The first eigenvalue of the Laplacian on orientable surfaces

2021/06/01 by Mikhail Karpukhin, Karpukhin, Mikhail, Denis Vinokurov +1 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2106.00627

openalex publication_date 2021/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The famous Yang-Yau inequality provides an upper bound for the first eigenvalue of the Laplacian on an orientable Riemannian surface solely in terms of its genus γ and the area. Its proof relies on the existence of holomorhic maps to \mathbbCP¹ of low degree. Very recently, A.~Ros was able to use certain holomorphic maps to \mathbbCP² in order to give a quantitative improvement of the Yang-Yau inequality for γ=3. In the present paper, we generalize Ros' argument to make use of holomorphic maps to \mathbbCPⁿ for any n>0. As an application, we obtain a quantitative improvement of the Yang-Yau inequality for all genera γ>3 except for γ= 4,6,8,10,14.

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• First-eigenvalue maximization and inflation of maps

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