We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces cannot exist in Euclidean space, but we show that the gradient of the ambient scalar curvature acts as a “friction term” which permits the usual analytic gluing construction to be carried out.
@article{ASNSP_2012_5_11_3_653_0,
author = {Butscher, Adrian and Mazzeo, Rafe},
title = {CMC hypersurfaces condensing to geodesic segments and rays in Riemannian manifolds},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 11},
number = {3},
year = {2012},
pages = {653-706},
zbl = {1260.53111},
mrnumber = {3059841},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2012_5_11_3_653_0}
}
Butscher, Adrian; Mazzeo, Rafe. CMC hypersurfaces condensing to geodesic segments and rays in Riemannian manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, pp. 653-706. http://www.numdam.org/item/ASNSP_2012_5_11_3_653_0/
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