Spectral convergence of high-dimensional spheres to Gaussian spaces

Abstract

We prove that the spectral structure on the $N$-dimensional standard sphere of radius $(N-1{)}^{1/2}$ compatible with a projection onto the first $n$-coordinates converges to the spectral structure on the $n$-dimensional Gaussian space with variance 1 as $N\to \infty$. We also show the analogue for the first Dirichlet eigenvalue problem on a ball in the sphere and that on a halfspace in the Gaussian space.