# Surfaces de Riemann parfaites en petit genre

Abstract : This thesis is devoted to the search of extremal compact Riemann surfaces (i.e. which are local maximals) for the systole, or at least perfect surfaces. In genus~4, we construct a new extremal surface and two perfect non-extremal surfaces (these are the first examples of such surfaces in genus $\leq 10$). The idea of the method consists to realise geometrically automorphism groups which act with 4 branching points. Indeed, the locus of fixed points (in Teichmüller space $T_g$) by such a group, depends on a complex paramater that we can then ajust to maximize the systole ; we study therefore the variational properties in $T_g$ of these candidates. Extending this method, we construct a new extremal surface of genus 6, as an infinite sequence of perfect non-extremal surfaces of genus $>3$. Moreover we rediscover by this way the already-known surfaces of genus $\leq 5$. The method employed to search perfect surfaces, also gives many eutactics surfaces which are interesting in that that they are the critical points of the systole function. Finally, the last chapter, concerns a completely different point of view and developps a purely algebraic approach which gives a new proof of the extremality of both surfaces, that of Bolza and that of Klein.
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https://tel.archives-ouvertes.fr/tel-00005446
Contributor : Alexandre Casamayou <>
Submitted on : Tuesday, March 23, 2004 - 4:43:08 PM
Last modification on : Thursday, January 11, 2018 - 6:12:18 AM
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• HAL Id : tel-00005446, version 1

### Citation

Alexandre Casamayou. Surfaces de Riemann parfaites en petit genre. Mathématiques [math]. Université Sciences et Technologies - Bordeaux I, 2000. Français. ⟨tel-00005446⟩

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