Dimension of polynomial growth harmonic functions on locally conformally flat manifolds with nonnegative Ricci curvature

Technical Summary: Dimension of Polynomial Growth Harmonic Functions on Locally Conformally Flat Manifolds

Problem Statement
The paper addresses two fundamental questions raised by S.-T. Yau regarding the space $H_d(M)$ of harmonic functions with polynomial growth of degree at most $d$ on a complete Riemannian manifold $(M, g)$ with nonnegative Ricci curvature ($\text{Ric} \ge 0$).

1. Finite Dimensionality: Is $\dim H_d(M) < \infty$? (Confirmed by Colding and Minicozzi).
2. Sharp Euclidean Comparison: Is $\dim H_d(M) \le \dim H_d(\mathbb{R}^n)$?

While the finite dimensionality is established, the sharp Euclidean comparison (Question 2) remains open for general non-negatively curved manifolds. Counterexamples by Donnelly show the inequality fails for non-integer degrees $d \in (1, 2)$ in dimensions $n \ge 5$. The paper investigates whether the inequality holds for integer degrees $d$ under the specific geometric constraint that $(M, g)$ is locally conformally flat (LCF).

Methodology
The authors employ a combination of geometric analysis, potential theory, and asymptotic analysis of conformal factors. The methodology is structured as follows:

1. Classification and Reduction:
   Utilizing classification theorems by Zhu and Carron–Herzlich, the authors reduce the problem to four cases for connected complete LCF manifolds with $\text{Ric} \ge 0$:

   • Globally conformally equivalent to $\mathbb{R}^n$ (non-flat).
   • Globally conformally equivalent to a spherical space form.
   • Locally isometric to the round cylinder $\mathbb{R} \times S^{n-1}$.
   • Flat manifolds.
     The core technical effort focuses on the first case: $(M, g) = (\mathbb{R}^n, e^{2f}g_0)$ with $\text{Ric} \ge 0$.

2. Asymptotic Analysis of the Conformal Factor:
   The analysis relies heavily on the work of Ma and Qing regarding the asymptotic behavior of the conformal factor $f$ (where $g = e^{2f}g_0$).

   • They utilize the existence of an asymptotic exponent $m \in [0, 1]$ such that $\liminf_{x\to\infty} \frac{f(x)}{\ln|x|} = -m$.
   • They establish a refined relationship between the intrinsic distance $r(x)$ and the Euclidean distance $|x|$: $\lim_{x\to\infty} \frac{\ln r(x)}{\ln |x|} = 1 - m$.
   • They relate the asymptotic volume ratio $\beta$ to the exponent $m$ via $\beta = (1-m)^{n-1}$.

3. Distance Comparison and Growth Translation:
   The authors prove that intrinsic polynomial growth on $(M, g)$ translates to a specific growth rate with respect to the Euclidean metric. If $h \in H_d(M)$, then $|h(x)| \le C(1 + |x|)^{d(1-m+\epsilon)}$ for large $|x|$.

4. Weighted G-Convergence and Blow-Down:
   A central technical tool is the development of a weighted G-convergence framework.

   • The authors define rescaled metrics and weights $P_R, Q_R$ derived from the conformal factor.
   • They prove a compactness theorem (Theorem 4.2) showing that sequences of rescaled harmonic functions converge to harmonic functions on a limiting metric cone $(\mathbb{R}^n, |y|^{-2m}g_0)$.
   • This limiting operator is the Laplace-Beltrami operator on a cone with a specific warping factor.

5. Spectral Analysis on the Limit Cone:
   The authors analyze the spectrum of the limiting cone operator. They determine the homogeneous growth rates $\sigma_\ell$ of harmonic functions on the cone, which satisfy a quadratic equation involving the parameter $a = 1-m$.

   • They establish a "uniform one-step propagation" (Proposition 5.4) using the log-convexity of the weighted $L^2$ mass on the cone.
   • This propagation allows them to bound the dimension of the space of harmonic functions on the original manifold by the dimension of the space of harmonic functions on the cone with growth strictly less than the next spectral gap.

Key Results

• Main Theorem: For a complete noncompact locally conformally flat manifold $(M^n, g)$ ($n \ge 3$) with $\text{Ric} \ge 0$, and for any fixed integer $d \ge 0$:
  $$\dim H_d(M) \le \dim H_d(\mathbb{R}^n).$$
  Furthermore, if equality holds for some integer $d \ge 1$, then $(M, g)$ is isometric to Euclidean space $\mathbb{R}^n$.

• Case $\beta = 0$ (Vanishing Asymptotic Volume Ratio):
  If the asymptotic volume ratio $\beta = 0$, then $H_d(M) = \mathbb{R}$ (only constant functions) for any finite $d$. This is proven using Carron's criterion on the diameter growth of geodesic spheres and ODE analysis of radial modes.

• Case $\beta > 0$ (Euclidean Volume Growth):
  If $\beta \in (0, 1)$, the authors prove the sharper bound:
  $$\dim H_d(M) \le \dim H_{d-1}(\mathbb{R}^n) < \dim H_d(\mathbb{R}^n).$$
  This strict inequality arises because the limiting cone has a "slower" growth spectrum than Euclidean space due to the parameter $m > 0$.

• Rigidity:
  The equality $\dim H_d(M) = \dim H_d(\mathbb{R}^n)$ for an integer $d \ge 1$ forces $m=0$, implying $\beta=1$ and the manifold is isometric to $\mathbb{R}^n$.

• Non-Integer Degrees (Remark 6.3):
  The paper notes that the sharp Euclidean bound holds for non-integer $d$ as well ($\dim H_d(M) \le \dim H_d(\mathbb{R}^n)$), but the rigidity statement fails. They construct examples of non-flat LCF manifolds where $\dim H_p(M) = \dim H_p(\mathbb{R}^n)$ for non-integer $p \in (1, 2)$.

Significance and Claims
The paper provides an affirmative answer to Yau's second question (sharp Euclidean comparison) specifically for the class of locally conformally flat manifolds with nonnegative Ricci curvature.

• Extension of Li-Tam: The work is presented as an extension of Li and Tam's results on surfaces with finite total curvature to higher dimensions within the LCF category.
• Resolution of the Integer Case: It resolves the open question of whether the Euclidean bound holds for integer degrees in the presence of non-negative curvature, a case where general counterexamples exist for non-integer degrees.
• Methodological Contribution: The paper introduces a robust framework of weighted G-convergence and blow-down analysis tailored to conformal metrics with nonnegative Ricci curvature, linking the asymptotic geometry of the manifold (via the conformal factor's exponent) directly to the spectral properties of harmonic functions.

The authors do not claim to solve the problem for general manifolds with nonnegative Ricci curvature, but rather establish the result under the strong structural assumption of local conformal flatness, leveraging the specific classification and asymptotic properties available in that setting.