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Unseen symplectic boundaries. (English) Zbl 0870.53020
de Bartolomeis, Paolo (ed.) et al., Manifolds and geometry. Proceedings of a conference, held in Pisa, Italy, September 1993. Cambridge: Cambridge University Press. Symp. Math. 36, 178-189 (1996).
In this paper the authors show that the boundary of a symplectic manifold cannot be completely reconstructed in terms of the interior alone. The main result is as follows: There exist two domains $$\Omega, \Omega'$$ with smooth boundaries in the standard symplectic space $$\mathbb{R}^{2n}$$, such that their interiors $$\text{Int} \Omega$$ and $$\text{Int} \Omega'$$ are symplectomorphic but their boundaries $$\partial\Omega$$ and $$\partial\Omega'$$ are not. The boundaries $$\partial \Omega$$ and $$\partial \Omega'$$ can be chosen diffeomorphic and, moreover, arbitrarily $$C^\infty$$-close to the round ball $$D^{2n} \subset \mathbb{R}^{2n}$$.
For the entire collection see [Zbl 0840.00037].

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 53C40 Global submanifolds
##### Keywords:
diffeomorphic boundaries; symplectomorphic domains