Journal of Differential Geometry

Min-max embedded geodesic lines in asymptotically conical surfaces

Alessandro Carlotto and Camillo De Lellis

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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.

Article information

Source
J. Differential Geom., Volume 112, Number 3 (2019), 411-445.

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

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1563242470

Digital Object Identifier
doi:10.4310/jdg/1563242470

Mathematical Reviews number (MathSciNet)
MR3981294

Zentralblatt MATH identifier
1420.53049

Citation

Carlotto, Alessandro; De Lellis, Camillo. Min-max embedded geodesic lines in asymptotically conical surfaces. J. Differential Geom. 112 (2019), no. 3, 411--445. doi:10.4310/jdg/1563242470. https://projecteuclid.org/euclid.jdg/1563242470


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