Variational characterizations of the total scalar curvature and eigenvalues of the Laplacian (Q1951198)

Let \((M,g)\) be a compact Riemannian manifold of dimension \(n\) without boundary. If \(h\) is a symmetric \(2\)-tensor, the variation of the scalar curvature is given by \(s_g^\prime(h)=-\Delta_t\text{Tr}(h)+\text{div}_g(\text{div}_g(h))-g(\text{Ric}_g,h)\). The \(L^2\) adjoint \(s_g^{\prime*}\) is given by \(= S_g^{\prime*}(f)=\text{Hess}(f)-(\Delta_gf)g-f\text{Ric}_g\). The authors consider the fourth-order elliptic differential operator \(A=s_g^\prime s_g^{\prime*}\) acting on \(C^\infty M\). Let \(\kappa:=\dim\{ker(s_g^{\prime*})\}\) and let \(\nu:=\inf\{\int_M\phi A\phi dx_g/\int_M\phi^2 dx_g\}\). The authors show that \(\nu\) vanishes if and only if \(\kappa\neq0\). They show that if the first eigenvalue of the Laplace operator is large compared to its scalar curvature, then \(\nu>0\) and \(\kappa=0\). A lower bound for \(\nu\) is computed in this setting.