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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.

It has been shown that in a mobile ad hoc wireless network per node throughput remains constant as the number of nodes approaches infinity, when the nodes lie in a disk of unit area. However, this requires that some packets are delivered indirectly by a relay node. In this paper, we prove, in the same setup, that D(k) 2 Â(log k), where D(k) is the expected delay time of a mobile ad hoc wireless network with k nodes. As consequence, we have concluded that mobile ad hoc wireless networks are not scalable for real-time applications. z Observe that the rôle played by the unitary disk in Reference [6] is not of importance for their results. For example, the estimate lim z!0 FðzÞ=z 2= ¼ , obtained in Reference [6], does not depend on the fact that the nodes lie on the unitary disk. This estimate only depends on the fact that the measure used on the unitary disk is the restriction of the Lebesgue measure. The same estimate can be obtained for the unitary square, see Reference [11].

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