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2017 Low energy canonical immersions into hyperbolic manifolds and standard spheres
Heberto del Rio, Walcy Santos, Santiago R. Simanca
Publ. Mat. 61(1): 135-151 (2017). DOI: 10.5565/PUBLMAT_61117_05

Abstract

We consider critical points of the global $L^2$-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.

Citation

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Heberto del Rio. Walcy Santos. Santiago R. Simanca. "Low energy canonical immersions into hyperbolic manifolds and standard spheres." Publ. Mat. 61 (1) 135 - 151, 2017. https://doi.org/10.5565/PUBLMAT_61117_05

Information

Received: 17 April 2015; Revised: 25 September 2015; Published: 2017
First available in Project Euclid: 22 December 2016

zbMATH: 1360.53062
MathSciNet: MR3590117
Digital Object Identifier: 10.5565/PUBLMAT_61117_05

Subjects:
Primary: 53C20
Secondary: 53C25 , 53C42 , 57R42 , 57R70

Keywords: canonically placed Riemannian manifold , critical point , embeddings , immersions , Mean curvature vector , second fundamental form

Rights: Copyright © 2017 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.61 • No. 1 • 2017
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