A new 3-parameter curvature condition preserved by Ricci flow (Q2870421)

If \((M,g)\) is a Riemannian manifold, then the Riemann curvature tensor \(R\) defines a self-adjoint linear map \(\mathcal{R}\) (called the curvature operator) from \(\Lambda^2(M)\) to itself characterized by the identity \(g(\mathcal{R}(X\wedge Y),Z\wedge W)=R(X,Y,Z,W)\).NEWLINENEWLINE Let \(\{\mu_1(\mathcal{R})\leq\dots\leq\mu_{n(n-1)/2}(\mathcal{R})\}\) be the eigenvalues of the curvature operator \(\mathcal{R}\) and let \(\{\omega_\alpha\}\) be a corresponding basis of eigen 2-forms. One says that \((M,g)\) has a ``2-positive curvature operator'' if for any \(\alpha\neq\beta\) one has \(\mu_\alpha(\mathcal{R})+\mu_\beta(\mathcal{R})>0\). In the current paper, the author defines a generalization of this condition which relies on four eigenvalues of \(\mathcal{R}\). Let \(\vec\lambda=(\lambda_1,\lambda_2,\lambda_3)\) whereNEWLINENEWLINE(1) \(0\leq\lambda_i\leq 1\).NEWLINENEWLINE(2) \(\lambda_3\leq\lambda_2\leq\lambda_1\).NEWLINENEWLINE(3) \(0<1-(\lambda_i+\lambda_j)\) for \(\lambda_j\leq \lambda_i\leq 1\).NEWLINENEWLINE(4) \(\lambda_i+\lambda_j\geq1\) for \(1\leq i\neq j\leq 3\).NEWLINENEWLINEOne says \((M,g)\) has \(\vec\lambda\)-nonnegative curvature operator if for \(1\leq\alpha<\beta<\gamma<\delta\leq n(n-1)/2\) and for \(1\leq i\neq j\leq 3\), the curvature operator \(\mathcal{R}\) satisfies:NEWLINENEWLINE(1) \(\mathcal{R}(\omega_\alpha,\omega_\alpha)+\lambda_1\mathcal{R}(\omega_\beta,\omega_\beta) +\lambda_2\mathcal{R}(\omega_\gamma,\omega_\gamma)+\lambda_3\mathcal{R}(\omega_\delta,\omega_\delta)\geq0\).NEWLINENEWLINE(2) \(\lambda_i\mathcal{R}(\omega_\alpha)+(1-(\lambda_i+\lambda_j)\lambda_j)\mathcal{R}(\omega_\beta,\omega_\beta)\geq0\).NEWLINENEWLINEThe author shows that the \(\vec\lambda\)-nonnegative curvature condition is preserved by the Ricci flow of Hamilton \(\partial_t\mathcal{R}=\mathcal{R}^2+\mathcal{R}^{\#}\) and derives the weak maximum principle for 3-parameter \(\lambda\)-nonnegativity.