The estimates of principle frequency of domains on Riemannian manifolds and minimal surfaces stability

Equilibrium surfaces originate from the mechanics of liquids and gases as the interface between two media that are in equilibrium. The equilibrium condition arises from the condition of minimum potential energy of the corresponding mechanical system. Equilibrium surfaces include the classes of minimal surfaces, surfaces of constant mean curvature and equilibrium capillary surfaces. The study of the stability of equilibrium surfaces is closely related to the questions of the existence of a solution to the variational multidimensional problem for the minimum of the potential energy functional. In particular, unstable solutions of the corresponding differential equations are not realizable in nature. Stability is characterized by the positivity of the form of the second variation of the corresponding functional (for example, the area functional for minimal surfaces). In most cases, this property means a lower bound for a quantity similar to the fundamental frequency of a region on a surface. In this article, I follow the approach of Sh.T. Yau obtained lower bounds for the quantity that generalizes the fundamental frequency of the region. Based on these estimates, the stability conditions for minimal surfaces and surfaces of constant mean curvature are proved.