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<p>In this paper, we obtain geometric upper bounds for the first eigenvalue <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\lambda _1(J)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>J</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the Jacobi operator for both closed hypersurfaces and compact hypersurfaces with boundary having constant mean curvature (CMC). As an application, we derive new rigidity results for the area of CMC hypersurfaces under suitable conditions on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\lambda _1(J)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>J</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the curvature of the ambient space. We also address the Jacobi–Steklov problem, proving geometric upper bounds for its first eigenvalue <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\sigma _1(J)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>J</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and deriving rigidity results related to the length of the boundary. Additionally, we present some results in higher dimensions related to the Yamabe invariants.</p>

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First Eigenvalue of Jacobi Operator and Rigidity Results for Constant Mean Curvature Hypersurfaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月24日

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  • Márcio Batista,
  • Marcos P. Cavalcante,
  • Luiz R. Melo

摘要

In this paper, we obtain geometric upper bounds for the first eigenvalue \(\lambda _1(J)\) λ 1 ( J ) of the Jacobi operator for both closed hypersurfaces and compact hypersurfaces with boundary having constant mean curvature (CMC). As an application, we derive new rigidity results for the area of CMC hypersurfaces under suitable conditions on \(\lambda _1(J)\) λ 1 ( J ) and the curvature of the ambient space. We also address the Jacobi–Steklov problem, proving geometric upper bounds for its first eigenvalue \(\sigma _1(J)\) σ 1 ( J ) and deriving rigidity results related to the length of the boundary. Additionally, we present some results in higher dimensions related to the Yamabe invariants.