Minimal rays on surfaces of genus greater than one

2014/04/02 by Jan Philipp Schröder, Schröder, Jan Philipp
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #math.DG #math.DS

paper · pdf · doi:10.48550/arxiv.1404.0573

25 pages, 4 figures

arxiv created 2014/04/02 · arxiv updated 2014/04/03

Abstract

For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all asymptotic directions the minimal rays with these directions laminate the universal cover and that the Busemann functions with these directions are unique up to adding constants. Moreover, using a kind of weak KAM theory, we show that for almost all types of minimal geodesics in the sense of Morse, there is precisely one minimal geodesic of this type.

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