Harmonic Dimension Comparison Conjecture

The harmonic dimension comparison conjecture asserts that a complete -dimensional Riemannian manifold with nonnegative Ricci curvature satisfies , where counts polynomial-growth harmonic functions of order at most the nonnegative integer .

The asymptotic volume ratio is , where is the Euclidean unit-ball volume.

See also

Polynomial-Growth Harmonic Function, Ricci Curvature Tensor, Riemannian Metric

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References

Axler, S.; Bourdon, P.; and Ramey, W. Harmonic Function Theory, 2nd ed.. New York: Springer, 2001. https://doi.org/10.1007/978-1-4757-8137-3.Sung, C.-J.; Tam, L.-F.; and Wang, J. "Spaces of Harmonic Functions." J. London Math. Soc. 61, 789-806, 2000. https://doi.org/10.1112/S0024610700008759.

Cite this as:

Weisstein, Eric W. "Harmonic Dimension Comparison Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HarmonicDimensionComparisonConjecture.html

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