Technical Summary: Eigenvalues on Spheres

1. Problem Statement and Motivation

The paper addresses two interconnected problems in spectral geometry and the analysis of manifolds with nonnegative curvature:

1. Spectral Comparison on Spheres: It investigates whether the Laplace eigenvalues of a Riemannian or Alexandrov two-sphere with Gaussian curvature bounded below by one ($K \ge 1$) are bounded from below by the corresponding eigenvalues of the unit round sphere ($S^2_{\text{round}}$). Specifically, it asks if the inequality $\lambda_i(S^2, g) \ge \lambda_i(S^2_{\text{round}})$ holds for all $i \ge 1$, and whether equality at any positive index forces the metric to be isometric to the round metric.
2. Dimension of Harmonic Functions: It seeks to resolve the "sharp" version of Yau's finite-dimensionality question for harmonic functions with polynomial growth on complete manifolds with nonnegative sectional curvature. The question asks if the dimension of the space of such functions, $H_d(M)$, is bounded by the dimension of the corresponding space on Euclidean space, $\dim H_d(\mathbb{R}^n)$, and if equality implies the manifold is isometric to $\mathbb{R}^n$.

The authors note that while Colding and Minicozzi established a Weyl-type growth bound ($\dim H_d(M) \le C(n)d^{n-1}$), the sharp integer-degree comparison remains open. A key obstruction is that for dimensions $n \ge 4$, counterexamples to the eigenvalue comparison exist (as shown by Aryan and constructed in Section 16 for $n=4$), suggesting the result relies on specific two-dimensional structures.

2. Methodology

The paper employs a unified "abstract ladder-counting mechanism" applied across four distinct geometric realizations, progressing from simple to singular settings.

The Abstract Mechanism (Section 2)

The core analytic tool involves a sequence of closed, densely defined operators $B_m: H_m \to H_{m+1}$ between Hilbert spaces. The mechanism relies on:

• Partner Spectra: The positive spectra of $B_m^* B_m$ and $B_m B_m^*$ coincide.
• Kernel Dimensions: The dimension of $\ker B_m$ (denoted $r_m$) and the vanishing of $\ker B_m^*$ ($s_m = 0$) create an index shift.
• Ladder Inequality: A quadratic form inequality $C_m \ge A_{m+1}$, where $A_m = B_m^* B_m + \alpha_m I$ and $C_m = B_m B_m^* + \alpha_m I$.
• Recursion: Using the Min-Max principle, these ingredients generate a recursion for the counting function $N(\Lambda)$, allowing the authors to bound the number of eigenvalues below specific thresholds.

Geometric Realizations

The authors instantiate this mechanism in four stages:

1. Rotationally Symmetric Case (Part 1):

   • For metrics of the form $g = dr^2 + f(r)^2 g_{S^{n-2}}$, the Laplacian decomposes into radial operators.
   • The authors construct a "radial ladder" where $B_m$ acts on radial functions.
   • They prove $K_{\text{rad}} \ge 1$ implies the necessary form inequality and compute $\dim \ker B_m = 1$.

2. Smooth Riemannian 2-Spheres (Part 2):

   • Without rotational symmetry, the authors replace Fourier modes with line bundles $E_m = (T^{1,0}S^2)^{\otimes m}$.
   • They define first-order operators $B_m: L^2(E_m) \to L^2(E_{m+1})$ using the Levi-Civita connection.
   • Key Identity: A Riemannian ladder identity $B_m B_m^* - B_{m+1}^* B_{m+1} = 2(m+1)K_g$ is established.
   • Kernel Calculation: Using conformal invariance and stereographic projection, they show $\dim_{\mathbb{C}} \ker B_m = 2m+1$ and $\ker B_m^* = \{0\}$.
   • The curvature condition $K_g \ge 1$ ensures the form inequality $C_m \ge A_{m+1}$.

3. Complex-Geometric Reformulation (Section 10):

   • The smooth case is reinterpreted on the Riemann surface $\Sigma = (S^2, J)$.
   • The bundles $E_m$ are identified with powers of the anticanonical bundle $K^{-m}$.
   • The operators $B_m$ become Dolbeault operators $\bar{\partial}_m$.
   • Kernel dimensions are derived via Riemann-Roch and Serre duality, while the ladder inequality follows from the Bochner-Kodaira identity. This provides a structural explanation for the smooth counting argument.

