On the degenerated Arnold-Givental conjecture

Abstract

We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M,\omega ,\tau )$ with nonempty and compact real part $L=\mathrm{Fix}(\tau )$. For given $\Lambda \in (0,+\infty ]$ and $m\in \mathbb{N}\cup \{0\}$ we show the equivalence of the following two claims: (i) $♯(L\cap {\varphi }_1^H(L))\ge m$ for any Hamiltonian function $H\in C_0^{\infty }([0,1]\times M)$ with Hofer's norm $\Vert H\Vert <\Lambda$; (ii) $♯\mathcal{P}(H,\tau )\ge m$ for every $H\in C_0^{\infty }(\mathbb{R}/\mathbb{Z}\times M)$ satisfying $H(t,x)=H(-t,\tau (x))\forall (t,x)\in \mathbb{R}\times M$ and with Hofer's norm $\Vert H\Vert <2\Lambda$, where $\mathcal{P}(H,\tau )$ is the set of all $1$-periodic solutions of $\dot{x}(t)=X_H(t,x(t))$ satisfying $x(-t)=\tau (x(t))\forall t\in \mathbb{R}$ (which are also called brake orbits sometimes). Suppose that $(M,\omega )$ is geometrical bounded for some $J\in \mathcal{J}(M,\omega )$ with ${\tau }^{\ast }J=-J$ and has a rationality index $r_{\omega }>0$ or $r_{\omega }=+\infty$. Using Hofer's method we prove that if the Hamiltonian $H$ in (ii) above has Hofer's norm $\Vert H\Vert <r_{\omega }$ then $\sharp(L\capϕ^H_1(L))\ge\sharp {\cal P}_0(H,τ)\ge {\rm Cuplength}_{\F}(L)$ for $\F=\Z_2$, and further for $\F=\Z$ if $L$ is orientable, where ${\mathcal{P}}_0(H,\tau )$ consists of all contractible solutions in $\mathcal{P}(H,\tau )$.

28 pages, this is a corrected version for the last one which had been withdrawn many years ago

References in corpus (2)

• Quantum Structures for Lagrangian Submanifolds
• Symplectic fixed points and Lagrangian intersections on weighted projective spaces

Cited by in corpus (2)

• The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems
• Representation formula for symmetric symplectic capacity and applications