Generating functions versus action functional—stable Morse theory versus Floer theory

1998 ◽  
pp. 107-125 ◽  
Author(s):  
Darko Milinković ◽  
Yong-Geun Oh
Keyword(s):  
Generating Functions ◽  
Morse Theory ◽  
Action Functional ◽  
Floer Theory ◽  
Theory Versus

scholarly journals HYPERKÄHLER ARNOLD CONJECTURE AND ITS GENERALIZATIONS

2012 ◽  
Vol 23 (08) ◽  
pp. 1250077 ◽  
Author(s):  
VIKTOR L. GINZBURG ◽  
DORIS HEIN
Keyword(s):  
Generating Functions ◽  
Dimensional Reduction ◽  
Morse Theory ◽  
Three Dimensional ◽  
Degenerate Case ◽  
Floer Theory ◽  
Arnold Conjecture ◽  
Finite Dimensional Reduction ◽  
Finite Dimensional ◽  
Degenerate Version

We generalize and refine the hyperkähler Arnold conjecture, which was originally established, in the non-degenerate case, for three-dimensional time by Hohloch, Noetzel and Salamon by means of hyperkähler Floer theory. In particular, we prove the conjecture in the case where the time manifold is a multidimensional torus and also establish the degenerate version of the conjecture. Our method relies on Morse theory for generating functions and a finite-dimensional reduction along the lines of the Conley–Zehnder proof of the Arnold conjecture for the torus.

scholarly journals Spectral invariants for monotone Lagrangians

2018 ◽  
Vol 10 (03) ◽  
pp. 627-700 ◽  
Author(s):  
Rémi Leclercq ◽  
Frol Zapolsky
Keyword(s):  
Generating Functions ◽  
Floer Homology ◽  
Spectral Invariants ◽  
Seminal Paper ◽  
Floer Theory ◽  
Cotangent Bundles

Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper [73], they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a “classical” Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which are crucial for their applications. Finally, as a demonstration, we apply these new invariants to symplectic rigidity of some specific monotone Lagrangians.

Generating functions

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Final Section ◽  
Lagrangian Submanifolds ◽  
Variational Problems ◽  
Action Functional ◽  
Arnold Conjecture ◽  
Cotangent Bundles ◽  
Symplectic Action

This chapter discusses generating functions in more detail. It shows how generating functions give rise to discrete-time analogues of the symplectic action functional and hence lead to discrete variational problems. The results of this chapter form the basis for the proofs in Chapter 11 of the Arnold conjecture for the torus and in Chapter 12 of the existence of the Hofer–Zehnder capacity. The final section examines generating functions for exact Lagrangian submanifolds of cotangent bundles.

scholarly journals Morse theory for the Hofer length functional

2014 ◽  
Vol 06 (02) ◽  
pp. 193-210 ◽  
Author(s):  
Yasha Savelyev
Keyword(s):  
Morse Theory ◽  
Symplectic Manifolds ◽  
Floer Theory

Following [16], we develop here a connection between Morse theory for the (positive) Hofer length functional L : Ω Ham (M, ω) → ℝ, with Gromov–Witten/Floer theory, for monotone symplectic manifolds (M, ω). This gives some immediate restrictions on the topology of the group of Hamiltonian symplectomorphisms (possibly relative to the Hofer length functional), and a criterion for non-existence of certain higher index geodesics for the Hofer length functional. The argument is based on a certain automatic transversality phenomenon which uses Hofer geometry to conclude transversality and may be useful in other contexts. Strangely the monotone assumption seems essential for this argument, as abstract perturbations necessary for the virtual moduli cycle, decouple us from underlying Hofer geometry, causing automatic transversality to break.

Theory versus Experiments in Two-Dimensional Surf Beats

1991 ◽  
Author(s):  
Hemming A. Schäffer ◽  
Ivar G. Jonsson
Keyword(s):  
Two Dimensional ◽  
Theory Versus
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