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The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 1001–1017 | Cite as

Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds

  • Lucas C. AmbrozioEmail author
Article

Abstract

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru (Splitting of 3-manifolds and rigidity of area-minimizing surfaces, arXiv:1107.5346, 2011). We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.

Keywords

Free boudary minimal surfaces Scalar curvature Mean curvature Rigidity 

Mathematics Subject Classification

53A10 53C24 

Notes

Acknowledgements

I am grateful to my Ph.D advisor at IMPA, Fernando Codá Marques, for his constant advice and encouragement. I also thank Ivaldo Nunes for enlightening discussions about free boundary surfaces. Finally, I am grateful to the hospitality of the Institut Henri Poincaré, where the first drafts of this work were written in October/November 2012. I was supported by CNPq-Brazil and FAPERJ.

References

  1. 1.
    Bray, H.: The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature. Thesis, Stanford University (1997) Google Scholar
  2. 2.
    Bray, H., Brendle, S., Neves, A.: Rigidity of area-minimizing two-spheres in three-manifolds. Commun. Anal. Geom. 18(4), 821–830 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cai, M., Galloway, G.: Rigidity of area-minimizing tori in 3-manifolds of nonnegative scalar curvature. Commun. Anal. Geom. 8(3), 565–573 (2000) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Chen, J., Fraser, A., Pang, C.: Minimal immersions of compact bordered Riemann surfaces with free boundary. arXiv:1209.1165
  5. 5.
    Fischer-Colbrie, D., Schoen, R.: The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Commun. Pure Appl. Math. 33(2), 199–211 (1980) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Huisken, G., Yau, S.-T.: Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124(1–3), 281–311 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kazdan, J., Warner, F.: Prescribing curvatures. In: Differential Geometry. Proc. Sympos. Pure Math., vol. 27, pp. 309–319. Am. Math. Soc., Providence (1975) CrossRefGoogle Scholar
  8. 8.
    Ladyzhenskaia, O., Uralt’seva, N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968), 495 pp. Google Scholar
  9. 9.
    Li, M.: Rigidity of area-minimizing disks in three-manifolds with boundary. Preprint Google Scholar
  10. 10.
    Meeks, W., Yau, S.T.: Topology of three-dimensional manifolds and the embedding problems in minimal surface theory. Ann. Math. (2) 112(3), 441–484 (1980) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Meeks, W., Yau, S.T.: The existence of embedded minimal surfaces and the problem of uniqueness. Math. Z. 179(2), 151–168 (1982) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Micallef, M., Moraru, V.: Splitting of 3-Manifolds and rigidity of area-minimizing surfaces. To appear in Proc. Am. Math. Soc. arXiv:1107.5346
  13. 13.
    Nunes, I.: Rigidity of area-minimizing hyperbolic surfaces in three-manifolds. J. Geom. Anal. (2011) doi: 10.1007/s12220-011-9287-8. Published electronically Google Scholar
  14. 14.
    Schoen, R., Yau, S.T.: Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature. Ann. Math. (2) 110(1), 127–142 (1979) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Shen, Y., Zhu, S.: Rigidity of stable minimal hypersurfaces. Math. Ann. 309(1), 107–116 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, Canberra (1983), vii+272 pp. zbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Instituto de Matemática Pura e Aplicada (IMPA)Rio de JaneiroBrazil

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