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Estimates of the first Steklov eigenvalue of properly embedded minimal hypersurfaces with free boundary

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Abstract

We consider a properly embedded minimal hypersurfacewith free boundary in a compact n-dimensional Riemannian manifold M be with nonnegative Ricci curvature and strictly convex boundary. Here, we obtain a new estimate from below for the first nonzero Steklov eigenvalue.

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References

  1. S. Agmon, A. Douglis and L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math., 12 (1959), 623–727. MR 0125307 (23 A2610).

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Azzam. On Dirichlet problem for elliptic equations in sectionally smooth ndimensional domains. SIAM J. Math. Anal., 11(2) (1980), 248–253. MR 559866 (82k:35032a).

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Azzam. On Dirichlet problem for elliptic equations in sectionally smooth ndimensional domains. II. SIAM J. Math. Anal., 12(2) (1981), 242. MR 605433 (82k:35032b).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Barros and G.P. Bessa. Estimates of the first eigenvalue ofminimal hypersurfaces of Sn+1. Mat. Contemp., 17 (1999), 42–47.

    Google Scholar 

  5. H. Choi and A.N. Wang. A first eingenvalue estimate for minimal hypersurfaces. J. Diff. Geom., 18 (1983), 559–562.

    Article  MATH  Google Scholar 

  6. J.F. Escobar. The Geometry of the First Non-zero Steklov Eigenvalue. Journal of Functional Analysis, 150 (1997), 544–556.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Fraser and M. Li. Compactness of the space of emnedded minimal srfaces with free boundary in three-manifolds with non-negative Ricci curvature and convex boundary. J. Differential Geom., 96(2) (2014), 183–200.

    Article  MathSciNet  Google Scholar 

  8. A. Fraser and R. Schoen. The first Steklov eigenvalue, conformal geometry, and minimal surfaces. Adv. Math., 226(5) (2011), 4011–4030. MR 2770439.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Fraser and R. Schoen. Minimal surfaces and eigenvalue problems. Geometric analysis, mathematical relativity, and nonlinear partial differential equations, 105-121, Contemp. Math., 599, AMS, rovidence, RI, (2013).

  10. G. Huisken. Asymptotic behaviour for singularities of the mean curvature flow. J. Differential Geom., 31(1) (1999), 285–299.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Hersch. Quatre propriétées isopérimétriqes de membranes sphériques homogènes. C.R. Acad. Sci. Paris Sér. A-B, 270 (1970), A1645–A1648.

    Google Scholar 

  12. H. Blaine Lawson, Jr. The unknottedness of minimal embeddings. Invent. Math., 11 (1970), 183–187.

    Article  MathSciNet  MATH  Google Scholar 

  13. M.C. Li. On a free boundary problem for embedded minimal surfaces and instability theorems for manifolds with positive isotropic curvature. Stanford University, doctor tese, (2011).

    Google Scholar 

  14. P.C. Yang and S.T. Yau. Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 7(1) (1980), 55–63.

    MathSciNet  MATH  Google Scholar 

  15. R. Schoen and S.T. Yau. Lectures on Differential Geometry. International Press, (1994).

    MATH  Google Scholar 

  16. R. Weinstock. Inequalities for a classical eigenvalue problem. J. Rational Mech. Anal., 3 (1954), 745–753.

    MathSciNet  MATH  Google Scholar 

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Correspondence to Rondinelle Batista.

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Partially supported by CNPq-BR.

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Batista, R., Cunha, A.W. Estimates of the first Steklov eigenvalue of properly embedded minimal hypersurfaces with free boundary. Bull Braz Math Soc, New Series 47, 871–881 (2016). https://doi.org/10.1007/s00574-016-0194-2

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  • DOI: https://doi.org/10.1007/s00574-016-0194-2

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