<p>In this article, we introduce a modification of the timelike Hausdorff measure <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {V}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">V</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> defined by McCann and Sämann on Lorentzian pre-length spaces. We write the modification of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {V}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">V</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {W}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">W</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>. We establish volume comparison inequalities by causality preserving and timelike Lipschitz maps for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {V}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">V</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {W}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">W</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>, and discuss basic properties of both <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {V}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">V</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {W}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">W</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>. Moreover, we show the coincidence of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {W}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">W</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {V}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">V</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> on smooth spacetimes and some Lorentzian pre-length spaces, and construct some examples of timelike Lipschitz maps and causality preserving maps.</p>

Volume comparison by timelike Lipschitz maps

期刊论文
发表日期：

2025年12月26日

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Hikaru Kubota

摘要

In this article, we introduce a modification of the timelike Hausdorff measure $\mathcal {V}^N$ $V^{N}$ defined by McCann and Sämann on Lorentzian pre-length spaces. We write the modification of $\mathcal {V}^N$ $V^{N}$ as $\mathcal {W}^N$ $W^{N}$ . We establish volume comparison inequalities by causality preserving and timelike Lipschitz maps for $\mathcal {V}^N$ $V^{N}$ and $\mathcal {W}^N$ $W^{N}$ , and discuss basic properties of both $\mathcal {V}^N$ $V^{N}$ and $\mathcal {W}^N$ $W^{N}$ . Moreover, we show the coincidence of $\mathcal {W}^N$ $W^{N}$ and $\mathcal {V}^N$ $V^{N}$ on smooth spacetimes and some Lorentzian pre-length spaces, and construct some examples of timelike Lipschitz maps and causality preserving maps.