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<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10161_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation> be a <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10161_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(n-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>-dimensional hyperbolic paraboloid in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10161_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and let <i>m</i> be the minimum between the number of positive and negative principal curvatures of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10161_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation>. We study the restriction problem for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10161_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation> in dimension <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10161_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\ge 4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> by the polynomial partitioning method and improve the result of Hickman and Iliopoulou (Math. Z. 301:1143–1189, 2022) when <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10161_Article_IEq7.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\ge 5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> is odd and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41_2025_10161_Article_IEq8.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">\(m=\frac{n-3}{2}.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mfrac> <mrow> <mi>n</mi> <mo>-</mo> <mn>3</mn> </mrow> <mn>2</mn> </mfrac> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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Restriction Estimates for Hyperbolic Paraboloids in Higher Dimensions

  • 期刊论文

  • 发表日期：

    2025年5月2日

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  • Zhuoran Li

摘要

Let \(\mathbb {H}\) H be a \(n-1\) n - 1 -dimensional hyperbolic paraboloid in \(\mathbb {R}^n\) R n and let m be the minimum between the number of positive and negative principal curvatures of \(\mathbb {H}\) H . We study the restriction problem for \(\mathbb {H}\) H in dimension \(n\ge 4\) n ≥ 4 by the polynomial partitioning method and improve the result of Hickman and Iliopoulou (Math. Z. 301:1143–1189, 2022) when \(n\ge 5\) n ≥ 5 is odd and \(m=\frac{n-3}{2}.\) m = n - 3 2 .