Geometric rigidity for analytic estimates of Müller–Šverák

Abstract

In the paper Müller–Šverák (J Differ Geom 42(2):229–258, 1995) conformally immersed surfaces with finite total curvature were studied. In particular it was shown that surfaces with total curvature ${\int_{\Sigma} |A|^2 < 8 \pi}$ in dimension three were embedded and conformal to the plane with one end. Here, using techniques from Kuwert–Li (W ^2,2-conformal immersions of a closed Riemann surface into R ⁿ. arXiv:1007.3967v2 [math.DG], 2010), we will show that if the total curvature ${ \int_{\Sigma}|A|^2\leq8\pi}$ , then we are either embedded and conformal to the plane, isometric to a catenoid or isometric to Enneper’s minimal surface. In fact the technique of our proof shows that if we are conformal to the plane, then if n ≥ 3 and ${ \int_{\Sigma} | A|^{2}\leq 16 \pi }$ then Σ is embedded or Σ is the image of a generalized catenoid inverted at a point on the catenoid. In order to prove these theorems, we prove a Gauss–Bonnet theorem for surfaces with complete ends and isolated finite area singularities which extends a theorem of Jorge-Meeks (Topology 22(2):203–221, 1983). Using this theorem, we then prove an inversion formula for the Willmore energy.