<p>Our purpose in this article is to study the geometric properties of complete spacelike hypersurfaces immersed in the anti-de Sitter space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {H}_1^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">H</mi> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </math></EquationSource> </InlineEquation>. In this context, we introduce the notion of a linearized curvature function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {F}_{r,s}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> associated to such a spacelike hypersurface and, by applying some maximum principles to a second order differential operator <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {L}_{r,s}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">L</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> naturally related to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {F}_{r,s}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, we obtain new characterizations of totally umbilical hypersurfaces of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {H}_1^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">H</mi> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </math></EquationSource> </InlineEquation> which are isometric to certain hyperbolic spaces. Our approach also allows us to obtain some nonexistence results concerning complete noncompact spacelike hypersurfaces of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {H}_1^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">H</mi> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </math></EquationSource> </InlineEquation>.</p>

Characterizing Hyperbolic Spaces in the Anti-de Sitter Space

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发表日期：

2025年10月2日

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Ary V. F. Leite,
Henrique F. de Lima,
Marco A. L. Velásquez

摘要

Our purpose in this article is to study the geometric properties of complete spacelike hypersurfaces immersed in the anti-de Sitter space $\mathbb {H}_1^{n+1}$ $H_{1}^{n + 1}$ . In this context, we introduce the notion of a linearized curvature function $\mathcal {F}_{r,s}$ $F_{r, s}$ associated to such a spacelike hypersurface and, by applying some maximum principles to a second order differential operator $\mathcal {L}_{r,s}$ $L_{r, s}$ naturally related to $\mathcal {F}_{r,s}$ $F_{r, s}$ , we obtain new characterizations of totally umbilical hypersurfaces of $\mathbb {H}_1^{n+1}$ $H_{1}^{n + 1}$ which are isometric to certain hyperbolic spaces. Our approach also allows us to obtain some nonexistence results concerning complete noncompact spacelike hypersurfaces of $\mathbb {H}_1^{n+1}$ $H_{1}^{n + 1}$ .