In this paper, we investigate the existence of convex, entire, spacelike hypersurfaces of constant <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3266_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> curvature with prescribed set of lightlike directions <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3266_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}\subset \mathbb {S}^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and perturbation q on <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3266_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation>. We prove that given a closed set <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3266_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> in the ideal boundary at infinity of hyperbolic space and a perturbation q that satisfies some mild conditions, there exists a complete entire spacelike constant <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3266_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma _k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>σ</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> curvature hypersurface <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3266_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {M}_u$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>u</mi> </msub> </math></EquationSource> </InlineEquation> with prescribed set of lightlike directions <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3266_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {F}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">F</mi> </math></EquationSource> </InlineEquation> satisfying when <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3266_Article_IEq11.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{x}{|x|}\in \mathcal {F},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mi>x</mi> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> </mfrac> <mo>∈</mo> <mi mathvariant="script">F</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3266_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$|x|\rightarrow \infty ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mi>∞</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3266_Article_IEq13.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="155" /> </InlineMediaObject> <EquationSource Format="TEX">$u(x)-|x|\rightarrow q\left( \frac{x}{|x|}\right) .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mi>q</mi> <mfenced close=")" open="("> <mfrac> <mi>x</mi> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> </mfrac> </mfenced> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> This result is new even for the case of constant Gauss curvature hypersurfaces.

Entire spacelike constant $\sigma _k$ curvature hypersurfaces with prescribed boundary data at infinity

期刊论文
发表日期：

2025年8月18日

查看全文

Zhizhang Wang,
Ling Xiao

摘要

In this paper, we investigate the existence of convex, entire, spacelike hypersurfaces of constant $\sigma _k$ $σ_{k}$ curvature with prescribed set of lightlike directions $\mathcal {F}\subset \mathbb {S}^{n-1}$ $F \subset S^{n - 1}$ and perturbation q on $\mathcal {F}$ $F$ . We prove that given a closed set $\mathcal {F}$ $F$ in the ideal boundary at infinity of hyperbolic space and a perturbation q that satisfies some mild conditions, there exists a complete entire spacelike constant $\sigma _k$ $σ_{k}$ curvature hypersurface $\mathcal {M}_u$ $M_{u}$ with prescribed set of lightlike directions $\mathcal {F}$ $F$ satisfying when $\frac{x}{|x|}\in \mathcal {F},$ $\frac{x}{| x |} \in F,$ as $|x|\rightarrow \infty ,$ $| x | \to \infty,$ $u(x)-|x|\rightarrow q\left( \frac{x}{|x|}\right) .$ $u (x) - | x | \to q (\frac{x}{| x |}) .$ This result is new even for the case of constant Gauss curvature hypersurfaces.

Entire spacelike constant \(\sigma _k\) curvature hypersurfaces with prescribed boundary data at infinity

摘要