On surfaces of finite total curvature

S. Müller, V. Šverák

Research output: Contribution to journalArticlepeer-review

80 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)229-258
Number of pages30
JournalJournal of Differential Geometry
Volume42
Issue number2
DOIs
StatePublished - Sep 1995

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