First stability eigenvalue characterization of Clifford hypersurfaces

2002/04/11 by Oscar Perdomo, Óscar Perdomo · 5 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology

paper · pdf · doi:10.1090/s0002-9939-02-06451-1

Abstract

The stability operator of a compact oriented minimal hypersurface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Superscript n minus 1 Baseline subset-of upper S Superscript n"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo> ⊂ </mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">M^n-1⊂ Sⁿ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J equals negative normal upper Delta minus double-vertical-bar upper A double-vertical-bar squared minus left-parenthesis n minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo> − </mml:mo> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>A</mml:mi> <mml:msup> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> − </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">J = -Δ -‖A‖²-(n-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper A double-vertical-bar"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>A</mml:mi> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">‖A‖</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the norm of the second fundamental form. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda 1"> <mml:semantics> <mml:msub> <mml:mi> λ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">λ ₁</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the first eigenvalue of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and define <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta equals minus lamda 1 minus 2 left-parenthesis n minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:msub> <mml:mi> λ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> − </mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">β = -λ ₁-2(n-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In 1968 Simons proved that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">β ≥ 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for any non-equatorial minimal hypersurface <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M subset-of upper S Superscript n"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">M⊂ Sⁿ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this paper we will show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta equals 0"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn>

Citations

• Local Rigidity Theorems for Minimal Hypersurfaces
• Minimal Varieties in Riemannian Manifolds

Cited by

• A spectral characterization of the H(r)-torus by the first stability eigenvalue
• First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature
• First stability eigenvalue of singular hypersurfaces with constant mean curvature in the unit sphere
• First Eigenvalue of Jacobi operator and Rigidity Results for Constant Mean Curvature Hypersurfaces
• Linearized equation and generic regularity in the Alt-Caffarelli problem

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