Let
(
M
,
g
)
(M,g)
be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature
K
≤
−
1
K\le -1
. If
f
f
is a compactly supported function of bounded variation on
M
M
, then
f
f
satisfies the Sobolev inequality
\[
4
π
∫
M
f
2
d
A
+
(
∫
M
|
f
|
d
A
)
2
≤
(
∫
M
‖
∇
f
‖
d
A
)
2
.
4\pi \int _M f^2\,dA+ \left (\int _M |f|\,dA \right )^2\le \left (\int _M\|\nabla f\|\,dA \right )^2.
\]
Conversely, letting
f
f
be the characteristic function of a domain
D
⊂
M
D\subset M
recovers the sharp form
4
π
A
(
D
)
+
A
(
D
)
2
≤
L
(
∂
D
)
2
4\pi A(D)+A(D)^2\le L(\partial D)^2
of the isoperimetric inequality for simply connected surfaces with
K
≤
−
1
K\le -1
. Therefore this is the Sobolev inequality “equivalent” to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces. Under the same assumptions on
(
M
,
g
)
(M,g)
, if
c
:
[
a
,
b
]
→
M
c\colon [a,b]\to M
is a closed curve and
w
c
(
x
)
w_c(x)
is the winding number of
c
c
about
x
x
, then the Sobolev inequality implies
\[
4
π
∫
M
w
c
2
d
A
+
(
∫
M
|
w
c
|
d
A
)
2
≤
L
(
c
)
2
,
4\pi \int _M w_c^2\,dA+ \left (\int _M|w_c|\,dA \right )^2\le L(c)^2,
\]
which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature
≤
−
1
\le -1
.