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 and its eigenfunction on a ball in the sphere and on a half-space in the Gaussian space.

22pages. Comments are welcome!

References in corpus (1)

• Obata's rigidity theorem for metric measure spaces