Riv. Mat. Univ. Parma, to appear

Santiago R. Simanca[a],

ISOMETRIC EMBEDDINGS I: GENERAL THEORY

Pages:
Accepted: 7 November 2016
Mathematics Subject Classification (2010): 53C20, 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.
[a]:University of Miami, Department of Mathematics , Coral Gables, FL 33124, U.S.A.

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 $${S}^3 \hookrightarrow Sp(2)\stackrel {\pi_{\circ}}{\rightarrow} {S}^7$$ is a totally geodesic canonical representative of the generator $$D$$ of $$H_3(Sp(2);{Z})$$, and that this representative is unique up to isometries. For the nonrepresentable generator class of $$H_7(Sp(2);{Z})$$, 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.

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