Wide short geodesic loops on closed Riemannian manifolds

2019/10/04 by Rotman, Regina
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1910.01772

Abstract

It is not known whether or not the lenth of the shortest periodic geodesic on a closed Riemannian manifold Mⁿ can be majorized by c(n) vol ^{1 \over n}, or c(n)d, where n is the dimension of Mⁿ, vol denotes the volume of Mⁿ, and d denotes its diameter. In this paper we will prove that for each ε>0 one can find such estimates for the length of a geodesic loop with with angle between π-ε and π with an explicit constant that depends both on n and ε. That is, let ε> 0, and let a = \lceil 1 \over sin (ε\over 2) \rceil+1 . We will prove that there exists a "wide" (i.e. with an angle that is wider than π-ε) geodesic loop on Mⁿ of length at most 2n!aⁿd. We will also show that there exists a "wide" geodesic loop of length at most 2(n+1)!²a⁽ⁿ⁺¹⁾³ FillRad ≤ 2 ⋅ n(n+1)!²a⁽ⁿ⁺¹⁾³ vol^{1 \over n}. Here FillRad is the Filling Radius of Mⁿ.

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