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

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

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

VibeMathed record: Yau's dimension comparison for harmonic functions of integer polynomial growth under nonnegative Ricci curvature evidence for this event

Lean formalization (HarmonicGrowth main) evidence for this event

Companion manuscript: A Three-Dimensional Counterexample to Integer-Degree Harmonic Dimension Comparison evidence for this event

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. parent of this event