Yamabe's conjecture

Let $X$ be a closed manifold of dimension $d\geq 3$, and let $g_0$ be a Riemannian metric on $X$. A metric $g$ on $X$ is conformally equivalent to $g_0$ if it lies in the conformal class of $g_0$.

Yamabe's conjecture. There exists a metric $g$ on $X$ conformally equivalent to $g_0$ and having constant scalar curvature.

Yamabe's conjecture asks for a constant-scalar-curvature representative in every conformal class on a closed manifold of dimension at least three. It was confirmed by the work of Trudinger, Aubin, and Schoen.