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<p>We study Perelman’s <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {W}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">W</mi> </math></EquationSource> </InlineEquation>-entropy functional on finite-dimensional <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textrm{RCD}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>RCD</mtext> </math></EquationSource> </InlineEquation> spaces, a synthetic generalization of spaces with Bakry–Émery Ricci curvature bounded from below. We rigorously justify the formula for the time derivative of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {W}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">W</mi> </math></EquationSource> </InlineEquation>-entropy and derive its monotonicity and rigidity properties. Additionally, we establish bounds for solutions of the heat equation, which are of independent interest.</p>

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Perelman’s entropy and heat kernel bounds on RCD spaces

  • 期刊论文

  • 发表日期：

    2025年8月18日

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  • Camillo Brena

摘要

We study Perelman’s \(\mathcal {W}\) W -entropy functional on finite-dimensional \(\textrm{RCD}\) RCD spaces, a synthetic generalization of spaces with Bakry–Émery Ricci curvature bounded from below. We rigorously justify the formula for the time derivative of the \(\mathcal {W}\) W -entropy and derive its monotonicity and rigidity properties. Additionally, we establish bounds for solutions of the heat equation, which are of independent interest.