The Euler characteristic and signature of four-dimensional closed manifolds and the normalized Ricci flow equation (Q2634856)

In the present paper the author shows the following results: Theorem 1. Let \(M\) be a closed manifold of dimension 4 and let \(g(t)\), \(t\in[0,T)\), be a solution of the normalized Ricci flow with initial data the metric \(g(0)=g_0\), where \(T\) is the maximal time of the solution. If \(T=\infty\) then \(\chi(M)\geq 0\). Theorem 2. Let \(M\) be a closed manifold of dimension 4 and let \(g(t)\), \(t\in[0,T)\), a solution of the normalized Ricci flow with initial data the metric \(g(0)=g_0\), where \(T\) is the maximal time of the solution. Suppose that \(T=\infty\) and the scalar curvature \(S\) of \(g(t)\) satisfies the inequality \[ \| S\|_{2,M\times [0,\infty)}:=\left(\int_0^\infty\left(\int_MS^2 \mathrm{dvol}\right)(\tau)d\tau\right)^{1/2}\leq C<\infty. \] Then the Euler characteristic \(\chi(M)\) and the signature \(\sigma(M)\) of \(M\) satisfy the inequality \[ 2\chi(M)\geq 3|\sigma(M)|. \]