CMC hypersurfaces condensing to geodesic segments and rays in Riemannian manifolds
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.

Publié le :
Classification : 53A10, 35J93, 35B25
Butscher, Adrian 1 ; Mazzeo, Rafe 1

1 Department of Mathematics Stanford University Stanford, CA 94305 USA
@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},
     pages = {653--706},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 11},
     number = {3},
     year = {2012},
     mrnumber = {3059841},
     zbl = {1260.53111},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2012_5_11_3_653_0/}
}
TY  - JOUR
AU  - Butscher, Adrian
AU  - Mazzeo, Rafe
TI  - CMC hypersurfaces condensing to geodesic segments and rays in Riemannian manifolds
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2012
SP  - 653
EP  - 706
VL  - 11
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2012_5_11_3_653_0/
LA  - en
ID  - ASNSP_2012_5_11_3_653_0
ER  - 
%0 Journal Article
%A Butscher, Adrian
%A Mazzeo, Rafe
%T CMC hypersurfaces condensing to geodesic segments and rays in Riemannian manifolds
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2012
%P 653-706
%V 11
%N 3
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2012_5_11_3_653_0/
%G en
%F 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, Série 5, Tome 11 (2012) no. 3, pp. 653-706. http://archive.numdam.org/item/ASNSP_2012_5_11_3_653_0/

[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/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