Low energy canonical immersions into hyperbolic manifolds and standard spheres

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 the...

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Detalles Bibliográficos
Autores: del Rio, Heberto, Santos, Walcy, Simanca, Santiago R.
Tipo de recurso: artículo
Fecha de publicación:2017
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:168347
Acceso en línea:https://ddd.uab.cat/record/168347
https://dx.doi.org/urn:doi:10.5565/PUBLMAT_61117_05
Access Level:acceso abierto
Palabra clave:Immersions
Embeddings
Second fundamental form
Mean curvature vector
Critical point, canonically placed riemannian manifold
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spelling Low energy canonical immersions into hyperbolic manifolds and standard spheresdel Rio, HebertoSantos, WalcySimanca, Santiago R.ImmersionsEmbeddingsSecond fundamental formMean curvature vectorCritical point, canonically placed riemannian manifoldWe 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. 22017-01-0120172017-01-01Articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/168347https://dx.doi.org/urn:doi:10.5565/PUBLMAT_61117_05reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:1683472026-06-06T12:50:31Z
dc.title.none.fl_str_mv Low energy canonical immersions into hyperbolic manifolds and standard spheres
title Low energy canonical immersions into hyperbolic manifolds and standard spheres
spellingShingle Low energy canonical immersions into hyperbolic manifolds and standard spheres
del Rio, Heberto
Immersions
Embeddings
Second fundamental form
Mean curvature vector
Critical point, canonically placed riemannian manifold
title_short Low energy canonical immersions into hyperbolic manifolds and standard spheres
title_full Low energy canonical immersions into hyperbolic manifolds and standard spheres
title_fullStr Low energy canonical immersions into hyperbolic manifolds and standard spheres
title_full_unstemmed Low energy canonical immersions into hyperbolic manifolds and standard spheres
title_sort Low energy canonical immersions into hyperbolic manifolds and standard spheres
dc.creator.none.fl_str_mv del Rio, Heberto
Santos, Walcy
Simanca, Santiago R.
author del Rio, Heberto
author_facet del Rio, Heberto
Santos, Walcy
Simanca, Santiago R.
author_role author
author2 Santos, Walcy
Simanca, Santiago R.
author2_role author
author
dc.subject.none.fl_str_mv Immersions
Embeddings
Second fundamental form
Mean curvature vector
Critical point, canonically placed riemannian manifold
topic Immersions
Embeddings
Second fundamental form
Mean curvature vector
Critical point, canonically placed riemannian manifold
description 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.
publishDate 2017
dc.date.none.fl_str_mv 2
2017-01-01
2017
2017-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/168347
https://dx.doi.org/urn:doi:10.5565/PUBLMAT_61117_05
url https://ddd.uab.cat/record/168347
https://dx.doi.org/urn:doi:10.5565/PUBLMAT_61117_05
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
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dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
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eu_rights_str_mv openAccess
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dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
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