Entropy of systolically extremal surfaces and asymptotic bounds

By Mikhail G. Katz, S. Sabourau. Ergodic Theory and Dynamical Systems, 2004

59 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eebc22570b1a504a19d5cf1206e079ba9ccb6c37

Citations in the corpus

• Homological Invariance For Asymptotic Invariants and Systolic Inequalities (2007, unknown): “The proof will cover the dimensions n 3. For surfaces, the theorem was shown in [ KS05 ] by Katz and Sabourau.”