Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds

Mouhamed Moustapha Fall; Fethi Mahmoudi

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2008)

  • Volume: 7, Issue: 3, page 407-446
  • ISSN: 0391-173X

Abstract

top
Given a domain Ω of ℝ m + 1 and a k -dimensional non-degenerate minimal submanifold K of ∂ Ω with 1 ≤ k ≤ m - 1 , we prove the existence of a family of embedded constant mean curvature hypersurfaces in Ω which as their mean curvature tends to infinity concentrate along K and intersecting ∂ Ω perpendicularly along their boundaries.

How to cite

top

Fall, Mouhamed Moustapha, and Mahmoudi, Fethi. "Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 7.3 (2008): 407-446. <http://eudml.org/doc/272255>.

@article{Fall2008,
abstract = {Given a domain $\Omega $ of $\mathbb \{R\}^\{m+1\}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\partial \Omega $ with $1\le k\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces in $\Omega $ which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial \Omega $ perpendicularly along their boundaries.},
author = {Fall, Mouhamed Moustapha, Mahmoudi, Fethi},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {407-446},
publisher = {Scuola Normale Superiore, Pisa},
title = {Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds},
url = {http://eudml.org/doc/272255},
volume = {7},
year = {2008},
}

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
PB - Scuola Normale Superiore, Pisa
VL - 7
IS - 3
SP - 407
EP - 446
AB - Given a domain $\Omega $ of $\mathbb {R}^{m+1}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\partial \Omega $ with $1\le k\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces in $\Omega $ which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial \Omega $ perpendicularly along their boundaries.
LA - eng
UR - http://eudml.org/doc/272255
ER -

References

top
  1. [1] W. Bürger and E. Kuwert, Area-minimizing disks with free boundary and prescribed enclosed volume, J. Reine Angew. Math., to appear. Zbl1166.53005MR2431248
  2. [2] M. M. Fall, Embedded disc-type surfaces with large constant mean curvature and free boundaries, Commun. Contemp. Math., to appear. Zbl1264.53011MR2989641
  3. [3] R. Finn, “Equilibrium Capillary Surfaces”, Springer-Verlag, New York, 1986. Zbl0583.35002MR816345
  4. [4] M. Grüter and J. Jost, On embedded minimal discs in convex bodies, Ann. Inst. H. Poincaré, Anal. Non Linéaire 3 (1986), 345–390. Zbl0617.49017MR868522
  5. [5] G. Huisken and S. T. Yau, 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. Zbl0858.53071MR1369419
  6. [6] J. Jost, Existence results for embedded minimal surfaces of controlled topological type I, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (4) 13 (1986), 15–50. Zbl0619.49019MR863634
  7. [7] T. Kato, “Perturbation Theory for Linear Operators”, GMW 132, Springer-Verlag, 1976. Zbl0342.47009MR407617
  8. [8] H. B. Lawson, Complete minimal surfaces in S 3 , Ann. of Math. (2) 92 (1970), 335–374. Zbl0205.52001MR270280
  9. [9] H. B. Lawson, “Lectures on Minimal Submanifolds”, Vol. I, second edition, Mathematics Lecture Series, 9, Publish or Perish, Wimington, Del., 1980. Zbl0434.53006MR576752
  10. [10] F. Mahmoudi, R. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing along a submanifold, Geom. Funct. Anal.16 (2006), 924–958. Zbl1108.53031MR2255386
  11. [11] F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math.209 (2007), 460–525. Zbl1160.35011MR2296306
  12. [12] A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, Geom. Funct. Anal., 15 (2005), 1162–1222. Zbl1087.35010MR2221246
  13. [13] A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math.55 (2002), 1507–1568. Zbl1124.35305MR1923818
  14. [14] A. Malchiodi and M. Montenegro, Multidimensional Boundary-layers for a singularly perturbed Neumann problem, Duke Math. J.124 (2004), 105–143. Zbl1065.35037MR2072213
  15. [15] R. Mazzeo and F. Pacard, Foliations by constant mean curvature tubes, Comm. Anal. Geom.13 (2005), 633–670. Zbl1096.53035MR2191902
  16. [16] M. Ritoré and C. Rosales, Existence and charaterization of regions minimizing perimeter under a volume constraint inside Euclidean cones, Trans. Amer. Math. Soc. 356, 4601–4622. Zbl1057.53023MR2067135
  17. [17] A. Ros, “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. Zbl1125.49034MR2167260
  18. [18] A. Ros and R. Souam, On stability of capillary surfaces in a ball, Pacific J. Math.178 (1997), 345–361. Zbl0930.53007MR1447419
  19. [19] A. Ros and E. Vergasta, Stability for hypersurfaces of constant mean curvature with free boundary, Geom. Dedicata56 (1995), 19–33. Zbl0912.53009MR1338315
  20. [20] R. Schoen and S. T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. (2) 110 (1979), 127–142. Zbl0431.53051MR541332
  21. [21] P. Sternberg and K. Zumbrun, Connectivity of phase boundaries in strictly convex domains, Arch. Ration. Mech. Anal.141 (1998), 375–400. Zbl0911.49025MR1620498
  22. [22] M. Struwe, Non-uniqueness in the Plateau problem for surfaces of constant mean curvature, Arch. Ration. Mech. Anal.93 (1986), 135–157. Zbl0603.49027MR823116
  23. [23] M. Struwe, The existence of surfaces of constant mean curvature with free boundaries, Acta Math.160 (1988), 19–64. Zbl0646.53005MR926524
  24. [24] M. Struwe, On a free boundary problem for minimal surfaces, Invent. Math.75 (1984), 547–560. Zbl0537.35037MR735340
  25. [25] R. Ye, Foliation by constant mean curvature spheres, Pacific J. Math.147 (1991), 381–396. Zbl0722.53022MR1084717
  26. [26] T. J. Willmore, “Riemannian Geometry”, Oxford Univ. Press. NY., 1993. Zbl0797.53002MR1261641

NotesEmbed ?

top

You must be logged in to post comments.