky property ........................................................... 615
Alexey Glutsyuk: Journal of Fixed Point Theory and Applications 2022,
2022:35 (12, April 2022) DOI: 10.1007/s11784-022-00948-7

Examples around the strong Viterbo conjecture ......................................... 677
Jean Gutt, Michael Hutchings, and Vinicius G. B. Ramos: Journal
of Fixed Point Theory and Applications 2022, 2022:41 (20, April 2022)
DOI: 10.1007/s11784-022-00949-6

Global surfaces of section with positive genus for dynamically
convex Reeb flows ........................................................................................... 699
Umberto L. Hryniewicz, Pedro A. S. Salomão, and Richard Siefring:
Journal of Fixed Point Theory and Applications 2022, 2022:45
(24, May 2022) DOI: 10.1007/s11784-022-00950-z

ECH capacities and the Ruelle invariant ...................................................... 721
Michael Hutchings: Journal of Fixed Point Theory and Applications
2022, 2022:50 (11, June 2022) DOI: 10.1007/s11784-022-00968-3
Capacities of billiard tables and S1-equivariant loop
space homology ................................................................ ............................... 747
Kei Irie: Journal of Fixed Point Theory and Applications 2022, 2022:51
(13, June 2022) DOI: 10.1007/s11784-022-00952-x

Remarks on the systoles of symmetric convex hypersurfaces
and symplectic capacities ............................................................................... 775
Joontae Kim, Seongchan Kim, and Myeonggi Kwon: Journal of Fixed
Point Theory and Applications 2022, 2022:28 (07, April 2022)
DOI: 10.1007/s11784-022-00953-w

Diffeomorphism type via aperiodicity in Reeb dynamics ............................ 801
Myeonggi Kwon, Kevin Wiegand, and Kai Zehmisch: Journal
of Fixed Point Theory and Applications 2022, 2022:21 (01, April 2022)
DOI: 10.1007/s11784-022-00954-9

Conservative surface homeomorphisms with finitely many
periodic points ................................................................................................. 827
Patrice Le Calvez: Journal of Fixed Point Theory and Applications 2022,
2022:20 (01, April 2022) DOI: 10.1007/s11784-022-00936-x
viii Contents
The Anosov–Katok method and pseudo-rotations
in symplectic dynamics ................................................................................... 863
Frédéric Le Roux and Sobhan Seyfaddini: Journal of Fixed
Point Theory and Applications 2022, 2022:36 (12, April 2022)
DOI: 10.1007/s11784-022-00955-8
Contact geometry in the restricted three-body problem: a survey ............ 903
Agustin Moreno, Journal of Fixed Point Theory and Applications 2022,
2022:29 (07, April 2022) DOI: 10.1007/s11784-022-00956-7

A generalized Poincaré–Birkhoff theorem ................................................... 981
Agustin Moreno and Otto van Koert: Journal of Fixed
Point Theory and Applications 2022, 2022:32 (08, April 2022)
DOI: 10.1007/s11784-022-00957-6

Covariant constancy of quantum Steenrod operations .............................. 1025
Paul Seidel and Nicholas Wilkins: Journal of Fixed
Point Theory and Applications 2022, 2022:52 (15, June 2022)
DOI: 10.1007/s11784-022-00967-4

An algebraic approach to the algebraic Weinstein conjecture .................. 1063

Vivek Shende: Journal of Fixed Point Theory and Applications 2022,
2022:25 (05, April 2022) DOI: 10.1007/s11784-022-00958-5

Quantum cohomology as a deformation of symplectic cohomology.......... 1073
Matthew Strom Borman, Nick Sheridan, and Umut Varolgunes:
Journal of Fixed Point Theory and Applications 2022, 2022:48
(07, June 2022) DOI: 10.1007/s11784-022-00965-6

A symplectic embedding of the cube with minimal sections
and a question by Schlenk ............................................................................. 1151
Fabian Ziltener: Journal of Fixed Point Theory and Applications 2022,
2022:40 (19, April 2022) DOI: 10.1007/s11784-022-00961-w
The two articles listed below were intended for inclusion in this volume, but unfortunately
they were not ready at the time of printing. They may be accessed at https://link.springer.
com/collections/jfjdhejgee.

