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<p>Thanks to an ambitious project initiated by Catino, Mastrolia, Monticelli, and Rigoli, which aims to provide a unified viewpoint for various geometric solitons, many classes, including Ricci solitons, Yamabe solitons, <i>k</i>-Yamabe solitons, quasi-Yamabe solitons, and conformal solitons, can now be studied under a unified framework known as Einstein-type manifolds. Einstein-type manifolds are characterized by four constants, denoted by <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1992_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha , \beta , \mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>μ</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1992_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>. In this paper, we completely classify all non-trivial, complete gradient Einstein-type Kähler manifolds with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1992_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha = 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. As a corollary, we obtain rotational symmetry for many classes. In particular, we show that any non-trivial complete quasi-Yamabe gradient soliton on Kähler manifolds is rotationally symmetric.</p>

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Classification of gradient Einstein-type Kähler manifolds with \(\alpha =0\)

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  • 发表日期：

    2025年9月30日

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  • Shun Maeta

摘要

Thanks to an ambitious project initiated by Catino, Mastrolia, Monticelli, and Rigoli, which aims to provide a unified viewpoint for various geometric solitons, many classes, including Ricci solitons, Yamabe solitons, k-Yamabe solitons, quasi-Yamabe solitons, and conformal solitons, can now be studied under a unified framework known as Einstein-type manifolds. Einstein-type manifolds are characterized by four constants, denoted by \(\alpha , \beta , \mu \) α , β , μ , and \(\rho \) ρ . In this paper, we completely classify all non-trivial, complete gradient Einstein-type Kähler manifolds with \(\alpha = 0\) α = 0 . As a corollary, we obtain rotational symmetry for many classes. In particular, we show that any non-trivial complete quasi-Yamabe gradient soliton on Kähler manifolds is rotationally symmetric.