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<p>We introduce an iterative scheme to solve the Yamabe equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( - a\Delta _{g} u + S u = \lambda u^{p-1} \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>a</mi> <msub> <mi mathvariant="normal">Δ</mi> <mi>g</mi> </msub> <mi>u</mi> <mo>+</mo> <mi>S</mi> <mi>u</mi> <mo>=</mo> <mi>λ</mi> <msup> <mi>u</mi> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> on small domains <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> inside a compact Riemannian manifold (<i>M</i>,&#xa0;<i>g</i>). Thus <i>g</i> admits a conformal change to a constant scalar curvature metric. The proof does not use the traditional functional minimization.</p>

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Solving the Yamabe problem by an iterative method on a small Riemannian domain

  • 期刊论文

  • 发表日期：

    2026年7月2日

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  • Steven Rosenberg,
  • Jie Xu

摘要

We introduce an iterative scheme to solve the Yamabe equation \( - a\Delta _{g} u + S u = \lambda u^{p-1} \) - a Δ g u + S u = λ u p - 1 on small domains \(\Omega \) Ω inside a compact Riemannian manifold (M, g). Thus g admits a conformal change to a constant scalar curvature metric. The proof does not use the traditional functional minimization.