MINIMAL HYPERSURFACES ASYMPTOTIC TO SIMONS CONES

Laurent Mazet

doi:10.1017/s1474748015000110

MINIMAL HYPERSURFACES ASYMPTOTIC TO SIMONS CONES

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748015000110 ◽

2015 ◽

Vol 16 (1) ◽

pp. 39-58

Author(s):

Laurent Mazet

Keyword(s):

Minimal Hypersurface ◽

Minimal Hypersurfaces ◽

Minimal Cone

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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THE SCALAR CURVATURE OF MINIMAL HYPERSURFACES IN A UNIT SPHERE

Communications in Contemporary Mathematics ◽

10.1142/s021919970700237x ◽

2007 ◽

Vol 09 (02) ◽

pp. 183-200 ◽

Cited By ~ 9

Author(s):

YOUNG JIN SUH ◽

HAE YOUNG YANG

Keyword(s):

Scalar Curvature ◽

Unit Sphere ◽

Constant Scalar Curvature ◽

Minimal Hypersurface ◽

Second Fundamental Form ◽

Geodesic Sphere ◽

Clifford Torus ◽

Totally Geodesic ◽

Minimal Hypersurfaces ◽

The Second Fundamental Form

In this paper, we study n-dimensional compact minimal hypersurfaces in a unit sphere Sn+1(1) and give an answer for S. S. Chern's conjecture. We have shown that [Formula: see text] if S > n, and prove that an n-dimensional compact minimal hypersurface with constant scalar curvature in Sn+1(1) is a totally geodesic sphere or a Clifford torus if [Formula: see text], where S denotes the squared norm of the second fundamental form of this hypersurface.

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THE STRUCTURE OF COMPLETE MANIFOLDS WITH WEIGHTED POINCARÉ INEQUALITY AND MINIMAL HYPERSURFACES

International Journal of Mathematics ◽

10.1142/s0129167x10006550 ◽

2010 ◽

Vol 21 (11) ◽

pp. 1421-1428 ◽

Cited By ~ 5

Author(s):

HAIPING FU ◽

ZHENQI LI

Keyword(s):

Euclidean Space ◽

Total Curvature ◽

Minimal Hypersurface ◽

Second Fundamental Form ◽

Dimensional Euclidean Space ◽

Minimal Hypersurfaces ◽

The Second Fundamental Form ◽

Weighted Poincaré Inequality ◽

Complete Manifolds ◽

Bounded Norm

In this paper, we refine some results of [arXiv: 0808.1185v1]. As an application, let M be a complete [Formula: see text]-stable minimal hypersurface in an (n + 1)-dimensional Euclidean space ℝn+1 with n ≥ 3, we prove that if M has bounded norm of the second fundamental form, then M must have only one end. Moreover, we also prove that if M has finite total curvature, then M is a hyperplane.

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Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2014-0147 ◽

2017 ◽

Vol 2017 (731) ◽

pp. 1-19 ◽

Cited By ~ 1

Author(s):

Stéphane Sabourau

Keyword(s):

Riemannian Manifold ◽

Level Set ◽

Ricci Curvature ◽

Morse Function ◽

Minimal Hypersurface ◽

Minimal Hypersurfaces ◽

Nonnegative Ricci Curvature ◽

Closed Riemannian Manifold

AbstractWe establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a consequence of this sweep-out estimate, there exists an embedded, closed (possibly singular) minimal hypersurface whose volume is bounded in terms of the volume of the manifold.

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STABLE MINIMAL HYPERSURFACES AND TANGENT CONE SINGULARITIES

International Journal of Mathematics ◽

10.1142/s0129167x9900015x ◽

1999 ◽

Vol 10 (03) ◽

pp. 407-413

Author(s):

HELEN MOORE

Keyword(s):

Tangent Cone ◽

Volume Growth ◽

Minimal Immersion ◽

Minimal Hypersurface ◽

Singular Set ◽

Minimal Hypersurfaces ◽

Bounded Volume

In this paper, I give an estimate on the dimension of the singular set of a tangent cone at infinity of a stable minimal hypersurface. Namely, let Mn ⊂ ℝn+1, n ≥ 2, be a complete orientable stable minimal immersion with bounded volume growth. Then n < 7 implies T∞(M) is smooth, and n ≥ 7 implies the singular set of T∞(M) has codimension at least seven.

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The structure of weakly stable minimal hypersurfaces

Anais da Academia Brasileira de Ciências ◽

10.1590/s0001-37652006000200001 ◽

2006 ◽

Vol 78 (2) ◽

pp. 195-201 ◽

Cited By ~ 2

Author(s):

Xu Cheng ◽

Leung-Fu Cheung ◽

Detang Zhou

Keyword(s):

Sectional Curvature ◽

Positive Constant ◽

Short Communication ◽

Minimal Hypersurface ◽

Minimal Hypersurfaces ◽

Nonnegative Sectional Curvature ◽

Ambient Manifold ◽

Total Scalar Curvature ◽

Weakly Stable ◽

Bounded Below

In this short communication, we announce results from our research on the structure of complete noncompact oriented weakly stable minimal hypersurfaces in a manifold of nonnegative sectional curvature. In particular, a complete oriented weakly stable minimal hypersurface in Rm, m > 4, must have only one end; any complete noncompact oriented weakly stable minimal hypersurface has only one end if the complete oriented ambient manifold Nm, m > 7, has nonnegative sectional curvature and Ricci curvature bounded below by a positive constant; a complete oriented weakly stable minimal hypersurface in Rm, m > 4, with finite total scalar curvature is a hyperplane.

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