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Splitting rigidity from polynomial growth harmonic functions under IV(M)<2

Determine whether, for every open n-manifold M with nonnegative Ricci curvature and IV(M)<2, the existence of a nonconstant polynomial growth harmonic function implies that M is isometric to the product R×N for some compact manifold N.

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Background

Sormani’s theorem shows that for manifolds with nonnegative Ricci curvature and linear volume growth, the existence of a nonconstant polynomial growth harmonic function forces an isometric splitting as R×N.

The paper proves a partial generalization (Theorem harmonicrigidity): under IV(M)<2, the existence of a nonconstant linear growth harmonic function implies an isometric splitting M≅R×N with N compact. The author then asks whether the same splitting conclusion holds under IV(M)<2 assuming the existence of a general (polynomial, not just linear) growth harmonic function.

References

Compared with Theorem \ref{polyharmonicrigidity}, we propose the following question:

Let $M$ be an open $n$-manifold with $\mathrm{Ric}\geq0$ and $\mathrm{IV}(M)<2$. If there exists a nonconstant polynomial growth harmonic function on $M$, is $M$ necessarily isometric to $\mathbb{R}\times N$ for some compact manifold $N$?

— On manifolds with nonnegative Ricci curvature and the infimum of volume growth order $<2$ (2405.00852 - Ye, 1 May 2024) in Question, Introduction (following Theorem harmonicrigidity)