[Paper Review] Minimal rays on surfaces of genus greater than one

Jan Philipp Schröder|arXiv (Cornell University)|Apr 2, 2014

Point processes and geometric inequalities5 references3 citations

TL;DR

This paper studies minimal geodesics and rays on Finsler metrics over closed orientable surfaces of genus greater than one, using weak KAM theory to show that for almost all asymptotic directions ξ ∈ S¹, the bounding minimal geodesics form a lamination in the universal cover (the Poincaré disk), and that for almost all geodesic types, there is exactly one minimal geodesic. The results establish uniqueness of Busemann functions up to constants and confirm the existence of unstable minimal geodesics.

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.

Motivation & Objective

• To understand the global structure of minimal rays and geodesics in the universal cover of surfaces of genus >1 under general Finsler metrics.
• To extend Morse's earlier results on minimal geodesics by analyzing asymptotic directions not fixed by deck transformations.
• To establish the uniqueness of minimal geodesics and Busemann functions for almost all directions using weak KAM theory.
• To characterize the dynamics of minimal geodesics via the Aubry and Mather sets in the context of Finsler geometry.

Proposed method

• Uses weak KAM theory to analyze the structure of minimal rays and geodesics in the universal cover, which is identified with the Poincaré disk.
• Applies the concept of weak KAM solutions and their calibrated curves to study the asymptotic behavior of minimal geodesics.
• Employs the notion of forward and backward calibrated curves to identify extremal geodesics in the lamination structure.
• Utilizes the convergence of weak KAM solutions along asymptotic directions to prove continuity and uniqueness properties.
• Leverages the group action of deck transformations Γ to relate dynamics in the universal cover to the base manifold M.
• Applies compactness and continuity arguments on the space of weak KAM solutions to deduce limit behavior and intersection properties.

Experimental results

Research questions

• RQ1 For which asymptotic directions ξ ∈ S¹ do the bounding minimal geodesics of rays with endpoint ξ form a lamination in the universal cover?
• RQ2 Are Busemann functions with almost all asymptotic directions uniquely determined up to additive constants?
• RQ3 Does every type of minimal geodesic (in the sense of Morse) admit exactly one representative in the universal cover?
• RQ4 How do the Aubry and Mather sets relate to the dynamics of minimal geodesics in the Finsler setting?
• RQ5 Can the non-wandering set of the geodesic flow be characterized using the calibrated graphs of weak KAM solutions?

Key findings

• For all but countably many ξ ∈ S¹, the union of bounding geodesics for rays ending at ξ forms a lamination in the universal cover, meaning no two curves intersect transversely.
• For almost every γ ∈ ℳ (in the Lebesgue measure sense), the set ℳ(γ) of minimal geodesics with endpoints γ(±∞) contains exactly one geodesic.
• The Busemann functions associated with almost all asymptotic directions ξ are unique up to additive constants, implying a strong rigidity in the asymptotic geometry.
• The unique minimal geodesic in ℳ(γ) for almost all γ is both forward and backward unstable, as defined in the paper.
• The non-wandering set of the geodesic flow on the base manifold is contained in the union of the images under the projection of the calibrated graphs of the weak KAM solutions u₊ⁱ(ξ) for all ξ ∈ S¹ and i ∈ {0,1}.
• The calibrated curves of the weak KAM solutions u₊ⁱ(ξ) correspond exactly to the bounding geodesics cᵠⁱ(γ) for γ ∈ ℳ₊(ξ), establishing a deep link between weak KAM theory and the geometric structure of minimal geodesics.