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Low energy canonical immersions into hyperbolic manifolds and standard spheres

Heberto del Rio, Walcy Santos, and Santiago R. Simanca

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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.

Article information

Publ. Mat., Volume 61, Number 1 (2017), 135-151.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 57R42: Immersions 57R70: Critical points and critical submanifolds

Immersions embeddings second fundamental form mean curvature vector critical point canonically placed Riemannian manifold


del Rio, Heberto; Santos, Walcy; Simanca, Santiago R. Low energy canonical immersions into hyperbolic manifolds and standard spheres. Publ. Mat. 61 (2017), no. 1, 135--151. doi:10.5565/PUBLMAT_61117_05.

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