# zbMATH — the first resource for mathematics

Riemann surfaces with shortest geodesic of maximal length. (English) Zbl 0810.53034
The following questions are treated: 1. For a fixed genus $$g$$, the Riemann surface of constant curvature $$-1$$ (called global maximal surface) is searched where the length of the systole, the shortest simple closed geodesic, is a global maximum in the Teichmüller space $$T_ g$$. 2. All surfaces (called maximal surfaces) should be found where the length of the systole is a local maximum in $$T_ g$$. The main result describing properties of these surfaces are the following: Theorem 1.1 A maximal surface of genus $$g$$ has at least $$6g - 5$$ shortest simple closed geodesics of equal length. Theorem 1.2. It is necessary and sufficient for a surface $$M$$ of genus $$g$$ to be a maximal surface: (i) $$M$$ is strongly $$F$$-minimal; (ii) $$M$$ is $$F$$-regular. Examples of such surfaces are given. So, maximal surfaces of signature $$(1,n)$$ and (2,2), maximal surfaces of genus 2 and maximal surfaces which are modelled upon Euclidean polyhedra are treated. Finally, two important maximal surfaces in genus 3 and 4 which may be global maximal surfaces are studied.

##### MSC:
 53C22 Geodesics in global differential geometry 30F60 Teichmüller theory for Riemann surfaces
##### Keywords:
Teichmüller space; systole; maximal surfaces
Full Text:
##### References:
 [1] [A]R.D.M. Accola, On the number of automorphisms of a closed Riemann surface, Trans. A.M.S. 131 (1968), 398–408. · Zbl 0179.11401 [2] [B1]C. Bavard, Inégalités isosystoliques conformes pour la bouteille de Klein, Geometriae Dedicata 27 (1988), 349–355. · Zbl 0667.53033 [3] [B2]C. Bavard, La systole des surfaces hyperelliptiques, Prépubl. no 71 ENS Lyon (1992). [4] [BeMaSi]A.-M. Bergé, J. Martinet, F. Siegrist, Une généralisation de l’algorithme de Voronoï pour les formes quadratiques, Astérisque 209 (1992), 137–158. [5] [Ber]M. Berger, Lectures on Geodesics in Riemannian Geometry, Tata Institute, Bombay (1965). · Zbl 0165.55601 [6] [Bers1]L. Bers, Nielsen extensions of Riemann surfaces, Ann. Aca. Scient. Fennicae, Series A.I.Mathematica Vol. 2 (1976), 29–34. · Zbl 0352.30014 [7] [Bers2]L. Bers, Spaces of degenerating Riemann surfaces, in ”Discontinuous Groups and Riemann Surfaces’, Princeton Univ. Press, Princeton (1974), 43–55. [8] [Br]H.R. Brahana, Regular maps and their groups, Amer. J. of Math. 48 (1927), 268–284. · JFM 53.0549.01 [9] [Bro]R. Brooks, Some relations between spectral geometry and number theory, Preprint. [10] [Bu]P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Birkhäuser Basel-Boston-New York 1992. · Zbl 0770.53001 [11] [BuS]P. Buser, P. Sarnak, On the period matrix of a Riemann surface of large genus, Preprint 1992. [12] [C]E. Calabi, Isosystolic Problems, Math. Forschungsinstitut Oberwolfach, Tagungsbericht 22 (1991). [13] [CoSl]J. Conway, N. Sloane, Sphere packings, lattices and groups, Springer Berlin Heidelberg New York Tokyo 1988. · Zbl 0634.52002 [14] [CoxMo]H.S.M. Coxeter, W.O.J. Moser, Generators and Relations for Discrete Groups, Springer Berlin-New York 1965. · Zbl 0133.28002 [15] [F]L. Fejes Tóth, Regular Figures, Pergamon Press Oxford (1964). [16] [G]M. Gromov, Filling Riemannian manifolds, J. Differential Geometry 18 (1983), 1–147. · Zbl 0515.53037 [17] [J1]F.W. Jenny, Ueber das Spektrum des Laplace-Operators auf einer Schar kompakter Riemannscher Flächen, Dissertation. Basel 1981. [18] [J2]F.W. Jenny, Ueber den ersten Eigenwert des Laplace-Operators auf ausgewählten Beispielen kompakter Riemannscher Flächen, Comment. Math. Helvetici 59 (1984), 193–203. · Zbl 0541.30034 [19] [K1]S. Kerckhoff, The Nielsen realization problem, Annals of Math. 117 (1983), 235–265. · Zbl 0528.57008 [20] [K2]S. Kerckhoff, Earthquakes are analytic, Comment. Math. Helvetici 60 (1985), 17–30. · Zbl 0575.32024 [21] [K3]S. Kerckhoff, Lines of minima in Teichmüller space, Duke Math. J. 65 (1992), 187–213. · Zbl 0771.30043 [22] [KiKu]H. Kimura, A. Kuribayashi, Automorphism groups of compact Riemann surfaces of genus five, J. of Algebra 134 (1990), 80–103. · Zbl 0709.30037 [23] [KuKur]A. Kuribayashi, I. Kuribayashi, Automorphism groups of compact Riemann surfaces of genera three and four, J. of Pure and Applied Algebra 65 (1990), 277–292. · Zbl 0709.30036 [24] [Kur]I. Kuribayashi, On an algebraization of the Riemann-Hurwitz relation, Kodai Math. J. 7 (1984), 222–237. · Zbl 0575.32025 [25] [M]C. Maclachlan, A bound for the number of automorphisms of a Compact Riemann Surface, J. London Math. Soc. 44 (1969), 265–272. · Zbl 0164.37904 [26] [Mu]D. Mumford, A remark on Mahler’s compactness theorem, Proc. AMS 28 (1971), 289–294. · Zbl 0215.23202 [27] [N]M. Näätänen, Regularn-Gons and Fuchsian Groups, Ann. Aca. Scient. Fennicae, Series A.I.Mathematica 7 (1982), 291–300. · Zbl 0475.30038 [28] [Ni]J. Nielsen, Collected Papers, vol. 2, Birkhäuser, Basel-Boston-New York 1986. [29] [Sc1]P. Schmutz, Die Parametrisierung des Teichmüllerraumes durch geodätische Längenfunktionen, Comment. Math. Helvetici 68 (1993), 278–288. · Zbl 0790.30036 [30] [Sc2]P. Schmutz, Congruence subgroups and maximal Riemann surfaces, to appear in The Journal of Geometric Analysis. [31] [T]W.P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, in ”Low-dimensional Topology and Kleinian Groups, Warwick and Durham, 1984” (D.B.A. Epstein, ed.), London Math. Soc. Lecture Note Ser. 112, Cambridge Univ. Press, Cambridge, 1986, 91–112. [32] [V]G. Voronoï, Sur quelques propriétés des formes quadratiques positives parfaites, J. reine angew. Math. 133 (1908), 97–178. · JFM 38.0261.01 [33] [W]S. Wolpert, Geodesic Length Functions and the Nielsen Problem, J. Differential Geometry 25 (1987), 275–296. · Zbl 0616.53039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.