Local middle dimensional symplectic non-squeezing in the analytic setting (Q2318837)

Let \(B_r(x)\) be the ball of radius \(r\) and center \(x\), \(B_r(0)\) is denoted by \(B_r\), let \(\omega^k\) be the standard symplectic form on \(\mathbb{R}^{2k}\) and let \(P\) be a symplectic projection onto a linear \(2k\)-dimensional symplectic subspace \(V\subset \mathbb{R}^{2n}\). Then, the following middle-dimensional extension of \textit{M. Gromov}'s non-squeezing theorem [Invent. Math. 82, 307--347 (1985; Zbl 0592.53025)] in the analytic setting is proved: Theorem 2. Let \(\phi:D\to \mathbb{R}^{2n}\) be an analytic symplectic embedding of a domain \(D\subset \mathbb{R}^{2n}\). Then there exists a function \(r_0; D\to (0,\infty)\) such that the inequality \[ \mathrm{Vol}_{2k}(P\phi(B_r(x)), \omega^k_{0|V})\ge \pi r^{2k}, \quad r<r_0(x), \] holds for every \(x\in D\). This theorem is derived adopting Lojasiewicz's Structure Theorem ([\textit{S. G. Krantz} and \textit{H. R. Parks}, A primer of real analytic functions. 2nd ed. Boston, MA: Birkhäuser (2002; Zbl 1015.26030)], Theorem 5.2.3. \S4. Theorem 25) to the following analytic local squeezing theorem Theorem 1. Let \(t:(0,1]\to \phi_t\) be an analytic path of symplectic embeddings \(\phi_t:\bar{B}_1\to \mathbb{R}^{2n}\), such that \(\phi_0\) is linear. Then the middle-dimensional squeezing inequality \[ \mathrm{Vol}_{2k}(P\phi_t(B_1),\omega^k_{0|V})\ge \pi^k \] holds for \(t\) small enough. When \(\phi\) is a symplectic linear map, this inequality was proved in [\textit{A. Abbondandolo} and \textit{P. Majer}, Calc. Var. Partial Differ. Equ. 54, No. 2, 1469--1506 (2015; Zbl 1330.53112)] (\S4. Theorem 21 with an addendum 22). To derive Theorem 1 from Theorem 21, implications of either formally trivial and non formally trivial deformation \(P\phi_t(B_1)\) are obtained as Prop. 24 (cf. [\textit{J. C. Álvarez Paiva} and \textit{F. Balacheff}, Geom. Funct. Anal. 24, No. 2, 648--669 (2014; Zbl 1292.53050)]. \S2. Proposition 18). Analyticity of \(\phi\) is used to derive Theorem 1 from Theorem 21 by using Proposition 24. Necessary facts from the contact geometry including Zoll contact manifolds (regular contact manifolds) and the function \[ A_{\mathrm{min}}(M,\alpha)=\min \{A(\gamma)\mid \gamma \text{ is a closed Reeb orbit on }(M,\alpha)\} \] \(\alpha\) is the contact form and \(A(\gamma)\) is the action (period) of a Reeb orbit \(\gamma\), are explained in \S1. \(A_{\mathrm{min}}(M,\alpha)\) is the main tool in this paper. Informations on how \(A_{\mathrm{min}}\) behaves in case of a contact deformation on the unit sphere is explained in Section 2, mainly according to [loc. cit.]. In \S3, analyticity of \(\mathrm{Vol}_k(P\phi_{t,y}(B_1),\rho)\) under a suitable condition is proved (Proposition 10). Using calculations from [\textit{A. Abbondandolo} and \textit{R. Matveyev}, J. Topol. Anal. 5, No. 1, 87--119 (2013; Zbl 1268.37074)], Theorem 21 and Proposition 24 are proved after these preparations.