Smooth solutions to asymptotic Plateau type problem in hyperbolic space

Zhenan Sui; Wei Sun

doi:10.3934/cpaa.2022103

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2022, Volume 21, Issue 10: 3353-3369. Doi: 10.3934/cpaa.2022103

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Smooth solutions to asymptotic Plateau type problem in hyperbolic space

Zhenan Sui^1, , and
Wei Sun^2,

1.
Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology, China
2.
Institute of Mathematical Sciences, ShanghaiTech University, China

^*Corresponding author
^*Corresponding author

Received: February 2022

Revised: May 2022

Early access: June 22, 2022

Published online: June 22, 2022

The first author is supported by NSFC grant 12001138, and the second author is supported by NSFC grant 11501119 and a start-up grant from ShanghaiTech University.

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Abstract

We investigate on the existence of smooth complete hypersurface with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under the assumption that there exists an asymptotic subsolution. We give an affirmative answer for the case $ k = n $ when the asymptotic boundary $ \Gamma $ bounds a uniformly convex domain, and for $ k < n $ when $ \Gamma $ bounds a disk, utilizing Pogorelov type interior second order estimate. Our result complements our previous work [12, 13], and generalizes the asymptotic Plateau type problem to non-constant prescribed curvature case.

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Mathematics Subject Classification: Primary: 53C21; Secondary: 35J66, 58J32.

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References

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