Regularity of the first eigenvalue of the p-laplacian and yamabe invariant along geometric flows

We first prove that the first eigenvalue of the p-Laplace operator and the Yamabe invariant are both locally Lipschitz along geometric flows under weak assumptions without assumptions on curvature. Secondly, the Yamabe invariant is found to be directionally differentiable along geometric flows. As an application, an open question about the Yamabe metric and Einstein metric is partially answered.