<p>We demonstrate that <i>n</i>-dimension closed Einstein manifolds, whose smallest eigenvalue of the curvature operator of the second kind of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathring{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>R</mi> <mo>˚</mo> </mover> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda _1 \ge -\theta (n) \bar{\lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>≥</mo> <mo>-</mo> <mi>θ</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mover accent="true"> <mrow> <mi>λ</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </mrow> </math></EquationSource> </InlineEquation>, are either flat or round spheres, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\bar{\lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mi>λ</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </math></EquationSource> </InlineEquation> is the average of the eigenvalues of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathring{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>R</mi> <mo>˚</mo> </mover> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\theta (n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is defined as in Eq. (<InternalRef RefID="Equ2">1.2</InternalRef>). Our result improves a celebrated result (Theorem <InternalRef RefID="FPar1">1.1</InternalRef>) concerning Einstein manifolds with nonnegative curvature operator of the second kind.</p>

Einstein manifolds of negative lower bounds on curvature operator of the second Kind

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2026年1月20日

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Haiqing Cheng,
Kui Wang

摘要

We demonstrate that n-dimension closed Einstein manifolds, whose smallest eigenvalue of the curvature operator of the second kind of $\mathring{R}$ $\overset{˚}{R}$ satisfies $\lambda _1 \ge -\theta (n) \bar{\lambda }$ $λ_{1} \geq - θ (n) \bar{λ}$ , are either flat or round spheres, where $\bar{\lambda }$ $\bar{λ}$ is the average of the eigenvalues of $\mathring{R}$ $\overset{˚}{R}$ , and $\theta (n)$ $θ (n)$ is defined as in Eq. (1.2). Our result improves a celebrated result (Theorem 1.1) concerning Einstein manifolds with nonnegative curvature operator of the second kind.