In this paper, a new metric <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$D_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> in the Ptolemy space (X, d) is introduced as <Equation ID="Equ12"> <EquationSource Format="TEX">$ D_p(x, y) = \log \left( 1 + \frac{d(x,y)}{\max \{{\sqrt{1 + d(x,p)},\sqrt{1 + d(y,p)}\}}}\right) $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>D</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>log</mo> <mfenced close=")" open="("> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo movablelimits="true">max</mo> <mo stretchy="false">{</mo> <mrow> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msqrt> <mo>,</mo> <msqrt> <mrow> <mn>1</mn> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </msqrt> <mrow> <mo stretchy="false">}</mo> </mrow> </mrow> </mrow> </mfrac> </mfenced> </mrow> </math></EquationSource> </Equation>for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$x,y\in X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p\in X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>. We establish the Gromov hyperbolicity of the metric <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$D_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>, and define a new metric <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$D_\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi mathvariant="normal">Ω</mi> </msub> </math></EquationSource> </InlineEquation> associated with the boundary of domains in Ptolemy spaces. Moreover, we investigate connections between bi-Lipschitzian mappings and quasiconformal mappings with respect to the metric <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$D_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>, derive comparison inequalities with the relative metric, and explore inclusion relations between metric balls in domains of proper Ptolemy spaces. Additionally, we study distortion properties of the metric <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$D_\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mi mathvariant="normal">Ω</mi> </msub> </math></EquationSource> </InlineEquation> under Möbius transformations of the unit ball.

Properties of a New Hyperbolic Type Metric on Ptolemy Spaces

期刊论文
发表日期：

2026年4月22日

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Liangyue Qu

摘要

In this paper, a new metric $D_p$ $D_{p}$ in the Ptolemy space (X, d) is introduced as $ D_p(x, y) = \log \left( 1 + \frac{d(x,y)}{\max \{{\sqrt{1 + d(x,p)},\sqrt{1 + d(y,p)}\}}}\right) $ $D_{p} (x, y) = log (1 + \frac{d (x, y)}{max {\sqrt{1 + d (x, p)}, \sqrt{1 + d (y, p)}}})$ for $x,y\in X$ $x, y \in X$ and $p\in X$ $p \in X$ . We establish the Gromov hyperbolicity of the metric $D_p$ $D_{p}$ , and define a new metric $D_\Omega $ $D_{Ω}$ associated with the boundary of domains in Ptolemy spaces. Moreover, we investigate connections between bi-Lipschitzian mappings and quasiconformal mappings with respect to the metric $D_p$ $D_{p}$ , derive comparison inequalities with the relative metric, and explore inclusion relations between metric balls in domains of proper Ptolemy spaces. Additionally, we study distortion properties of the metric $D_\Omega $ $D_{Ω}$ under Möbius transformations of the unit ball.