Estimate Of The First Eigenvalue Of The Laplace Operator On Complete Surfaces

Isoperimetric ineqiality is an older and the most beautiful inequality in geome-try, there are so many difference proves give by difference mathematicians. Isoperi-metric inequality as a bridge connecting geometry and analysis. On the one hand, isoperimetric inequality is equivalent to the well-know Sobolev inequality. With the development of isoperimetric inequality, there are so many new Sobolev-style inequalities. On the other hand, Cheeger obtain some estimations of the first eigen-value by use the volume and area of the domain in1979, moreover, give a new method to study the first eigenvalue. Meanwhile, this also show another connection between isoperimetric inequality and analysis.In this paper, we investigate Bonnesen-style isoperimetric inequality and the first eigenvalue of Laplace on the complete surfaces. We show some Bonnesen-style isoperimetric inequalities and the lower and upper bounds of the first eigenvalue on complete surfaces which obtained by Osserman in [32]. We obtain some bounds of the first eigenvalue by using Cheeger isoperimetric constant and the Osserman’s Bonnesen-style isoperimetric inequalities. Moreover, those bounds obtained stronger then Osserman’s results.