Survey on optimal isosystolic inequalities on the real projective plane

• Autores: Unai Lejarza Alonso
• Localización: Reports@SCM: an electronic journal of the Societat Catalana de Matemàtiques, ISSN-e 2385-4227, Vol. 9, Nº. 1, 2024, págs. 21-30
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen
  • català

    Es revisen totes les desigualtats isosistòliques òptimes conegudes al pla projectiu real RP2, comparant-les amb el cas del 2-tor T2. Primer s’introdueixen nocions bàsiques de mètriques de Finsler. Després s’enuncien totes les desigualtats isosistòliques conegudes pel cas reversible i se’n dona la idea de prova. Finalment es tracten les desigualtats òptimes pel cas no-reversible. Actualment es coneixen totes les desigualtats òptimes per T2, tot i que no és així per RP2. S’hi presenten alguns petits progressos i arguments a favor de la desigualtat conjecturada en el cas encara obert.

  • English

    All known optimal isosystolic inequalities on the real projective plane RP2 are surveyed, comparing them to the case of the 2-torus T2. First, basic notions on Finsler metrics are introduced. Then, all previously known isosystolic inequalities are stated and a sketch of proof is given in the reversible case. Finally, optimal inequalities in the non-reversible case are discussed. All optimal inequalities are currently known for T2, although this is not the case for RP2. Some recent minor advances for RP2 are presented, and some arguments are given in favour of the conjectured inequality in the remaining open case.

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