On the spectra of geometric operators evolving with geometric flows

In this work we generalise various recent results on the evolution and monotonicity of the eigenvalues of certain geometric operators under specified geometric flows. Given a closed, compact Riemannian manifold $\big(M^n,g(t)\big)$ and a smooth function $\eta\in C^{\infty}(M)$ we consider the family of operators $\mathbb{L}=\Delta - g(\nabla\eta,\nabla\cdot)+cR$, where $R$ is the scalar curvature and $c$ is some real constant. We define a geometric flow on $M$ which encompasses the Ricci, the Ricci - Bourguignon and the Yamabe flows. Supposing that the metric $g(t)$ evolves along this general geometric flow we derive a formula for the evolution of the eigenvalues of $-\mathbb{L}$ and prove monotonicity results for the eigenvalues of both $-\Delta + g(\nabla\eta,\nabla\cdot)$ and $-\mathbb{L}$. We then prove Reilly-type formula for the operator $\mathbb{L}$ and employ it to establish an upper bound for the first variation of the eigenvalues of $-\mathbb{L}$. Finally, in the pursuit of a theoretical explanation of our generalisations, we formulate two conjectures on the monotonicity of the eigenvalues of Schr\"{o}dinger operators.