Low energy canonical immersions into hyperbolic manifolds and standard spheres
del Rio, Heberto (Barry University (Estats Units d'Amèrica). Department of Mathematics & Computer Sciences)
Santos, Walcy (Universidade Federal do Rio de Janeiro. Instituto de Matemática)
Simanca, Santiago R. (University of Miami (Estats Units d'Amèrica). Department of Mathematics)

Fecha:	2017
Resumen:	We consider critical points of the global L2-norm of the second fundamental form, and of the mean curvature vector of isometric immersions of compact Riemannian manifolds into a fixed background Riemannian manifold, as functionals over the space of deformations of the immersion. We prove new gap theorems for these functionals into hyperbolic manifolds, and show that the celebrated gap theorem for minimal immersions into the standard sphere can be cast as a theorem about their critical points having constant mean curvature function, and whose second fundamental form is suitably small in relation to it. In this case, the various minimal submanifolds that occur at the pointwise upper bound on the norm of the second fundamental form are realized by manifolds of nonnegative Ricci curvature, and of these, the Einstein ones are distinguished from the others by being those that are immersed on the sphere as critical points of the first of the functionals mentioned.
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Lengua:	Anglès
Documento:	Article ; recerca ; Versió publicada
Materia:	Immersions ; Embeddings ; Second fundamental form ; Mean curvature vector ; Critical point, canonically placed riemannian manifold
Publicado en:	Publicacions matemàtiques, Vol. 61 Núm. 1 (2017) , p. 135-151 (Articles) , ISSN 2014-4350

Adreça alternativa: https://raco.cat/index.php/PublicacionsMatematiques/article/view/316081
DOI: 10.5565/PUBLMAT_61117_05

17 p, 341.3 KB

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