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MINIMAL HYPERSURFACES ASYMPTOTIC TO SIMONS CONES

  • Laurent Mazet (a1)

Abstract

In this paper, we prove that, up to similarity, there are only two minimal hypersurfaces in $\mathbb{R}^{n+2}$ that are asymptotic to a Simons cone, i.e., the minimal cone over the minimal hypersurface $\sqrt{\frac{p}{n}}\mathbb{S}^{p}\times \sqrt{\frac{n-p}{n}}\mathbb{S}^{n-p}$ of $\mathbb{S}^{n+1}$ .

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1. Alencar, H., Barros, A., Palmas, O., Guadalupe Reyes, J. and Santos, W., O (m) × O (n)-invariant minimal hypersurfaces in ℝ m+n , Ann. Global Anal. Geom. 27 (2005), 179199.
2. Alexandrov, A. D., A characteristic property of spheres, Ann. Mat. Pura Appl. (4) 58 (1962), 303315.
3. Allard, W. K. and Almgren, F. J. Jr, On the radial behavior of minimal surfaces and the uniqueness of their tangent cones, Ann. of Math. (2) 113 (1981), 215265.
4. Almgren, F. J. Jr, Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem, Ann. of Math. (2) 84 (1966), 277292.
5. Bombieri, E., De Giorgi, E. and Giusti, E., Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243268.
6. Ilmanen, T. and White, B., Sharp lower bounds on density of area-minimizing cones, preprint, 2010, arXiv:1010.5068.
7. Marques, F. C. and Neves, A., Min-max theory and the Willmore conjecture, Ann. of Math. (2) 179 (2014), 683782.
8. Meeks, W. H. III and Wolf, M., Minimal surfaces with the area growth of two planes: the case of infinite symmetry, J. Amer. Math. Soc. 20 (2007), 441465.
9. Schoen, R. M., Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18(1984) (1983), 791809.
10. Simon, L., Isolated singularities of extrema of geometric variational problems, in Harmonic Mappings and Minimal Immersions (Montecatini 1984), Lecture Notes in Mathematics, Volume 1161, pp. 206277 (Springer, Berlin, 1985).
11. Simon, L. and Solomon, B., Minimal hypersurfaces asymptotic to quadratic cones in R n+1 , Invent. Math. 86 (1986), 535551.
12. Simons, J., Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62105.
13. Sogge, C. D., Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), 4365.
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MINIMAL HYPERSURFACES ASYMPTOTIC TO SIMONS CONES

  • Laurent Mazet (a1)

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