2015/09/27 by Jingyi Chen, Yuxiang Li, Chen, Jingyi +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG
paper · pdf · doi:10.48550/arxiv.1509.08061
19 pages
arxiv created 2015/09/27 · arxiv updated 2015/09/29
For a smooth immersion f from the punctured disk D\backslash\0\ into ℝn extendable continuously at the puncture, if its mean curvature is square integrable and the measure of f(D)∩ Brk=o(rk) for a sequence rk→ 0, we show that the Riemannian surface (Dr\backslash\0\,g) where g is the induced metric is conformally equivalent to the unit Euclidean punctured disk, for any r∈(0,1). For a locally W2,2 Lipschitz immersion f from the punctured disk D2\backslash\0\ into ℝn, if ‖∇ f‖L^∞ is finite and the second fundamental form of f is in L2, we show that there exists a homeomorphism ϕ:D→ D such that f∘ϕ is a branched W2,2-conformal immersion from the Euclidean unit disk D into ℝn.