An extension theorem in symplectic geometry (Q1865595)

Let \({\mathbb R}^{2n}\) be endowed with the canonical symplectic form, let \(U\) be a domain (an open connected subset) of \({\mathbb R}^{2n}\) and let \(\varphi\colon U\longrightarrow{\mathbb R}^{2n}\) be a symplectic embedding. One says that the pair \((U,\varphi)\) has the extension property if for each \(A\subset\overline{A}\subset U,\) there exists a symplectomorphism \(\Phi_A\) of \({\mathbb R}^{2n}\) such that \(\Phi_A|_A=\varphi|_A.\) In this paper the author gives conditions which ensure that a given pair \((U,\varphi)\) as above has such extension property. The class of domains \(U\) that he considers is the class of Lipschitz domains, that is, for the path-distance \(\text{dist}_U\colon U\times U\longrightarrow{\mathbb R}\) in \(U\) there exists a constant \(\lambda>0\) such that \(\text{dist}_U(z,z')\leq\lambda|z-z'|,\) for all \(z,z'\in U.\) The main theorem is the following. Theorem. Assume that \(\varphi\colon U\rightarrow{\mathbb R}^{2n}\) is a symplectic embedding of a starlike Lipschitz domain \(U\subset{\mathbb R}^{2n}\). Suppose there exists \(L>0\) such that \(|\varphi(z)-\varphi(z')|\geq L|z-z'|\) for all \(z,z'\in U.\) Then the pair \((U,\varphi)\) has the extension property. When \(n=1,\) the assumptions can be weakened so as to obtain that if \(\varphi\colon U\rightarrow{\mathbb R}^2\) is a symplectic embedding of a bounded simply-connected domain \(U\subset{\mathbb R}^{2n},\) then the pair \((U,\varphi)\) has the extension property. The author shows, by two counterexamples, that neither the boundedness condition nor the simple-connectivity condition can be avoided in this case. The above theorem generalizes the well-known ``Extension after Restriction Principle'' for symplectic embeddings of bounded starlike domains given in [\textit{I. Ekeland} and \textit{H. Hofer}, Math. Z. 200, 355-378 (1989; Zbl 0641.53035)].