Constant mean curvature hypersurfaces condensing along a submanifold
نویسندگان
چکیده
Constant mean curvature (CMC) hypersurfaces in a compact Riemannian manifold (Mm+1, g) constitute an important class of submanifolds and have been studied extensively. In this paper we are interested in degenerating families of such submanifolds which ‘condense’ to a submanifold Kk ⊂ Mm+1 of codimension greater than 1. It is not hard to see that the closer a CMC hypersurface is (e.g. in the Hausdorff metric) to such a submanifold, the larger its mean curvature must be; in other words, the mean curvatures of the elements of a condensing family of CMC hypersurfaces must tend to infinity. Less obvious is the fact that under fairly mild geometric assumptions, cf. [10], the existence of such a family implies that K is minimal. Two cases have been studied previously: Ye [14], [15] proved the existence of a local foliation by constant mean curvature hypersurfaces condensing to a point (which is required to be a nondegenerate critical point of the scalar curvature function); more recently, the second and third authors [10] proved the existence of a ‘partial foliation’ by constant mean curvature hypersurfaces in a neighborhood of a nondegenerate closed geodesic. In this paper we extend the result and methods of [10] to handle the general case, when K is an arbitrary nondegenerate minimal submanifold (no extra curvature hypotheses are required). As we explain below, this more general problem has a number of new analytic and geometric features, and despite the apparent similarities with the case when K is one-dimensional, is considerably more subtle to analyze. We now describe our result in more detail. Let Kk be a closed (embedded or immersed) submanifold in Mm+1, 1 ≤ k ≤ m− 1; the geodesic tube of radius ρ about K is
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