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July 2019 Min-max embedded geodesic lines in asymptotically conical surfaces
Alessandro Carlotto, Camillo De Lellis
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J. Differential Geom. 112(3): 411-445 (July 2019). DOI: 10.4310/jdg/1563242470

Abstract

We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we produce, which will be always less or equal than one (with equality under suitable curvature or genericity assumptions), as well as of their precise asymptotic behavior. In fact, we can prove that in any such surface for every couple of opposite half-lines there exists an embedded geodesic line whose two ends are asymptotic, in a suitable sense, to those half-lines.

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Alessandro Carlotto. Camillo De Lellis. "Min-max embedded geodesic lines in asymptotically conical surfaces." J. Differential Geom. 112 (3) 411 - 445, July 2019. https://doi.org/10.4310/jdg/1563242470

Information

Received: 3 November 2016; Published: July 2019
First available in Project Euclid: 16 July 2019

zbMATH: 1420.53049
MathSciNet: MR3981294
Digital Object Identifier: 10.4310/jdg/1563242470

Rights: Copyright © 2019 Lehigh University

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Vol.112 • No. 3 • July 2019
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