Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$h^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>h</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$h^{-}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>h</mi> <mo>-</mo> </msup> </math></EquationSource> </InlineEquation> be two complete, conformal metrics on the disc <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation>. Assume moreover that the derivatives of the conformal factors of metrics <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$h^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>h</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$h^{-}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>h</mi> <mo>-</mo> </msup> </math></EquationSource> </InlineEquation> are bounded at any order with respect to the hyperbolic metric, and that the metrics have curvatures in the interval <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\left( -\frac{1}{\epsilon }, -1 - \epsilon \right) $</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mo>-</mo> <mfrac> <mn>1</mn> <mi>ϵ</mi> </mfrac> <mo>,</mo> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mi>ϵ</mi> </mfenced> </math></EquationSource> </InlineEquation>, for some <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\epsilon > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Let f be a quasi-symmetric map. We show the existence of a globally hyperbolic convex subset <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> (see Definition <InternalRef RefID="FPar21">4.1</InternalRef>) of the three-dimensional anti-de Sitter space, such that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> has <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$h^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>h</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation> (respectively <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$h^{-}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>h</mi> <mo>-</mo> </msup> </math></EquationSource> </InlineEquation>) as the induced metric on its future boundary (respectively on its past boundary) and has a gluing map <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\Phi _{\Omega }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Φ</mi> <mi mathvariant="normal">Ω</mi> </msub> </math></EquationSource> </InlineEquation> (see Definition <InternalRef RefID="FPar34">5.7</InternalRef>) equal to f.

The Prescribed Metric on Convex Subsets of Anti-de Sitter Space with Quasi-Circle Ideal Boundaries

期刊论文
发表日期：

2025年11月1日

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Abderrahim Mesbah

摘要

Let $h^{+}$ $h^{+}$ and $h^{-}$ $h^{-}$ be two complete, conformal metrics on the disc $\mathbb {D}$ $D$ . Assume moreover that the derivatives of the conformal factors of metrics $h^{+}$ $h^{+}$ and $h^{-}$ $h^{-}$ are bounded at any order with respect to the hyperbolic metric, and that the metrics have curvatures in the interval $\left( -\frac{1}{\epsilon }, -1 - \epsilon \right) $ $(- \frac{1}{ϵ}, - 1 - ϵ)$ , for some $\epsilon > 0$ $ϵ > 0$ . Let f be a quasi-symmetric map. We show the existence of a globally hyperbolic convex subset $\Omega $ $Ω$ (see Definition 4.1) of the three-dimensional anti-de Sitter space, such that $\Omega $ $Ω$ has $h^{+}$ $h^{+}$ (respectively $h^{-}$ $h^{-}$ ) as the induced metric on its future boundary (respectively on its past boundary) and has a gluing map $\Phi _{\Omega }$ $Φ_{Ω}$ (see Definition 5.7) equal to f.