# Positive solutions to Yamabe problem?

Yamabe problem ensures that for any Riemannian metric $g$, in its conformal class $[g]$ there always exists a metric $\bar g$ whose scalar curvature $\bar R$ is constant.

I was wondering whether this is a possible way to find metrics with positive scalar curvature? i.e., for a Riemannian metric $g$, in its conformal class $[g]$ does there exist a metric $\bar g$ whose scalar curvature $\bar R$ is positive? If $R$ is positive, this trivial. If $R$ is negative, this is not possible by the conformal change of scalar curvature. If $R$ is positive somewhere and negative somewhere, is it possible to find a solution?

In other words, suppose $\phi>0$, is there a sufficient condition for $f$ such that $$-\Delta u+fu=\phi$$ has a positive solution?

• I don't understand how your displayed equation has anything to do with the paragraphs above. The conformal change equation for scalar curvature is well-known (see Wikipedia or this article), and is definitely non-linear. It differs from your linear equation. Did you mean to have a term $u^q$ for some $q$ on the right hand side? Jun 22, 2015 at 9:09
• I use $e^{2h}$ as the conformal factor, and after some change of variable we get an equation of the form that I wrote. Jun 24, 2015 at 16:23

First observe that by the positive solution to the Yamabe problem we can assume that the background metric $g$ has constant scalar curvature $S$. The conformal change of metric formula gives that the new scalar curvature $\tilde{S}$ is given by $$\frac{4(n-1)}{n-2} \triangle_g u + S u = \tilde{S} u^{p-1}$$ where $p = 2n/(n-2)$. When $M$ is compact, we can integrate over $M$ and obtain that $$\int S u = \int \tilde{S} u^{p-1}$$ Observing that the conformal factor needs to be positive, this means that
Theorem: if $g$ has zero or negative Yamabe invariant, there does not exist a conformal metric with positive scalar curvature.
Theorem (p289 of Aubin's Yamabe paper) If the compact manifold $(M,g)$ is not conformal to the sphere with the standard metric, there exists a constant $k > 1$ (which depends on the manifold) such that any smooth $f$ satisfying $$0 < \sup f \leq k \inf f$$ is the scalar curvature of some metric conformal to $g$.
Theorem (Escobar-Schoen) When $n = 3$ and $(M,g)$ not conformal to the sphere with the standard metric, if $\sup f > 0$ then $f$ is the scalar curvature of some metric conformal to $g$.