AbstractFor a closed, connected direct product Riemannian manifold $$(M, g)=(M_1, g_1) \times \cdots \times (M_l, g_l)$$
(
M
,
g
)
=
(
M
1
,
g
1
)
×
⋯
×
(
M
l
,
g
l
)
, we define its multiconformal class $$ [\![ g ]\!]$$
[
[
g
]
]
as the totality $$\{f_1^2g_1\oplus \cdots \oplus f_l^2g_l\}$$
{
f
1
2
g
1
⊕
⋯
⊕
f
l
2
g
l
}
of all Riemannian metrics obtained from multiplying the metric $$g_i$$
g
i
of each factor $$M_i$$
M
i
by a positive function $$f_i$$
f
i
on the total space M. A multiconformal class $$ [\![ g ]\!]$$
[
[
g
]
]
contains not only all warped product type deformations of g but also the whole conformal class $$[\tilde{g}]$$
[
g
~
]
of every $$\tilde{g}\in [\![ g ]\!]$$
g
~
∈
[
[
g
]
]
. In this article, we prove that $$ [\![ g ]\!]$$
[
[
g
]
]
contains a metric of positive scalar curvature if and only if the conformal class of some factor $$(M_i, g_i)$$
(
M
i
,
g
i
)
does, under the technical assumption $$\dim M_i\ge 2$$
dim
M
i
≥
2
. We also show that, even in the case where every factor $$(M_i, g_i)$$
(
M
i
,
g
i
)
has positive scalar curvature, $$ [\![ g ]\!]$$
[
[
g
]
]
contains a metric of scalar curvature constantly equal to $$-1$$
-
1
and with arbitrarily large volume, provided $$l\ge 2$$
l
≥
2
and $$\dim M\ge 3$$
dim
M
≥
3
.
On isospectral compactness in conformal class for 4-manifolds
