We study the deformation theory of Einstein-Yang-Mills fields over conformally compact, asymptotically locally hyperbolic manifolds. We prove that if an Einstein-Yang-Mills field <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2104_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$(g_0,\omega _0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>ω</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is trivial (which means that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2104_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$g_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>g</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is a Poincaré-Einstein metric and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2104_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega _0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> is a flat connection on a principal bundle over the underlying manifold) and non-degenerate in the appropriate sense then any sufficiently small perturbation of its boundary data at infinity may be realized as the boundary data of some Einstein-Yang-Mills field. This result is obtained as an application of the 0-calculus of Mazzeo and Melrose Journal of Functional Analysis 75(2), 260–310 (1987), Mazzeo Journal of Differential Geometry 28(2), 309–339 (1988) and may be viewed as a natural extension of previous results by Graham-Lee Advances in Mathematics 87(2), 186–225 (1991) , Lee American Mathematical Soc, Providende, Rhode Island (2006) and Usula Letters in Mathematical Physics 111, 1–23 (2021).

Einstein-Yang-Mills fields in conformally compact manifolds

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发表日期：

2025年7月15日

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Levi Lopes de Lima

摘要

We study the deformation theory of Einstein-Yang-Mills fields over conformally compact, asymptotically locally hyperbolic manifolds. We prove that if an Einstein-Yang-Mills field $(g_0,\omega _0)$ $(g_{0}, ω_{0})$ is trivial (which means that $g_0$ $g_{0}$ is a Poincaré-Einstein metric and $\omega _0$ $ω_{0}$ is a flat connection on a principal bundle over the underlying manifold) and non-degenerate in the appropriate sense then any sufficiently small perturbation of its boundary data at infinity may be realized as the boundary data of some Einstein-Yang-Mills field. This result is obtained as an application of the 0-calculus of Mazzeo and Melrose Journal of Functional Analysis 75(2), 260–310 (1987), Mazzeo Journal of Differential Geometry 28(2), 309–339 (1988) and may be viewed as a natural extension of previous results by Graham-Lee Advances in Mathematics 87(2), 186–225 (1991) , Lee American Mathematical Soc, Providende, Rhode Island (2006) and Usula Letters in Mathematical Physics 111, 1–23 (2021).