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Robert E. Gompf studies four-dimensional topology, symplectic topology and contact topology. His University of Texas research profile also identifies Stein surfaces as a focus.^[ref]

His symplectic sum construction glues symplectic manifolds along suitable codimension-two submanifolds. Among its consequences, every finitely presented group occurs as the fundamental group of a compact symplectic manifold in each even dimension at least four. This shows how flexible symplectic geometry can be compared with the additional restrictions imposed by a Kähler manifold.^[ref]

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.^[ref]