Calculus of Variations and Geometric Measure Theory

A. Carlotto - C. De Lellis

Min-max embedded geodesic lines on asymptotically conical surfaces

created by delellis on 02 Nov 2016
modified on 01 Dec 2020


Published Paper

Inserted: 2 nov 2016
Last Updated: 1 dec 2020

Journal: J. Differential Geom.
Volume: 112
Number: 3
Pages: 411-445
Year: 2019


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.