Santiago R. Simanca^[a],

Isometric embeddings I: General theory

Pages: 307-343
Received: 16 September 2016
Accepted: 7 November 2016
Mathematics Subject Classification (2010): Primary: 53C20, Secondary: 53C21, 53C25, 53C42, 57R40, 57R70.
Keywords: Immersions, embeddings, second fundamental form, mean curvature vector, critical point, canonically placed Riemannian manifold, shape of a homology class
Author address:
[a]:University of Miami, Department of Mathematics, Coral Gables, FL 33124, U.S.A.

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Abstract: We consider critical points of the global squared \(L^2\)-norms of the second fundamental form, \(\Pi(M)\), and the mean curvature vector, \(\Psi(M)\), of isometric immersions of \((M,g)\) into a fixed background Riemannian manifold \((\tilde{M},\tilde{g})\) under deformations of the immersion. We use the critical points of \(\Pi\) to define canonical representatives of a given integer homology class of \(\tilde{M}\). With a suitable set of left-invariant metrics on \(Sp(2)\), we prove that any fiber of the fibration \(\mathbb{S}^3 \hookrightarrow Sp(2)\stackrel {\pi_{\circ}}{\rightarrow} \mathbb{S}^7\) is a totally geodesic canonical representative of the generator \(D\) of \(H_3(Sp(2);\mathbb{Z})\), and that this representative is unique up to isometries. For the nonrepresentable generator class of \(H_7(Sp(2);\mathbb{Z})\), we prove also that the absolute minimum of \(\Pi\) is achieved by immersed representatives that are not embedded. Finally, for the functional \(\Pi-\Psi\), we exhibit examples of background manifolds \((\tilde{M},\tilde{g})\) admitting isotopically equivalent critical hypersurfaces of distinct critical values.

To the memory of our beloved Gracie