Stéphane Sabourau

Entropy of systolically extremal surfaces and asymptotic bounds

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.

Hyperelliptic surfaces are Loewner

We prove that C. Loewners inequality for the torus is satisfied by conformal metrics on hyperelliptic surfaces X as well. In genus 2, we first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to X away from Weierstrass points. The loops are then transplanted to X, and surgered to obtain a Loewner loop on X. In higher genus, we exploit M. Gromovs area estimates for e-regular metrics on X.

Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions

Abstract We hereby introduce and study the notion of self-contracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded self-contracted planar curves have a finite length. We also give an example of a convex function defined in the plane whose gradient orbits spiral infinitely many times around the unique minimizer of the function.

Systoles of 2-complexes, Reeb graph, and Grushko decomposition

Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov’s inequality in this dimension. The argument relies on the Reeb graph and the coarea formula, combined with an induction on the number of freely indecomposable factors in Grushko’s decomposition of the fundamental group. More specifically, we construct a kind of a Reeb space “minimal model” for X, reminiscent of the “chopping off long fingers” construction used by Gromov in the context of surfaces. As a consequence, we prove the agreement of the Lusternik-Schnirelmann and systolic categories of a 2-complex.

Entropy and systoles on surfaces

We show that the volume entropy of surfaces with unit systole is bounded from above by a constant which does not depend on the metric, which answers a question raised by A. Katok. This upper bound follows from a comparison between the geometric and algebraic lengths on the fundamental groups of surfaces.

Dyck’s surfaces, systoles, and capacities

We prove an optimal systolic inequality for nonpositively curved Dycks surfaces. The extremal surface is flat with eight conical singularities, six of angle theta and two of angle 9pi - theta, for a suitable theta with cos(theta) in Q(sqrt{19}). Relying on some delicate capacity estimates, we also show that the extremal surface is not conformally equivalent to the hyperbolic surface with maximal systole, yielding a first example of systolic extremality with this behavior.

Hyperellipticity and systoles of Klein surfaces

Given a hyperelliptic Klein surface, we construct companion Klein bottles, extending our technique of companion tori already exploited by the authors in the genus 2 case. Bavard’s short loops on such companion surfaces are studied in relation to the original surface so to improve a systolic inequality of Gromov’s. A basic idea is to use length bounds for loops on a companion Klein bottle, and then analyze how curves transplant to the original non-orientable surface. We exploit the real structure on the orientable double cover by applying the coarea inequality to the distance function from the real locus. Of particular interest is the case of Dyck’s surface. We also exploit an optimal systolic bound for the Möbius band, due to Blatter.

Optimal systolic inequalities on Finsler Möbius bands

We prove optimal systolic inequalities on Finsler Mobius bands relating the systole and the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimal systolic in- equality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric families both on the Mobius band and the Klein bottle are also presented.

A systolic-like extremal genus two surface

It is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved metrics. We prove that this piecewise flat metric is also critical for slow metric variations, without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in different free homotopy classes. The free homotopy classes considered correspond to those of the systolic loops and the second systolic loops of the extremal surface.