2004/10/13 by Mikhail G. Katz, Katz, Mikhail G., Stephane Sabourau +2
Mathematics · #53C23 #Analytic and geometric function theory #Combinatorics (math.CO) #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #math.CO #math.DG #math.DS #math.GT #math.MG #msc:53C23
paper · pdf · doi:10.48550/arxiv.math/0410312
14 pages. Ergodic Theory and Dynamical Systems, to appear
openalex publication_date 2004/10/13 · arxiv created 2004/12/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of M. Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermore, we improve the multiplicative constant in Gromov's theorem. We show that every surface of genus at least 20 is Loewner. Finally, we relate, in higher dimension, the isoembolic ratio to the minimal entropy.