AbstractWe show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$
(
W
,
g
¯
)
with smooth boundary $$\partial W$$
∂
W
. Assume that the boundary $$\partial W$$
∂
W
is minimal in $$(W, \bar{g})$$
(
W
,
g
¯
)
. If $$(\partial W, \bar{g}|_{\partial W})$$
(
∂
W
,
g
¯
|
∂
W
)
is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$
(
S
n
-
1
,
g
S
)
, then for any Einstein metric $$\check{g} \in [\bar{g}]$$
g
ˇ
∈
[
g
¯
]
with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$
g
ˇ
=
g
¯
. Here, $$g_S$$
g
S
and $$[\bar{g}]$$
[
g
¯
]
denote respectively the standard round metric on the $$(n-1)$$
(
n
-
1
)
-sphere $$S^{n-1}$$
S
n
-
1
and the conformal class of $$\bar{g}$$
g
¯
. Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$
∂
W
⊂
(
W
,
g
¯
)
is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$
(
W
,
∂
W
,
[
g
¯
]
)
.
Four-manifolds with positive Yamabe constant