Affiliation:
1. Department of Mathematical Sciences , Tsinghua University , Beijing 100084 , P. R. China
2. School of Mathematical Sciences , Peking University , Beijing 100871 , P. R. China
Abstract
Abstract
In this paper, we study the following prescribed Gaussian curvature problem:
K
=
f
~
(
θ
)
ϕ
(
ρ
)
α
−
2
ϕ
(
ρ
)
2
+
|
∇
¯
ρ
|
2
,
K=\frac{\tilde{f}(\theta)}{\phi(\rho)^{\alpha-2}\sqrt{\phi(\rho)^{2}+\lvert\overline{\nabla}\rho\rvert^{2}}},
a generalization of the Alexandrov problem (
α
=
n
+
1
\alpha=n+1
) in hyperbolic space, where
f
~
\tilde{f}
is a smooth positive function on
S
n
\mathbb{S}^{n}
, 𝜌 is the radial function of the hypersurface,
ϕ
(
ρ
)
=
sinh
ρ
\phi(\rho)=\sinh\rho
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
\alpha\geq n+1
.
Our argument provides a parabolic proof in smooth category for the Alexandrov problem in
H
n
+
1
\mathbb{H}^{n+1}
.
We also consider the cases
2
<
α
≤
n
+
1
2<\alpha\leq n+1
under the evenness assumption of
f
~
\tilde{f}
and prove the existence of solutions to the above equations.
Funder
National Natural Science Foundation of China
Subject
Applied Mathematics,Analysis
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