Index estimates for closed minimal submanifolds of the sphere

2021 ◽  
pp. 1-15
Author(s):  
Diego Adauto ◽  
Márcio Batista
Keyword(s):  
Ricci Curvature ◽  
Unit Sphere ◽  
Normal Bundle ◽  
Morse Index ◽  
Minimal Submanifolds ◽  
Laplacian Operator ◽  
Minimal Hypersurfaces ◽  
Hodge Laplacian ◽  
The Stability ◽  
Stability Operator

In this paper we are interested in comparing the spectra of two elliptic operators acting on a closed minimal submanifold of the Euclidean unit sphere. Using an approach introduced by Savo in [A Savo. Index Bounds for Minimal Hypersurfaces of the Sphere. Indiana Univ. Math. J. 59 (2010), 823-837.], we are able to compare the eigenvalues of the stability operator acting on sections of the normal bundle and the Hodge Laplacian operator acting on $1$ -forms. As a byproduct of the technique and under a suitable hypothesis on the Ricci curvature of the submanifold we obtain that its first Betti's number is bounded from above by a multiple of the Morse index, which provide evidence for a well-known conjecture of Schoen and Marques & Neves in the setting of higher codimension.

scholarly journals A characterisation of compact minimal hypersurfaces in a unit sphere

1993 ◽  
Vol 47 (2) ◽  
pp. 213-216
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Geodesic Sphere ◽  
Clifford Torus ◽  
Totally Geodesic ◽  
Minimal Hypersurfaces

In this note, we show that the totally geodesic sphere, Clifford torus and Cartan hypersurface are the only compact minimal hypersurfaces in S4(1) with constant scalar curvature if the Ricci curvature is not less than −1.

scholarly journals Minimal hypersurfaces of unit sphere

1997 ◽  
Vol 49 (3) ◽  
pp. 423-429 ◽  
Author(s):  
Huafei Sun ◽  
Koichi Ogiue
Keyword(s):  
Unit Sphere ◽  
Minimal Hypersurfaces

THE SCALAR CURVATURE OF MINIMAL HYPERSURFACES IN A UNIT SPHERE

2007 ◽  
Vol 09 (02) ◽  
pp. 183-200 ◽  
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.

scholarly journals Curvature estimates for stable free boundary minimal hypersurfaces

2020 ◽  
Vol 2020 (759) ◽  
pp. 245-264 ◽  
Author(s):  
Qiang Guang ◽  
Martin Man-chun Li ◽  
Xin Zhou
Keyword(s):  
Free Boundary ◽  
Riemannian Manifolds ◽  
A Priori ◽  
Minimal Submanifolds ◽  
Manifolds With Boundary ◽  
Curvature Estimate ◽  
Minimal Hypersurfaces ◽  
Curvature Estimates ◽  
Minimal Laminations ◽  
Uniform Area

AbstractIn this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalize the celebrated Schoen–Simon–Yau interior curvature estimates up to the free boundary. Our curvature estimates imply a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For 3-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-priori area bound. This generalizes Schoen’s interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi.

Rigidity of compact minimal submanifolds in a unit sphere

1993 ◽  
Vol 45 (1) ◽  
pp. 83-88 ◽  
Author(s):  
Qing Chen ◽  
Senlin Xu
Keyword(s):  
Unit Sphere ◽  
Minimal Submanifolds

On Frankel’s Theorem

2003 ◽  
Vol 46 (1) ◽  
pp. 130-139 ◽  
Author(s):  
Peter Petersen ◽  
Frederick Wilhelm
Keyword(s):  
Ricci Curvature ◽  
Positive Ricci Curvature ◽  
Minimal Hypersurfaces ◽  
Nonnegative Ricci Curvature

AbstractIn this paper we show that two minimal hypersurfaces in a manifold with positive Ricci curvature must intersect. This is then generalized to show that in manifolds with positive Ricci curvature in the integral sense two minimal hypersurfaces must be close to each other. We also show what happens if a manifold with nonnegative Ricci curvature admits two nonintersecting minimal hypersurfaces.

scholarly journals Width, Ricci curvature, and minimal hypersurfaces

2017 ◽  
Vol 105 (1) ◽  
pp. 33-54 ◽  
Author(s):  
Parker Glynn-Adey ◽  
Yevgeny Liokumovich
Keyword(s):  
Ricci Curvature ◽  
Minimal Hypersurfaces

scholarly journals Stability of Non-Linear Dirichlet Problems with ϕ-Laplacian

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 647
Author(s):  
Michał Bełdziński ◽  
Marek Galewski ◽  
Igor Kossowski
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Dirichlet Boundary Conditions ◽  
Laplacian Operator ◽  
Dirichlet Problems ◽  
Non Linear ◽  
The Stability

We study the stability and the solvability of a family of problems −(ϕ(x′))′=g(t,x,x′,u)+f* with Dirichlet boundary conditions, where ϕ, u, f* are allowed to vary as well. Applications for boundary value problems involving the p-Laplacian operator are highlighted.

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