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Min-max embedded geodesic lines in asymptotically conical surfaces


Carlotto, Alessandro; De Lellis, Camillo (2019). Min-max embedded geodesic lines in asymptotically conical surfaces. Journal of Differential Geometry, 112(3):411-445.

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 behaviour. 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.

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 behaviour. 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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Analysis
Physical Sciences > Algebra and Number Theory
Physical Sciences > Geometry and Topology
Uncontrolled Keywords:Geometry and Topology, Algebra and Number Theory, Analysis
Language:English
Date:1 July 2019
Deposited On:16 Dec 2019 08:48
Last Modified:29 Jul 2020 11:29
Publisher:International Press
ISSN:0022-040X
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.4310/jdg/1563242470
Project Information:
  • : FunderSNSF
  • : Grant ID200021_159403
  • : Project TitleRegularity questions in geometric measure theory

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