CMC hypersurfaces condensing to geodesic segments and rays in Riemannian manifolds

Butscher, Adrian; Mazzeo, Rafe

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 3, pp. 653-706.

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.

Classification : 53A10, 35J93, 35B25

[1] A. Butscher and F. Pacard, Doubling constant mean curvature tori in $S^3$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), 611–638. | EuDML | Numdam | MR | Zbl

[2] A. Butscher and F. Pacard, Generalized doubling constructions for constant mean curvature hypersurfaces in $S^{n+1}$, Ann. Global Anal. Geom. 32 (2007), 103–123. | MR | Zbl

[3] K. Grosse-Brauckmann, N. Korevaar, R. Kusner, J. Ratzkin and J. Sullivan, Coplanar k-unduloids are nondegenerate, Preprint: arXiv:0712.1865. | MR | Zbl

[4] K. Grosse-Brauckmann, R. Kusner and J. Sullivan, Triunduloids: embedded constant mean curvature surfaces with three ends and genus zero, J. Reine Angew. Math. 564 (2003), 35–61. | MR | Zbl

[5] K. Grosse-Brauckmann, R. Kusner and J. Sullivan, Coplanar constant mean curvature surfaces, Comm. Anal. Geom. 15 (2007), no. 5, 985–1023. | MR | Zbl

[6] N. Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (1990), 239–330. | MR | Zbl

[7] N. Kapouleas, Compact constant mean curvature surfaces in Euclidean three-space, J. Differential Geom. 33 (1991), 683–715. | MR | Zbl

[8] N. Korevaar, R. Kusner and B. Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), 465–503. | MR | Zbl

[9] R. Kusner, R. Mazzeo and D. Pollack, The moduli space of complete embedded constant mean curvature surfaces, Geom. Func. Anal. 6 (1996), 120–137. | EuDML | MR | Zbl

[10] R. Mazzeo and F. Pacard, Constant mean curvature surfaces with Delaunay ends, Comm. Anal. Geom. 9 (2001), 169–237. | MR | Zbl

[11] R. Mazzeo and F. Pacard, Foliations by constant mean curvature tubes, Comm. Anal. Geom. 13 (2005), 633–670. | MR | Zbl

[12] R. Mazzeo, F. Pacard and D. Pollack, Connected sums of constant mean curvature surfaces in Euclidean 3 space, J. Reine Angew. Math. 536 (2001), 115–165. | MR | Zbl

[13] R. Mazzeo, Recent advances in the global theory of constant mean curvature surfaces, In: “Noncompact Problems at the Intersection of Geometry, Analysis, and Topology”, Contemp. Math., Vol. 350, Amer. Math. Soc., Providence, RI, 2004, 179–199. | MR | Zbl

[14] W. H. Meeks, III, The topology and geometry of embedded surfaces of constant mean curvature, J. Differential Geom. 27 (1988), 539–552. | MR | Zbl

[15] F. Pacard, Connected Sum Constructions in Geometry and Nonlinear Analysis, preprint: http://perso-math.univ-mlv.fr/users/pacard.frank/Lecture-Part-I.pdf.

[16] F. Pacard, Surfaces à courbure moyenne constante, In: “Image des mathématiques 2006”, Publications of the CNRS, CNRS, Paris, 2006, 107–112.

[17] J. Ratzkin, “An End-to-End Construction for Constant Mean Curvature Surfaces”, Ph.D. thesis, University of Washington, 2001, preprint: http://www.math.uga.edu/$\sim$jratzkin/papers/thesis.pdf. | MR

[18] H. Rosenberg, Constant mean curvature surfaces in homogeneously regular 3-manifolds, Bull. Austral. Math. Soc. 74 (2006), 227–238. | MR | Zbl

[19] R. M. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure Appl. Math. 41 (1988), 317–392. | MR | Zbl

[20] R. Ye, Foliation by constant mean curvature spheres, Pacific J. Math. 147 (1991), 381–396. | MR | Zbl