Geometric rigidity for analytic estimates of Müller–Šverák

In the paper Muller–Sverak (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 A28 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 A28 , 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 A216 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.