Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities

• S. Semmes ^[1]
  1. [1] Rice University

     Rice University

     Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 2, Nº. 2, 1996, págs. 154-155
• Idioma: inglés
• DOI: 10.1007/bf01587936
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• Resumen

  • In many metric spaces one can connect an arbitrary pair of points with a curve of finite length, but in Euclidean spaces one can connect a pair of points with a lot of rectifiable curves, curves that are well distributed across a region. In the present paper we give geometric criteria on a metric space under which we can find similar families of curves. We shall find these curves by first solving a “dual” problem of building Lipschitz maps from our metric space into a sphere with good topological properties. These families of curves can be used to control the values of a function in terms of its gradient (suitably interpreted on a general metric space), and to derive Sobolev and Poincaré inequalities.