Min-max embedded geodesic lines in asymptotically conical surfaces

TL;DR

This paper uses min-max methods to construct many distinct geodesic lines in asymptotically conical surfaces with non-negative scalar curvature, controlling their Morse index and asymptotic behavior, where minimization fails.

Contribution

It introduces a min-max approach to produce uncountably many geodesic lines with controlled Morse index in settings where minimization is ineffective.

Findings

Constructed uncountably many geodesic lines in asymptotically conical surfaces.

Controlled the Morse index of the geodesic lines, often equal to one.

Proved existence of geodesic lines asymptotic to given half-lines in the surface.

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.