Finite energy foliations of tight three-spheres and Hamiltonian dynamics

By Helmut Hofer, K. Wysocki, E. Zehnder. 2003

209 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cc594d9edf77aec3bfa16f4dc78b80ed906a5109

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In Applied Mathematics

1. Y. Long
2. Huagui Duan
3. Michael Hutchings
4. Wei Wang
5. Helmut Hofer
6. Richard Siefring
7. Otto van Koert
8. Urs Frauenfelder

In Geometry and Topology

1. Michael Hutchings
2. Richard Siefring
3. Helmut Hofer
4. K. Wysocki
5. E. Zehnder
6. Otto van Koert
7. Urs Frauenfelder
8. Frederick J. Bourgeois III

Citations in the corpus

• The Conley Conjecture and Beyond (2014, unknown): “-or-innitely-many" alternative should hold for all contact forms supporting the standard contact structure on S3, which could be thought of as a three-dimensional analog of Franks’ theorem; see [HWZ03] for some other related results. 28 VIKTOR GINZBURG AND BAS˘AK GUREL The existence of innitely”
• The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds (2015, result): “and 1.6 are sharp. Note that bumpy is a generic condition, and our above theorems give generic results on the above mentioned manifolds. (ii) Based on our above theorems, the works [HWZ1] in 1998 and [HWZ2] in 2003 of Hofer, Wysocki and Zehnder, [BaL] in 2010 of Bangert and Long, [LoW] in 2007 of Lo”
• Polyfold and Fredholm Theory (2014, background): “The analytical difficulties in dealing with SFT become apparent in the series of papers [29, 33, 34, 41] culminating in the SFT-compactness paper [4].”
• Intersection theory of punctured pseudoholomorphic curves (2009, background): “These results and ideas are useful in the study of finite energy foliations, as studied in [12].”
• Stability of closed geodesics on Finsler 2-spheres (2008, background): “Zehnder in [12] imply that there exist either two or infinitely many prime closed geodesics on every Finsler 2-sphere provided all the closed geodesics together with their iterations are non-degenerate and the stable and unstable manifolds of every hyperbolic closed geodesic intersect transversally.”
• The existence of two closed geodesics on every Finsler 2-sphere (2007, background): “Wysocki and E. Zehnder in [ HWZ2 ] from 2003 imply the existence of either two or infinitely many”
• Torsion contact forms in three dimensions have two or infinitely many Reeb orbits (2017, unknown): “Theorem 1.2 (Hofer, Wyoscki and Zehnder [9, Corollary 1.10]) Let be a non-degenerate contact form on S 3 .”
• Taubes’s proof of the Weinstein conjecture in dimension three (2009, background): “It is shown in [30] that for a large class of contact forms on S(3) there are either two or infinitely many embedded Reeb orbits.”
• Multiple closed geodesics on bumpy Finsler n-spheres (2007, background): “In [13] of 2003, Hofer, Wysocki and Zehnder proved that there exist either two or infinitely many distinct closed geodesics on every bumpy Finsler 2-sphere if the stable and unstable manifolds of every hyperbolic closed geodesics intersect transversally.”
• THE WEINSTEIN CONJECTURE FOR PLANAR CONTACT STRUCTURES IN DIMENSION THREE (2004, unknown): “Finite energy foliations are known to be a useful tool in studying the dynamics of Reeb vector fields as well as topological applications, see [18, 16, 13].”