Berger's isoperimetric problem and minimal immersions of surfaces
The author establishes inequalities between eigenvalues of the Laplacian of a compact surface and the geometrical characteristics of this surface, and he considers the case of an orientable surface of genus one, \(T^2\). It can be shown that in the class of flat tori of fixed area the first nonzero eigenvalue of the Beltrami-Laplace operator \(\lambda_1\) attains its supremum on the equilateral torus, i.e., on the torus \(\mathbb{R}^2/\Gamma\), where \(\Gamma\) is the lattice generated by (1,0) and \((1/2,\sqrt{3/2})\). The main result of this paper is that the flat equilateral torus is \(\lambda_1\)-maximal. It means that the author gives affirmative answers to Berger's problem.
scientific article; zbMATH DE number 954723
scientific article