Given a domain of and a -dimensional non-degenerate minimal submanifold of with , we prove the existence of a family of embedded constant mean curvature hypersurfaces in which as their mean curvature tends to infinity concentrate along and intersecting perpendicularly along their boundaries.
@article{ASNSP_2008_5_7_3_407_0, author = {Fall, Mouhamed Moustapha and Mahmoudi, Fethi}, title = {Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {407--446}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 7}, number = {3}, year = {2008}, mrnumber = {2466435}, zbl = {1171.53010}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2008_5_7_3_407_0/} }
TY - JOUR AU - Fall, Mouhamed Moustapha AU - Mahmoudi, Fethi TI - Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2008 SP - 407 EP - 446 VL - 7 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2008_5_7_3_407_0/ LA - en ID - ASNSP_2008_5_7_3_407_0 ER -
%0 Journal Article %A Fall, Mouhamed Moustapha %A Mahmoudi, Fethi %T Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2008 %P 407-446 %V 7 %N 3 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2008_5_7_3_407_0/ %G en %F ASNSP_2008_5_7_3_407_0
Fall, Mouhamed Moustapha; Mahmoudi, Fethi. Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 7 (2008) no. 3, pp. 407-446. http://www.numdam.org/item/ASNSP_2008_5_7_3_407_0/
[1] Area-minimizing disks with free boundary and prescribed enclosed volume, J. Reine Angew. Math., to appear. | MR | Zbl
and ,[2] Embedded disc-type surfaces with large constant mean curvature and free boundaries, Commun. Contemp. Math., to appear. | MR | Zbl
,[3] “Equilibrium Capillary Surfaces”, Springer-Verlag, New York, 1986. | MR | Zbl
,[4] On embedded minimal discs in convex bodies, Ann. Inst. H. Poincaré, Anal. Non Linéaire 3 (1986), 345-390. | Numdam | MR | Zbl
and ,[5] Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature, Invent. Math. 124 (1996), 281-311. | MR | Zbl
and ,[6] Existence results for embedded minimal surfaces of controlled topological type I, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (4) 13 (1986), 15-50. | Numdam | MR | Zbl
,[7] “Perturbation Theory for Linear Operators”, GMW 132, Springer-Verlag, 1976. | MR | Zbl
,[8] Complete minimal surfaces in , Ann. of Math. (2) 92 (1970), 335-374. | MR | Zbl
,[9] “Lectures on Minimal Submanifolds”, Vol. I, second edition, Mathematics Lecture Series, 9, Publish or Perish, Wimington, Del., 1980. | MR | Zbl
,[10] Constant mean curvature hypersurfaces condensing along a submanifold, Geom. Funct. Anal. 16 (2006), 924-958. | MR | Zbl
, and ,[11] Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math. 209 (2007), 460-525. | MR | Zbl
and ,[12] Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, Geom. Funct. Anal., 15 (2005), 1162-1222. | MR | Zbl
,[13] Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math. 55 (2002), 1507-1568. | MR | Zbl
and ,[14] Multidimensional Boundary-layers for a singularly perturbed Neumann problem, Duke Math. J. 124 (2004), 105-143. | MR | Zbl
and ,[15] Foliations by constant mean curvature tubes, Comm. Anal. Geom. 13 (2005), 633-670. | MR | Zbl
and ,[16] Existence and charaterization of regions minimizing perimeter under a volume constraint inside Euclidean cones, Trans. Amer. Math. Soc. 356, 4601-4622. | MR | Zbl
and ,[17] “The Isoperimetric Problem”, Lecture series given during the Caley Mathematics Institute Summer School on the Global Theory of Minimal Surfaces at the MSRI, Berkley, California, 2001. | MR | Zbl
,[18] On stability of capillary surfaces in a ball, Pacific J. Math. 178 (1997), 345-361. | MR | Zbl
and ,[19] Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata 56 (1995), 19-33. | MR | Zbl
and ,[20] Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), 127-142. | MR | Zbl
and ,[21] Connectivity of phase boundaries in strictly convex domains, Arch. Ration. Mech. Anal. 141 (1998), 375-400. | MR | Zbl
and ,[22] Non-uniqueness in the Plateau problem for surfaces of constant mean curvature, Arch. Ration. Mech. Anal. 93 (1986), 135-157. | MR | Zbl
,[23] The existence of surfaces of constant mean curvature with free boundaries, Acta Math. 160 (1988), 19-64. | MR | Zbl
,[24] On a free boundary problem for minimal surfaces, Invent. Math. 75 (1984), 547-560. | MR | Zbl
,[25] Foliation by constant mean curvature spheres, Pacific J. Math. 147 (1991), 381-396. | MR | Zbl
,[26] “Riemannian Geometry”, Oxford Univ. Press. NY., 1993. | MR | Zbl
,