Let (M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function f on M such that the Riemannian metric fg has constant scalar curvature.

Let (M,g) be a closed smooth Riemannian manifold. Then there exists a positive and smooth function φ on M, and a number c, such that

4
(
n
−
1
)


n
−
2




Δ

g


φ
+

R

g


φ
+
c

φ

(
n
+
2
)

/

(
n
−
2
)


=
0.


{\\displaystyle {\rac {4(n-1)}{n-2}}\\Delta ^{g}\\varphi +R^{g}\\varphi +c\\varphi ^{(n+2)/(n-2)}=0.}

Here n denotes the dimension of M, R^g denotes the scalar curvature of g, and ∆^g denotes the Laplace-Beltrami operator of g.