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

Abstract In this paper, we obtain geometric upper bounds for the first eigenvalue $$\lambda _1(J)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>J</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 $$\lambda _1(J)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>J</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and the curvature of the ambient space. We also address the Jacobi–Steklov problem, proving geometric upper bounds for its first eigenvalue $$\sigma _1(J)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>J</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and deriving rigidity results related to the length of the boundary. Additionally, we present some results in higher dimensions related to the Yamabe invariants.