Yau's dimension comparison for harmonic functions of integer polynomial growth under nonnegative Ricci curvature

Theorem 1.1: there are an even $n\ge8$ (the paper allows $n=16$), an integer $k\ge2$ ($k=50000$) and a complete smooth metric on $\mathbb R^n$ with $\mathrm{Ric}\ge0$, Euclidean near the origin, asymptotic volume ratio strictly between 0 and 1, and $h_k>h_k(\mathbb R^n)$; the metric has nonunique tangent cones at infinity. Companion: in dimension three, for $4/9<v<1$, $1<c<9v/4$ and all large $k$, metrics with $\mathrm{Ric}\ge0$, AVR $v$ and $h_k\ge c(k+1)^2$, which can be made $(1+\epsilon)$-bi-Lipschitz to Euclidean. The metric depends on $k$. Not shown: a single metric violating the bound for all large $k$, or anything about the finite-dimensionality results, which stand.

Theorem 1.1: there are an even $n\ge8$ (the paper allows $n=16$), an integer $k\ge2$ ($k=50000$) and a complete smooth metric on $\mathbb R^n$ with $\mathrm{Ric}\ge0$, Euclidean near the origin, asymptotic volume ratio strictly between 0 and 1, and $h_k>h_k(\mathbb R^n)$; the metric has nonunique tangent cones at infinity. Companion: in dimension three, for $4/9<v<1$, $1<c<9v/4$ and all large $k$, metrics with $\mathrm{Ric}\ge0$, AVR $v$ and $h_k\ge c(k+1)^2$, which can be made $(1+\epsilon)$-bi-Lipschitz to Euclidean. The metric depends on $k$. Not shown: a single metric violating the bound for all large $k$, or anything about the finite-dimensionality results, which stand.