Generalized isoperimetric inequalities for extrinsic balls in minimal submanifolds

Steen Markvorsen, Vicente Palmer

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    The volume of an extrinsic ball in a minimal submanifold has a well defined lower bound when the ambient manifold has an upper bound on its sectional curvatures, see e.g. [2] and [10]. When this upper bound is non-positive, the second named author has shown an isoperimetric inequality for such domains, see [11]. This result again gives the comparison result for volumes alluded to above together with a characterization of the totally geodesic submanifolds of hyperbolic space forms. In the present paper we find a corresponding sharp isoperimetric inequality for minimal submanifolds in spaces with sectional curvatures bounded from above by any constant. As a corollary we find again a characterization of the totally geodesic submanifolds of spherical space forms.
    Original languageEnglish
    JournalJournal fuer die Reine und Angewandte Mathematik
    Pages (from-to)101-121
    Publication statusPublished - 2002

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