Isosystolic genus three surfaces critical for slow metric variations

Sabourau, Stéphane¹

¹ Laboratoire de Mathématiques et Physique Théorique, Université François-Rabelais Tours, Parc de Grandmont, 37200 Tours, France

Geometry & topology, Tome 15 (2011) no. 3, pp. 1477-1508.

Résumé

We show that the two piecewise flat surfaces with conical singularities conjectured by E Calabi as extremal surfaces for the isosystolic problem in genus $3$ are critical with respect to some metric variations. The proof relies on a new approach to study isosystolic extremal surfaces.

Détail

DOI : 10.2140/gt.2011.15.1477

Keywords: systole, systolic inequality, extremal surface, flat surface with conical singularities, Calabi surface, calibration, Busemann function

Bibliographie

[1] J C Álvarez Paiva, F Balacheff, Infinitesimal systolic rigidity of metrics all of whose geodesics are closed and of the same length

[2] F Balacheff, Sur la systole de la sphère au voisinage de la métrique standard, Geom. Dedicata 121 (2006) 61

[3] F Balacheff, A local optimal diastolic inequality on the two-sphere, J. Topol. Anal. 2 (2010) 109

[4] C Bavard, Inégalité isosystolique pour la bouteille de Klein, Math. Ann. 274 (1986) 439

[5] C Bavard, Inégalités isosystoliques conformes pour la bouteille de Klein, Geom. Dedicata 27 (1988) 349

[6] C Bavard, Inégalités isosystoliques conformes, Comment. Math. Helv. 67 (1992) 146

[7] G Besson, G Courtois, S Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative, Geom. Funct. Anal. 5 (1995) 731

[8] R L Bryant, On extremals with prescribed Lagrangian densities, from: "Manifolds and geometry (Pisa, 1993)" (editors P de Bartolomeis, F Tricerri, E Vesentini), Sympos. Math. XXXVI, Cambridge Univ. Press (1996) 86

[9] D Burago, S Ivanov, On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume, Ann. of Math. $(2)$ 156 (2002) 891

[10] D Burago, S Ivanov, Boundary rigidity and filling volume minimality of metrics close to a flat one, Ann. of Math. $(2)$ 171 (2010) 1183

[11] E Calabi, Extremal isosystolic metrics for compact surfaces, from: "Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992)" (editor A L Besse), Sémin. Congr. 1, Soc. Math. France (1996) 167

[12] C B Croke, M Katz, Universal volume bounds in Riemannian manifolds, from: "Surveys in differential geometry, Vol. VIII (Boston, MA, 2002)" (editor S T Yau), Surv. Differ. Geom. VIII, Int. Press (2003) 109

[13] J Eells, L Lemaire, Deformations of metrics and associated harmonic maps, Proc. Indian Acad. Sci. Math. Sci. 90 (1981) 33

[14] A El Soufi, S Ilias, Riemannian manifolds admitting isometric immersions by their first eigenfunctions, Pacific J. Math. 195 (2000) 91

[15] M Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983) 1

[16] M Gromov, Metric structures for Riemannian and non-Riemannian spaces, Progress in Math. 152, Birkhäuser (1999)

[17] C Horowitz, K Usadi Katz, M G Katz, Loewner's torus inequality with isosystolic defect, J. Geom. Anal. 19 (2009) 796

[18] S V Ivanov, On two-dimensional minimal fillings, Algebra i Analiz 13 (2001) 26

[19] M G Katz, Systolic geometry and topology, Math. Surveys and Monogr. 137, Amer. Math. Soc. (2007)

[20] M G Katz, S Sabourau, Entropy of systolically extremal surfaces and asymptotic bounds, Ergodic Theory Dynam. Systems 25 (2005) 1209

[21] M G Katz, S Sabourau, Hyperelliptic surfaces are Loewner, Proc. Amer. Math. Soc. 134 (2006) 1189

[22] M G Katz, S Sabourau, An optimal systolic inequality for CAT$(0)$ metrics in genus two, Pacific J. Math. 227 (2006) 95

[23] N Nadirashvili, Berger's isoperimetric problem and minimal immersions of surfaces, Geom. Funct. Anal. 6 (1996) 877

[24] P M Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952) 55

[25] S Sabourau, Systoles des surfaces plates singulières de genre deux, Math. Z. 247 (2004) 693

[26] S Sabourau, Local extremality of the Calabi–Croke sphere for the length of the shortest closed geodesic, J. Lond. Math. Soc. $(2)$ 82 (2010) 549

[27] M Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793