An Arzela-Ascoli theorem for immersed submanifolds
Graham Smith[1]
- [1] Equipe de topologie et dynamique, Laboratoire de mathématiques, Bâtiment 425, UFR des sciences d’Orsay, UMR 8628 du CNRS, 91405 Orsay cedex (France)
Annales de la faculté des sciences de Toulouse Mathématiques (2007)
- Volume: 16, Issue: 4, page 817-866
- ISSN: 0240-2963
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topSmith, Graham. "An Arzela-Ascoli theorem for immersed submanifolds." Annales de la faculté des sciences de Toulouse Mathématiques 16.4 (2007): 817-866. <http://eudml.org/doc/10071>.
@article{Smith2007,
abstract = {The classical Arzela-Ascoli theorem is a compactness result for families of functions depending on bounds on the derivatives of the functions, and is of invaluable use in many fields of mathematics. In this paper, inspired by a result of Corlette, we prove an analogous compactness result for families of immersed submanifolds which depends only on bounds on the derivatives of the second fundamental forms of these submanifolds. We then show how the result of Corlette may be obtained as an immediate corollary.},
affiliation = {Equipe de topologie et dynamique, Laboratoire de mathématiques, Bâtiment 425, UFR des sciences d’Orsay, UMR 8628 du CNRS, 91405 Orsay cedex (France)},
author = {Smith, Graham},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Arzela-Ascoli theorem; finiteness theorem; immersions; Cheeger-Gromov topology; compactness result},
language = {eng},
number = {4},
pages = {817-866},
publisher = {Université Paul Sabatier, Toulouse},
title = {An Arzela-Ascoli theorem for immersed submanifolds},
url = {http://eudml.org/doc/10071},
volume = {16},
year = {2007},
}
TY - JOUR
AU - Smith, Graham
TI - An Arzela-Ascoli theorem for immersed submanifolds
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2007
PB - Université Paul Sabatier, Toulouse
VL - 16
IS - 4
SP - 817
EP - 866
AB - The classical Arzela-Ascoli theorem is a compactness result for families of functions depending on bounds on the derivatives of the functions, and is of invaluable use in many fields of mathematics. In this paper, inspired by a result of Corlette, we prove an analogous compactness result for families of immersed submanifolds which depends only on bounds on the derivatives of the second fundamental forms of these submanifolds. We then show how the result of Corlette may be obtained as an immediate corollary.
LA - eng
KW - Arzela-Ascoli theorem; finiteness theorem; immersions; Cheeger-Gromov topology; compactness result
UR - http://eudml.org/doc/10071
ER -
References
top- Cheeger (J.).— Finiteness theorems for Riemannian manifolds, Amer. J. Math.92, p. 61-74 (1970). Zbl0194.52902MR263092
- Corlette (K.).— Immersions with bounded curvature, Geom. Dedicata33, no. 2, p. 153-161 (1990). Zbl0717.53035MR1050607
- Gromov (M.).— Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics, 152, Birkhäuser, Boston, (1998). Zbl0953.53002MR1699320
- Petersen (P.).— Riemannian Geometry, Graduate Texts in Mathematics, 171, Springer Verlag, New York, (1998). Zbl0914.53001MR1480173
- Smith (G.).— Special Legendrian structures and Weingarten problems, Preprint, Orsay (2005).
- Smith (G.).— Thèse de doctorat, Paris (2004).
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