TY - JOUR

T1 - On surfaces of finite total curvature

AU - Müller, S.

AU - Šverák, V.

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1995/9

Y1 - 1995/9

N2 - We consider surfaces M immersed into Rnand we prove that the quantity ∫M|A|2(where A is the second fundamental form) controls in many ways the behaviour of conformal parametrizations of M. If M is complete, connected, noncompact and ∫M|A|2< ∞ we obtain a more or less complete picture of the behaviour of the immersions. In particular we prove that under these assumptions the immersions are proper. Moreover, if ∫M|A|2≤ 4π or if n = 3 and ∫M|A|2< 8π, then M is embedded. We also prove that conformal parametrizations of graphs of W2, 2functions on R2exist, are bilipschitz and the conformal metric is continuous. The paper was inspired by recent results of T.Toro.

AB - We consider surfaces M immersed into Rnand we prove that the quantity ∫M|A|2(where A is the second fundamental form) controls in many ways the behaviour of conformal parametrizations of M. If M is complete, connected, noncompact and ∫M|A|2< ∞ we obtain a more or less complete picture of the behaviour of the immersions. In particular we prove that under these assumptions the immersions are proper. Moreover, if ∫M|A|2≤ 4π or if n = 3 and ∫M|A|2< 8π, then M is embedded. We also prove that conformal parametrizations of graphs of W2, 2functions on R2exist, are bilipschitz and the conformal metric is continuous. The paper was inspired by recent results of T.Toro.

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U2 - 10.4310/jdg/1214457233

DO - 10.4310/jdg/1214457233

M3 - Article

AN - SCOPUS:0029679718

VL - 42

SP - 229

EP - 258

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 2

ER -