Geometric rigidity for analytic estimates of Müller-Šverák
The author improves results by \textit{S. Müller} and \textit{V. Šverák} [J. Differ. Geom. 42, No. 2, 229--258 (1995; Zbl 0853.53003)] on complete, connected, non-compact surfaces \(\Sigma\) immersed into \(\mathbb R^n\). He presents two new main results: Firstly, in dimension \(n=3\) for surfaces with total curvature \(\int_\Sigma \left|A\right|^2 < 8 \pi\) the author shows that the surface is either conformal to a plane and embedded, isometric to a catenoid and embedded or is isometric to Enneper's minimal surface and not embedded. In dimension \(n=4\) a surface with total curvature \(\int_\Sigma\left|A\right|^2 \leq 4 \pi\) is embedded and either conformal to the plane or isometric to a minimal surface. The second main result is devoted to surfaces with \(\int_\Sigma K =0\) and \(\int_\Sigma \left|A\right|^2 \leq 16 \pi - 4 \delta\) with some \(\delta > 0\). These surfaces are embedded and conformal to a plane. For \(\delta = 0\) the surface is either embedded or \(\Sigma\) is isometric to the Hoffmann-Ossermann cylinder inverted with respect to a point on that cylinder.
scientific article; zbMATH DE number 6116863
scientific article