{
  "id": "2109.07534",
  "version": "v1",
  "published": "2021-09-15T19:02:26.000Z",
  "updated": "2021-09-15T19:02:26.000Z",
  "title": "Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature",
  "authors": [
    "Xian-Tao Huang"
  ],
  "comment": "Comments are welcome!",
  "categories": [
    "math.DG"
  ],
  "abstract": "Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the cross-section $X$. Denote by $\\alpha=\\lim_{r\\rightarrow\\infty}\\frac{\\mathrm{Vol}(B_{r}(p))}{r^{n}}$ the asymptotic volume ratio. Let $h_{k}=h_{k}(M)$ be the dimension of the space of harmonic functions with polynomial growth of growth order at most $k$. In this paper, we prove a upper bound of $h_{k}$ in terms of the counting function of eigenvalues of $X$. As a corollary, we obtain $\\lim_{k\\rightarrow\\infty}k^{1-n}h_{k}=\\frac{2\\alpha}{(n-1)!\\omega_{n}}$. These results are sharp, as they recover the corresponding well-known properties of $h_{k}(\\mathbb{R}^{n})$. In particular, these results hold on manifolds with nonnegative sectional curvature and maximal volume growth.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-15T19:02:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonnegative ricci curvature",
      "polynomial growth",
      "harmonic functions",
      "maximal volume growth",
      "unique tangent cone"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}