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I am reading papers about yamabe flow. I have a problem about how people derive it as a gradient flow. Suppose we have $(M,g_0)$, $g(t)=u^{\frac{4}{n-2}}(t)g_0$ is another conformal metric. Let $R=R( …

I am reading Richard Schoen's classical example on the multiplicity of solutions of yamabe problem. He says on $S^1(T)\times S^{n-1}$, there exists a critical number $T_0$ such that if $T\leq T_0$, th …

Suppose we have a two-dimensional surface $M$ with smooth boundary $\partial M$. Equip $M$ with a metric $g$ such that the Gauss curvature $K$ of $M$ and geodesic curvature $\kappa$ of $\partial M$ ar …

I am reading the paper of Convergence of the Yamabe flow for arbitrary initial energy I am stuck by one part of the paper. Suppose $u_\infty>0$ is a smooth function on $(M, g_0)$ and $$L_0=\frac{4(n-1 …

I am reading some stuff on orbifolds. I am particularly interested in the metrics on orbifolds. The famous example of one orbifold is the "American football", which is $\mathbb{S}^2$ quotient by the g …