Large dimension behavior of the Hessian eigenvalues of the unit balls

Abstract

We show that a sequence of $k$-Hessian eigenvalues of the unit ball in ${\mathbb{R}}^n$ stays bounded as long as the ratio $n/k$ stays bounded. Moreover, we identify their growth of order at least $(2-1/k)$ in $n/k$. In the case $k=n$, we show that the Monge--Ampère eigenvalues of the unit balls tend to $4$ in the large dimension limit.

v2: added Remark 1.6 and a reference. v3: final version incorporating suggestions from the referee report; to be published in Kodai. Math. J