It is established in [Cao, X., Gursky, M.J., Tran, H.: Comment. Math. Helv. 98(1), 195–216 (2023), Li, X.: J. Geom. Anal., 32(11):Paper No. 281, 14, (2022), Nienhaus, J., Petersen, P., Wink, M.: J. Lond. Math. Soc. (2), 108(4):1642–1668, (2023)] that any closed Einstein manifold with two-nonnegative curvature operator of the second kind is either flat or a round sphere. In this paper, we refine this result by relaxing the curvature condition to a cone condition (strictly weaker than two nonnegativity) proposed by Li [Li, X.: <a href="http://arxiv.org/abs/2407.13847">arXiv:2407.13847</a>, (2024)]. Precisely, we prove that any closed Einstein manifold of dimension <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2227_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n=4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2227_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$n=5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2227_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>8</mn> </mrow> </math></EquationSource> </InlineEquation>, if the curvature operator of the second kind <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2227_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathring{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>R</mi> <mo>˚</mo> </mover> </math></EquationSource> </InlineEquation> satisfies <Equation ID="Equ30"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2227_Article_Equ30.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="165" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} (\lambda _1+\lambda _2)/2 \ge -\theta (n) \bar{\lambda }, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mn>2</mn> <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> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>then the manifold is either flat or a round sphere. Here, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2227_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="207" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _1\le \lambda _2\le \cdots \le \lambda _{(n-1)(n+2)/2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>≤</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> are the eigenvalues of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2227_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathring{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>R</mi> <mo>˚</mo> </mover> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2227_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$ \bar{\lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mrow> <mi>λ</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </math></EquationSource> </InlineEquation> is their average, and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2227_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$\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 a positive constant defined as in (<InternalRef RefID="Equ2">1.2</InternalRef>).

Einstein Manifolds Under Cone Conditions for the Curvature Operator of the Second Kind

期刊论文
发表日期：

2025年10月15日

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Haiqing Cheng,
Kui Wang

摘要

It is established in [Cao, X., Gursky, M.J., Tran, H.: Comment. Math. Helv. 98(1), 195–216 (2023), Li, X.: J. Geom. Anal., 32(11):Paper No. 281, 14, (2022), Nienhaus, J., Petersen, P., Wink, M.: J. Lond. Math. Soc. (2), 108(4):1642–1668, (2023)] that any closed Einstein manifold with two-nonnegative curvature operator of the second kind is either flat or a round sphere. In this paper, we refine this result by relaxing the curvature condition to a cone condition (strictly weaker than two nonnegativity) proposed by Li [Li, X.: arXiv:2407.13847, (2024)]. Precisely, we prove that any closed Einstein manifold of dimension $n=4$ $n = 4$ or $n=5$ $n = 5$ or $n\ge 8$ $n \geq 8$ , if the curvature operator of the second kind $\mathring{R}$ $\overset{˚}{R}$ satisfies $\begin{aligned} (\lambda _1+\lambda _2)/2 \ge -\theta (n) \bar{\lambda }, \end{aligned}$ $\begin{matrix} (λ_{1} + λ_{2}) / 2 \geq - θ (n) \bar{λ}, \end{matrix}$ then the manifold is either flat or a round sphere. Here, $\lambda _1\le \lambda _2\le \cdots \le \lambda _{(n-1)(n+2)/2}$ $λ_{1} \leq λ_{2} \leq \dots \leq λ_{(n - 1) (n + 2) / 2}$ are the eigenvalues of $\mathring{R}$ $\overset{˚}{R}$ , $ \bar{\lambda }$ $\bar{λ}$ is their average, and $\theta (n)$ $θ (n)$ is a positive constant defined as in (1.2).