Yamabe flow

Remarks

Example: on the unit sphere $S^n$ with $n \ge 2$ and round metric $g_0$, put $g(t) = \rho(t) g_0$. The scalar curvature of $g_0$ is $n(n-1)$, so that of $g(t)$ is $n(n-1)/\rho(t)$. The equation becomes $\rho'(t) g_0 = -n(n-1) g_0$, which holds for $\rho(t) = 1 - n(n-1)t$ with $0 \le t < 1/(n(n-1))$. Non-example: the family $g(t) = (1 + t) g_0$ on the flat torus fails, since the scalar curvature is $0$ but the metric changes. The right side is a multiple of $g$. So a solution has the form $g(t) = u(\cdot, t)\, g(0)$ for a positive function $u$ on $M \times [0, T)$, and it stays in the conformal class of $g(0)$. In dimension $2$ the Ricci curvature equals $\tfrac{1}{2} R g$, so the Yamabe flow coincides with the Ricci flow $\partial g/\partial t = -2 \operatorname{Ric}$.