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<p>Harmonic manifolds of hypergeometric type form a class of non-compact harmonic manifolds that includes rank one symmetric spaces of non-compact type and Damek-Ricci spaces. When normalizing the metric of a harmonic manifold of hypergeometric type to satisfy the Ricci curvature <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2521_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textrm{Ric} = -(n-1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Ric</mtext> <mo>=</mo> <mo>-</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we show that the volume entropy of this manifold satisfies a certain inequality. Additionally, we show that manifolds yielding the upper bound of volume entropy are only real hyperbolic spaces with sectional curvature <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2521_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, while examples of Damek-Ricci spaces yielding the lower bound exist in only four cases.</p>

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A note on the volume entropy of harmonic manifolds of hypergeometric type

  • 期刊论文

  • 发表日期：

    2025年9月19日

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  • Hiroyasu Satoh

摘要

Harmonic manifolds of hypergeometric type form a class of non-compact harmonic manifolds that includes rank one symmetric spaces of non-compact type and Damek-Ricci spaces. When normalizing the metric of a harmonic manifold of hypergeometric type to satisfy the Ricci curvature \(\textrm{Ric} = -(n-1)\) Ric = - ( n - 1 ) , we show that the volume entropy of this manifold satisfies a certain inequality. Additionally, we show that manifolds yielding the upper bound of volume entropy are only real hyperbolic spaces with sectional curvature \(-1\) - 1 , while examples of Damek-Ricci spaces yielding the lower bound exist in only four cases.