Eigenvalues of Laplacians on a closed Riemannian manifold and its nets

Abstract

We show that the eigenvalues of the Laplacian of a closed manifold <italic>M</italic> is approximated in a certain sense by the eigenvalues of the Laplacian of the graph of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over n EndFraction"> <mml:semantics> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>n</mml:mi> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {1}{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-net in <italic>M</italic> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n \to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our approximation needs no assumption on <italic>M</italic> except for dimension.