Embedded minimal surfaces and total curvature of curves in a manifold

Jaigyoung Choe, Robert Gulliver

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

Let Mn be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant -κ2. It is proved that every branched minimal surface in M bounded by a smooth Jordan curve Γ with total curvature ≤ 4π + κ2infp∈M Area(p×Γ) is embedded. p×Γ denotes the geodesic cone over Γ with vertex p. It follows that a Jordan curve Γ in M3 with total curvature ≤ 4π + κ2infp∈M Area(p×Γ) is unknotted. In the hemisphere S+n, we prove the embeddedness of any minimal surface whose boundary curve has total curvature ≤ 4π - supp∈S+n Area(p×Γ).

Original languageEnglish (US)
Pages (from-to)343-362
Number of pages20
JournalMathematical Research Letters
Volume10
Issue number2-3
DOIs
StatePublished - 2003

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