Startseite A flow approach to the prescribed Gaussian curvature problem in ℍ𝑛+1
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A flow approach to the prescribed Gaussian curvature problem in ℍ𝑛+1

  • Haizhong Li und Ruijia Zhang ORCID logo EMAIL logo
Veröffentlicht/Copyright: 6. Mai 2023

Abstract

In this paper, we study the following prescribed Gaussian curvature problem:

K = f ~ ( θ ) ϕ ( ρ ) α 2 ϕ ( ρ ) 2 + | ¯ ρ | 2 ,

a generalization of the Alexandrov problem ( α = n + 1 ) in hyperbolic space, where f ~ is a smooth positive function on S n , 𝜌 is the radial function of the hypersurface, ϕ ( ρ ) = sinh ρ and 𝐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α n + 1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 . We also consider the cases 2 < α n + 1 under the evenness assumption of f ~ and prove the existence of solutions to the above equations.

MSC 2010: 35K55; 53E40

Award Identifier / Grant number: 11831005

Award Identifier / Grant number: 12126405

Funding statement: The authors were partially supported by NSFC grants No. 11831005 and No. 12126405.

Acknowledgements

The authors would like to thank Professor Julian Scheuer, Professor Xianfeng Wang and Professor Yong Wei for helpful discussions and useful comments. The authors would like to thank the referee for his/her helpful suggestions.

  1. Communicated by: Guofang Wang

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Received: 2022-05-04
Accepted: 2023-03-10
Published Online: 2023-05-06
Published in Print: 2024-07-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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