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<p>We utilise the two principles of decoupling introduced in Li and Yang (Two principles of decoupling. <a href="http://arxiv.org/abs/2407.16108">arXiv:2407.16108</a>, 2024) to prove the following conditional result: assuming uniform decoupling for graphs of all <i>n</i>-variate polynomials with identically zero Gaussian curvature, we can prove decoupling for all smooth hypersurfaces in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {R}^{n+1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. Moreover, we are able to prove (unconditional) decoupling for all smooth hypersurfaces in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb {R}^4\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation> and graphs of homogeneous polynomials in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathbb {R}^5\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>5</mn> </msup> </math></EquationSource> </InlineEquation>.</p>

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Decoupling for smooth hypersurfaces in \(\mathbb {R}^4\)

  • 期刊论文

  • 发表日期：

    2026年3月2日

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  • Jianhui Li,
  • Tongou Yang

摘要

We utilise the two principles of decoupling introduced in Li and Yang (Two principles of decoupling. arXiv:2407.16108, 2024) to prove the following conditional result: assuming uniform decoupling for graphs of all n-variate polynomials with identically zero Gaussian curvature, we can prove decoupling for all smooth hypersurfaces in \(\mathbb {R}^{n+1}\) R n + 1 . Moreover, we are able to prove (unconditional) decoupling for all smooth hypersurfaces in \(\mathbb {R}^4\) R 4 and graphs of homogeneous polynomials in \(\mathbb {R}^5\) R 5 .