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COMPLETE HYPERSURFACES WITH LINEARLY RELATED HIGHER ORDER MEAN CURVATURES IN THE HYPERBOLIC SPACE

Part of: Classical differential geometry Global differential geometry

Published online by Cambridge University Press:  10 March 2025

ARY V. F. LEITE Open the ORCID record for ARY V. F. LEITE [Opens in a new window] ,
HENRIQUE F. DE LIMA Open the ORCID record for HENRIQUE F. DE LIMA [Opens in a new window]  and
MARCO A. L. VELÁSQUEZ Open the ORCID record for MARCO A. L. VELÁSQUEZ [Opens in a new window]
ARY V. F. LEITE*
Affiliation:
Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil
HENRIQUE F. DE LIMA
Affiliation:
Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil e-mail: henriquedelima74@gmail.com
MARCO A. L. VELÁSQUEZ
Affiliation:
Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil e-mail: marcolazarovelasquez@gmail.com
*
e-mail: ary.v.l.f@gmail.com
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Abstract

In this article, we study the behavior of complete two-sided hypersurfaces immersed in the hyperbolic space $\mathbb H^{n+1}$. Initially, we introduce the concept of the linearized curvature function $\mathcal {F}_{r,s}$ of a two-sided hypersurface, its associated modified Newton transformation $\mathcal {P}_{r,s}$ and its naturally attached differential operator $\mathcal {L}_{r,s}$. Then, we obtain two formulas for differential operator $\mathcal {L}_{r,s}$ acting on the height function of a two-sided hypersurface and, for the case where their support functions are related by a negative constant, we derive two new formulas for the Newton transformation $P_{r}$ and the modified Newton transformation $\mathcal {P}_{r,s}$ acting on a gradient of the height function. Finally, these formulas, jointly with suitable maximum principles, enable us to establish our rigidity and nonexistence results concerning complete two-sided hypersurfaces in $\mathbb H^{n+1}$.

Keywords

hyperbolic spacecomplete two-sided hypersurfaceshorospheresclosed horoballhigher order mean curvatures

MSC classification

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.)
Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature 53C20: Global Riemannian geometry, including pinching

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 119 , Issue 1 , August 2025 , pp. 82 - 105
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by James McCoy

The first author is partially supported by CAPES, Brazil. The second and third authors are partially supported by CNPq, Brazil, grants 305608/2023-1 and 304891/2021-5, respectively.

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