Construction of surfaces with large systolic ratio

2013/11/06 by Hugo Akrout, Akrout, Hugo, Bjoern Muetzel +1
Mathematics · #53B21 #53C22 and 53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1311.1449

openalex publication_date 2013/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (M,g) be a closed, oriented, Riemannian manifold of dimension m. We call a systole a shortest non-contractible loop in (M,g) and denote by sys(M,g) its length. Let SR(M,g)=\fracsys(M,g)^mvol(M,g) be the systolic ratio of (M,g). Denote by SR(k) the supremum of SR(S,g) among the surfaces of fixed genus k ≠ 0. In Section 2 we construct surfaces with large systolic ratio from surfaces with systolic ratio close to the optimal value SR(k) using cutting and pasting techniques. For all k_i ≥ 1, this enables us to prove: (1)/(SR(k₁ + k₂)) ≤ (1)/(SR(k₁)) + (1)/(SR(k₂)). We furthermore derive the equivalent intersystolic inequality for SR_h(k), the supremum of the homological systolic ratio. As a consequence we greatly enlarge the number of genera k for which the bound SR_h(k) ≥ SR(k) \gtrsim (4)/(9π) (log(k)²)/(k) is valid and show that that SR_h(k) ≤ ((log(195k)+8)²)/(π(k-1)) for all k ≥ 76. In Section 3 we expand on this idea. There we construct product manifolds with large systolic ratio from lower dimensional manifolds.

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