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In this paper, we consider some evolution equations of generalized Ricci curvature and generalized scalar curvature under the List’s flow. As applications, we obtain $L^2$-estimates for generalized scalar curvature and the first variational formulae for non-negative eigenvalues with respect to the Laplacian.

Let $(M,g)$ be a complete noncompact Riemannian manifold. We consider gradient estimates on positive solutions to the following nonlinear equation ${\Delta }_{f}u+c{u}^{-\alpha }=0$ in $M$, where $\alpha$, $c$ are two real constants and $\alpha >0$, $f$ is a smooth real valued function on $M$ and ${\Delta }_f=\Delta -\nabla f\nabla$. When $N$ is finite and the $N$-Bakry-Emery Ricci tensor is bounded from below, we obtain a gradient estimate for positive solutions of the above equation. Moreover, under the assumption that $\infty$-Bakry-Emery Ricci tensor is bounded from below and $\left|\nabla f\right|$ is bounded from above,...

For compact hypersurfaces with constant mean curvature in the unit sphere, we give a comparison theorem between eigenvalues of the stability operator and that of the Hodge Laplacian on 1-forms. Furthermore, we also establish a comparison theorem between eigenvalues of the stability operator and that of the rough Laplacian.