Kai Cieliebak, Nancy Hingston, and Alexandru Oancea, “Loop coproduct in Morse and Floer
homology”

Yakov Eliashberg and Thomas Kragh, “Non-trivial families of Legendrian submanifolds”
J.FixedPointTheoryAppl.(2022)24:54
https://doi.org/10.1007/s11784-022-00969-2
Publishedonline 11July2022
(cid:2)c The Author(s), under exclusive licence to JournalofFixedPointTheory
andApplications
Springer Nature Switzerland AG 2022, corrected
publication2022
Dedication
Helmut Hofer, Alberto Abbondandolo, Urs Frauenfelder and
Felix Schlenk
This special volume on the occasion of Claude Viterbo’s 60th birthday
is a tribute to his mathematical work.
Claude was born in Geneva on April 20, 1961, and later grew up in
Paris. After studies at the E´cole Normale Sup´erieure, he received his Ph.D.
working with Ivar Ekeland and Franc¸ois Laudenbach in 1985.
After postdoctoral positions at the Courant Institute in New York
(1986–1988) and the Mathematical Sciences Research Institute (MSRI) in
Berkeley (1988–1989), he returned to Paris where he held professorships at
the Universit´e de Paris-Sud, the E´cole Polytechnique and the E´cole Normale
Sup´erieure. He is now a professor at the Universit´e Paris-Saclay in Orsay.
Thisarticleispartofthetopicalcollection“Symplecticgeometry-AFestschriftinhonour
of Claude Viterbo’s 60th birthday” edited by Helmut Hofer, Alberto Abbondandolo, Urs
FrauenfelderandFelixSchlenk.
Reprinted from the journal 1
A.Abbondandoloetal. JFPTA
Besidesbeingabrilliantmathematician,Claudealwaysshowedastrong
senseofservicetothescientificcommunity.Henotonlyhasbeenamemberof
numerousscientificboardsandhiringcommitteesinFranceaswellasinterna-
tionally, but also he has been the chairman of the Centre de Math´ematiques
del’E´colePolytechnique (2000–2006), PresidentoftheMathematicsDepart-
ment of the E´cole Polytechnique (2009–2010) and chairman of the Mathe-
matics Department of E´cole Normale Sup´erieure (2013–2017).
Claude has broadly impacted the development of symplectic geome-
try/topology and Hamiltonian dynamics as a mathematician, as a mentor,
andasafriend.Healwaysconsideredmathematicalresearchasbothasocial
activity and a solitary one—a deep mathematical discussion with friends fol-
lowed by a quiet contemplation.
He played a significant role in growing the symplectic community in
Europebyhisworkwithstudentsandhisservice.Thisisreflectedinthewide
range of the contributions to this Festschrift, in which the authors express
their appreciation, gratitude and friendship.
Showing the same exquisite taste as his Ph.D. advisors, Claude worked
successfully on carefully chosen problems opening doors to important devel-
opments. In 1987, he proved the Weinstein conjecture for R2n, which was
followed by joint work with Andreas Floer and Helmut Hofer utilizing pseu-
doholomorphic curve methods for more general cases of Weinstein’s conjec-
ture. In work with Hofer, before the existence of Gromov–Witten invariants,
it was shown in 1992 that the Weinstein conjecture holds in compact sym-
plectic manifolds provided the moduli spaces of rational curves are suitably
structured.
Inanotherworkin1992,heusedgeneratingfunctionstoconstructspec-
tral invariants leading to an alternative construction of a bi-invariant metric
on the group of compactly generated Hamiltonian diffeomorphisms in R2n.
Thiscontributionledtoapowerfultoolinsymplecticgeometry,inparticular
in the reformulation in terms of Floer homology by Schwarz and Oh.
Another important contribution is the ‘Viterbo Transfer Map’ intro-
duced in 1999 in his paper on functors and computations in Floer homology.
A year later, Claude formulated what is now called the ‘Viterbo Conjec-
ture’, an intriguing relation between convex and symplectic geometry with
far-reaching consequences.
There is also some unpublished work which had significant impact.
For example, Claude’s work in real algebraic geometry described in V.
Kharlamov’s S´eminaire Bourbaki talk “Vari´et´es de Fano R´eelles [d’apr`es C.
Viterbo]”.ItisprovedthatthereallocusofastronglyFanomanifoldcannot
carryametricofnegativesectionalcurvature.Analternativeshorterproofby
Eliashberg, based on SFT neck stretching, extends this to the uniruled case
(negatively solving the higher dimensional Nash conjecture in the smooth
case).TheresultisnowcalledtheViterbo–EliashbergTheorem.Inanunpub-
lishedsequeltothe‘functorsandcomputations’paper,Claudeexplainedthat
the Floer homology of a cotangent bundle is equal to the homology of the
loop space of the underlying manifold. Many improvements were later given
2 Reprinted from the journal