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Are there any nontrivial compact Einstein four-manifolds of negative or nonpositive sectional curvature? by nontrivial we mean not quotients of $\mathbb{H}^4$, $\mathbb{C}H^2$, $\mathbb{H}^2\times\mat …

Maybe this is a vague question. In Besse's book Einstein manifolds, $\delta W^{\pm}=0$ is considered as a generalization of Einstein metrics on four-manifolds. I was wondering what is the difference b …

I am not sure whether I should ask for help here or math stackexchange. I got trouble with an inequality involving the Schrodinger operator on manifolds. Any suggestion is appreciated! Let $(M,g)$ be …

It is well known (see Derdzinski) that for a Kaehler metric on a four-manifold, its self-dual Weyl curvature has only two distinct eigenvalues: $$-\frac{R}{12},\ -\frac{R}{12},\ \frac{R}{6}.$$ I was …

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 thi …