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Entropy of systolically extremal surfaces and asymptotic bounds. (English) Zbl 1097.53027
Summary: We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with Katok’s lower bound in terms of the volume, to obtain a simpler alternative proof of 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.

MSC:
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
37C35 Orbit growth in dynamical systems
53C20 Global Riemannian geometry, including pinching
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