Upper bounds for the first eigenvalue of the Laplacian on non-orientable surfaces

2015/03/29 by Mikhail A. Karpukhin, Karpukhin, Mikhail A.
Mathematics · #35P15 #58E11 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP) #math.DG #math.SP #msc:35P15 #msc:58E11 #msc:58J50

paper · pdf · doi:10.48550/arxiv.1503.08493

7 pages

arxiv created 2015/03/29 · arxiv updated 2015/03/31

Abstract

In 1980 Yang and Yau~\citeYY proved the celebrated upper bound for the first eigenvalue on an orientable surface of genus γ. Later Li and Yau~\citeLY gave a simple proof of this bound by introducing the concept of conformal volume of a Riemannian manifold. In the same paper they proposed an approach for obtaining a similar estimate for non-orientable surfaces. In the present paper we formalize their argument and improve the bounds stated in~\citeLY.

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