= Stein surface
A <Stein surface> is a <Stein manifold> of complex <dimension> two, hence real <dimension> four. Its global <holomorphic functions> separate points and provide local <coordinates>, and the holomorphic hull of every <compact subset> is compact. The word “surface” here refers to complex <dimension>, rather than to a real two-dimensional <surface>.
The affine complex plane $\mathbb C^2$ is a <Stein surface>, as is the smooth closed hypersurface $xy-z^2=1$ in $\mathbb C^3$: its gradient $(y,x,-2z)$ never vanishes on the hypersurface. A compact connected positive-dimensional <complex manifold> cannot be Stein, because its <holomorphic functions> are constant. In real <dimension> four, the topology of <Stein surfaces> has especially rich links with <handle decompositions> and <contact structures> on three-dimensional boundaries.<Handlebody Construction of Stein Surfaces by Robert E. Gompf>[Abstract and introduction, Stein surfaces and contact three-manifolds]{ref}
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