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<p>We provide a new proof of the equality case of the spacetime positive mass theorem, which states that if a complete asymptotically flat initial data set (<i>M</i>,&#xa0;<i>g</i>,&#xa0;<i>k</i>) satisfying the dominant energy condition has null ADM energy-momentum (that is, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2947_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">\(|E|=|P|\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>E</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mo stretchy="false">|</mo> <mi>P</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>), then (<i>M</i>,&#xa0;<i>g</i>) must isometrically embed into Minkowski space with <i>k</i> as its second fundamental form. Previous proofs either used spinor methods [<CitationRef CitationID="CR2">2</CitationRef>, <CitationRef CitationID="CR5">5</CitationRef>, <CitationRef CitationID="CR26">26</CitationRef>], relied on the Jang equation [<CitationRef CitationID="CR10">10</CitationRef>, <CitationRef CitationID="CR16">16</CitationRef>], or assumed three spatial dimensions [<CitationRef CitationID="CR17">17</CitationRef>]. In contrast, our new proof only requires knowing that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2947_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">\(E\ge |P|\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>≥</mo> <mo stretchy="false">|</mo> <mi>P</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation> for all complete initial data sets near (<i>g</i>,&#xa0;<i>k</i>) on <i>M</i> satisfying the dominant energy condition.</p>

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Equality in the spacetime positive mass theorem II

  • 期刊论文

  • 发表日期：

    2025年2月27日

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  • Lan-Hsuan Huang,
  • Dan A. Lee

摘要

We provide a new proof of the equality case of the spacetime positive mass theorem, which states that if a complete asymptotically flat initial data set (M, g, k) satisfying the dominant energy condition has null ADM energy-momentum (that is, \(|E|=|P|\) | E | = | P | ), then (M, g) must isometrically embed into Minkowski space with k as its second fundamental form. Previous proofs either used spinor methods [2, 5, 26], relied on the Jang equation [10, 16], or assumed three spatial dimensions [17]. In contrast, our new proof only requires knowing that \(E\ge |P|\) E ≥ | P | for all complete initial data sets near (g, k) on M satisfying the dominant energy condition.