On Regularity Of The First Eigenvalue Of Some Operators Along Geometry Flow

In this paper, we mainly investigate the regularity of the first eigenvalue of some operators. We first study the relationship between Dini derivatives and monotone of semi-continuous real function, then drive a sufficient condition of locally Lipschitz property of a continuous function, which plays an important role in the study of the regularity of first eigenvalue of general operators along geometry flow. We then consider the regularity of first eigenvalue of p-Laplace operator along general geometry flow, we show that it is locally Lipschitz along the flow. And as an application, we come to a comparative type theorem concerning it. Next, we consider the regularity of Yamabe constant along general geometry flow, we prove that it is locally Lipschitz and direction differential along the flow. In the last, as an application, we give a partial answer to the problem whether a Yamabe metric is Einstein metric if it is a local maximum of scalar curvature on the space of constant scalar curvature metrics.