The asymptotic growth of first eigenvalue of the Laplacian on compact surfaces in terms of the genus

Author: Antonio Ros

Abstract: Hersch (1970) proved that for any compact Riemannian metric on the sphere \((S^2,ds^2)\) the product of the first eigenvalue of the Laplacian \(\lambda_1(ds^2)\) and the area \(Area(ds^2),\) is bounded above by \(8\pi\) and equality is achieved only for metrics of constant curvature. In the case \(\Sigma\) is a compact orientable surface of genus \(g\), Yang and Yau shown that for any metric on \(\Sigma\) \[ \lambda_1(ds^2)Area(ds^2)\leq \big[\frac{g+3}{2}\big]8\pi, \] \([x]\) being the integer part of \(x\). In particular, the supremum of the normalized first eigenvalue \[ \Lambda_1(g)= {\mathrm Supremum} \, \big\{\lambda_1(ds^2) Area(ds^2) \, / \, ds^2 \, {\rm is\, a \,Riemannian\, metric\, on}\, \Sigma \, \big\} \] satisfies the asymptotic inequality \(\limsup_g \, \Lambda_1(g)/g\leq 4\pi\). A number of interesting related questions have been understood but many others remain unsolved. Concerning the asymptotic behaviour of the sequence \(\big\{\Lambda_1(g)\big\}\), as a first progress, Karpukhin and Vinokurov prove that \(\limsup_g \, \Lambda_1(g)/g\leq 3.416\pi\). In this talk we will present the following improved result \[ \limsup_{g\, \rightarrow\, \infty} \, \frac{1}{g}\Lambda_1(g) \leq 4(3-\sqrt{5})\pi \approx 3.056\pi. \]