Stein and Weinstein structures on disk cotangent bundles of surfaces
A complex manifold \(X\) is called a Stein manifold if it admits an exhausting strictly plurisubharmonic function, which is essentially characterized as being a proper function \(f: X \rightarrow\mathbb{R} \) that is bounded below and can be assumed a Morse function whose level sets \(f^{-1}(c)\) are strictly pseudoconvex, where \(f^{-1}(c)\) is oriented as the boundary of the complex manifold \(f^{-1}((- \infty, c])\). For a 4-dimensional manifold \(X\), strict pseudoconvexity implies that \( f^{-1} (c) \) inherits a contact structure determining its given orientation. A compact complex \( X \) with boundary is called a Stein domain if it admits a strictly plurisubharmonic function such that the boundary \(\partial X\) is a level set. If \(S\) is a closed, connected, and smooth surface, then the disk cotangent bundle \(\mathbb{D}T^*S\) of \(S\) carries the canonical symplectic structure \(\omega_{\mathrm{can}}=d\lambda_{\mathrm{can}}\), and the unit cotangent bundle \(\partial\mathbb{D}T^*S=\mathbb{S}T^*S\) carries the canonical contact structure \(\xi_{\mathrm{can}}= \ker({\lambda_{\mathrm{can}}}_{|\mathbb{S}T^*S})\), where \(\lambda_{\mathrm{can}}\) is the Liouville 1-form on \(T^*S\). In [Ann. Math. (2) 148, No. 2, 619--693 (1998; Zbl 0919.57012)], \textit{R. E. Gompf} showed that \(\mathbb{D}T^*S\) admits the structure of a Stein domain by explicitly exhibiting \(\mathbb{D}T^*S\) as a Legendrian handlebody diagram. In this paper, the author prove that Gompf's Stein domain is symplectomorphic to \((\mathbb{D}T^*S,\omega_{\mathrm{can}})\) and the boundary contact 3-manifold is contactomorphic to \((\mathbb{S}T^*S,\xi_{\mathrm{can}})\). The author shows that if \(\Sigma_g\) is a closed, connected, smooth, and orientable surface of genus \(g\ge 1\), then the presented Stein handlebody diagram is symplectomorphic to \((\mathbb{D}T^*\Sigma_g,\omega_{\mathrm{can}})\) and the boundary contact 3-manifold is contactomorphic to \((\mathbb{S}T^*N_k,\xi_{\mathrm{can}})\), and if \(N_k\) is a closed, connected, smooth, and nonorientable surface of genus \(k\ge 1\), i.e., \(N_k=\#_k\mathbb{RP}^2\), then the presented Stein handlebody diagram is symplectomorphic to \((\mathbb{D}T^*\Sigma_g,\omega_{\mathrm{can}})\) and the boundary contact 3-manifold is contactomorphic to \((\mathbb{S}T^*N_k,\xi_{\mathrm{can}})\). Finally, the author obtains a contact surgery diagram for the canonical contact structure on the unit cotangent bundle \(\mathbb{S}T^*S\) of \(S\).
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