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<p>The goal of this paper is to provide an expository description of a result of Carrasco Piaggio[8] connecting the Ahlfors regular conformal dimension of a compact uniformly perfect doubling metric space with the combinatorial <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p$</EquationSource> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> </InlineEquation>-moduli of the metric space. We give detailed construction of a metric associated with the <InlineEquation ID="IEq101"> <EquationSource Format="TEX">$p$</EquationSource> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> </InlineEquation>-modulus of the space when the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p$</EquationSource> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> </InlineEquation>-modulus is zero, so that the constructed metric is in the Ahlfors regular conformal gauge of the metric space. To do so, we utilize the tools of hyperbolic filling, developed first in[10,6].</p>

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On Carrasco Piaggio's theorem connecting combinatorial modulus and Ahlfors regular conformal dimension

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  • 发表日期：

    2025年11月6日

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  • B. Esmayli,
  • R. Schardine,
  • N. Shanmugalingam

摘要

The goal of this paper is to provide an expository description of a result of Carrasco Piaggio[8] connecting the Ahlfors regular conformal dimension of a compact uniformly perfect doubling metric space with the combinatorial $p$ p -moduli of the metric space. We give detailed construction of a metric associated with the $p$ p -modulus of the space when the $p$ p -modulus is zero, so that the constructed metric is in the Ahlfors regular conformal gauge of the metric space. To do so, we utilize the tools of hyperbolic filling, developed first in[10,6].