Examples around the strong Viterbo conjecture

A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that certain specific normalized symplectic capacities agree on convex domains. We also review why all normalized symplectic capacities agree on S 1...

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Published in:Journal of fixed point theory and applications Vol. 24; no. 2
Main Authors: Gutt, Jean, Hutchings, Michael, Ramos, Vinicius G. B.
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.06.2022
Springer Verlag
Series:Symplectic geometry - A Festschrift in honour of Claude Viterbo’s 60th birthday
Subjects:
Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Symplectic geometry - A Festschrift in honour of Claude Viterbo’s 60th birthday
53D35
Viterbo’s conjecture
Symplectic capacities
53D42
toric domains
ISSN:1661-7738, 1661-7746
Online Access:Get full text
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Abstract A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that certain specific normalized symplectic capacities agree on convex domains. We also review why all normalized symplectic capacities agree on S 1 -invariant convex domains. We introduce a new class of examples called “monotone toric domains”, which are not necessarily convex, and which include all dynamically convex toric domains in four dimensions. We prove that for monotone toric domains in four dimensions, all normalized symplectic capacities agree. For monotone toric domains in arbitrary dimension, we prove that the Gromov width agrees with the first equivariant capacity. We also study a family of examples of non-monotone toric domains and determine when the conclusion of the strong Viterbo conjecture holds for these examples. Along the way, we compute the cylindrical capacity of a large class of “weakly convex toric domains” in four dimensions.
AbstractList A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that certain specific normalized symplectic capacities agree on convex domains. We also review why all normalized symplectic capacities agree on S 1-invariant convex domains. We introduce a new class of examples called "monotone toric domains", which are not necessarily convex, and which include all dynamically convex toric domains in four dimensions. We prove that for monotone toric domains in four dimensions, all normalized symplectic capacities agree. For monotone toric domains in arbitrary dimension, we prove that the Gromov width agrees with the first equivariant capacity. We also study a family of examples of non-monotone toric domains and determine when the conclusion of the strong Viterbo conjecture holds for these examples. Along the way we compute the cylindrical capacity of a large class of "weakly convex toric domains" in four dimensions.
A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that certain specific normalized symplectic capacities agree on convex domains. We also review why all normalized symplectic capacities agree on S 1 -invariant convex domains. We introduce a new class of examples called “monotone toric domains”, which are not necessarily convex, and which include all dynamically convex toric domains in four dimensions. We prove that for monotone toric domains in four dimensions, all normalized symplectic capacities agree. For monotone toric domains in arbitrary dimension, we prove that the Gromov width agrees with the first equivariant capacity. We also study a family of examples of non-monotone toric domains and determine when the conclusion of the strong Viterbo conjecture holds for these examples. Along the way, we compute the cylindrical capacity of a large class of “weakly convex toric domains” in four dimensions.
ArticleNumber 41
Author Gutt, Jean
Ramos, Vinicius G. B.
Hutchings, Michael
Author_xml – sequence: 1
  givenname: Jean
  surname: Gutt
  fullname: Gutt, Jean
  email: jean.gutt@univ-jfc.fr
  organization: Université Toulouse III - Paul Sabatier, Institut National Universitaire Champollion
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  givenname: Michael
  surname: Hutchings
  fullname: Hutchings, Michael
  organization: University of California
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  givenname: Vinicius G. B.
  surname: Ramos
  fullname: Ramos, Vinicius G. B.
  organization: Instituto de Matemática Pura e Aplicada
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Issue 2
Keywords 53D35
Viterbo’s conjecture
Symplectic capacities
53D42
toric domains
Language English
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PublicationSeriesTitle Symplectic geometry - A Festschrift in honour of Claude Viterbo’s 60th birthday
PublicationTitle Journal of fixed point theory and applications
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Snippet A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that...
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SubjectTerms Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Symplectic geometry - A Festschrift in honour of Claude Viterbo’s 60th birthday
Title Examples around the strong Viterbo conjecture
URI https://link.springer.com/article/10.1007/s11784-022-00949-6
https://hal.science/hal-04085700
Volume 24
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