Abstract
Abstract
In this article, we prove that on any compact Riemann surface
(
M
,
∂
M
,
g
)
with non-empty smooth boundary
∂
M
and a Riemannian metric g, (i) any
K
∈
C
∞
(
M
)
is the Gaussian curvature function of some Riemannian metric on M; (ii) any
σ
∈
C
∞
(
∂
M
)
is the geodesic curvature of some Riemannian metric on M. These geometric results are obtained analytically by solving a semi-linear elliptic equation
−
Δ
g
u
=
K
e
2
u
on M with oblique boundary condition
∂
u
∂
ν
=
σ
e
u
. One essential tool is the existence results of Brezis–Merle type equations
−
Δ
g
u
+
A
u
=
K
e
2
u
in
M
and
∂
u
∂
ν
+
κ
u
=
σ
e
u
on
∂
M
with given functions
K
,
σ
and some constants
A
,
κ
. In addition, we rely on the extension of the uniformization theorem given by Osgood, Phillips and Sarnak.