4. Alexandrov 2-Spheres (Part 3):

   • The authors extend the results to singular spaces (Alexandrov surfaces) with curvature $\ge 1$.
   • They use a conformal parametrization $g_X = e^{2u}g_0$ where the curvature is a measure $\omega_X$.
   • Approximation: They employ heat regularization ($g_\tau$) to approximate the singular metric by smooth metrics, preserving the curvature bound $K_{g_\tau} \ge 1$.
   • Mosco Convergence: They prove the convergence of the graphs of the operators and their quadratic forms, allowing the transfer of spectral inequalities to the limit.
   • Rigidity Analysis: To handle the equality case, they analyze the "defect measure" $\nu_X = \omega_X - dA_X$. They prove that if the counting equality holds at a round threshold, $\nu_X$ must be atomless. A subsequent "defect identity" then forces $\nu_X = 0$, implying the space is isometric to the round sphere.

3. Key Contributions and Results

Main Theorems

• Theorem 1.8 (Smooth Rigidity): For any smooth Riemannian metric on $S^2$ with $K_g \ge 1$, $\lambda_i(S^2, g) \ge \lambda_i(S^2_{\text{round}})$ for all $i \ge 1$. Equality at any $i \ge 1$ implies $(S^2, g)$ is isometric to the unit round sphere.
• Theorem 1.10 (Alexandrov Counting): For any Alexandrov two-sphere $X$ with curvature $\ge 1$, the counting function satisfies $N_{-\Delta_X}(l(l+1)) \le (l+1)^2$ for all $l \ge 0$. If equality holds for some $l \in \mathbb{Z}^+$, then $X$ is isometric to the unit round sphere.
  • Note: The authors emphasize that equality of a single eigenvalue (e.g., $\lambda_1$) is insufficient for rigidity in the Alexandrov setting (counterexamples are "football" metrics), necessitating the stronger condition of equality at a full spectral cluster threshold.
• Theorem 1.12 (Sharp Dimension Bound): Let $(M^3, g)$ be a complete Riemannian manifold with nonnegative sectional curvature ($K_g \ge 0$) and positive asymptotic volume ratio ($AVR(M) > 0$). Then:
  $$\dim H_d(M) \le \dim H_d(\mathbb{R}^3) = (d+1)^2$$
  for all $d \ge 0$. If equality holds for some $d \in \mathbb{Z}^+$, then $(M^3, g)$ is isometric to $\mathbb{R}^3$.

Counterexamples and Limitations

• Theorem 16.1: The authors construct an explicit conformal perturbation of the round metric on $S^3$ satisfying $\text{Ric} \ge 2g$ where a specific high eigenvalue is strictly smaller than the corresponding round eigenvalue. This demonstrates that the spectral comparison theorem does not generalize to higher dimensions via a simple Ricci lower bound, highlighting the unique role of the two-dimensional line-bundle structure.

4. Significance and Claims

The paper claims to provide a complete solution to the sharp dimension question for polynomial-growth harmonic functions in dimension 3, contingent on the non-collapsed condition ($AVR > 0$).

• Methodological Significance: The work unifies spectral geometry, complex analysis, and Alexandrov geometry through a single "ladder-counting" framework. It demonstrates how complex-geometric tools (Dolbeault operators, Riemann-Roch) can be adapted to singular metric spaces via Mosco convergence.
• Rigidity Insight: The paper clarifies that for singular spaces, rigidity requires the saturation of the entire counting function at a round threshold, not just the coincidence of individual eigenvalues. This distinction is crucial for the application to tangent cones at infinity.
• Dimensional Boundary: By providing a counterexample in dimension 3 (for the cross-section) and citing recent work for higher dimensions, the paper delineates the precise boundary where the "Ricci lower bound implies spectral rigidity" intuition fails, attributing the success in dimension 2 to the specific interplay between the curvature bound and the line-bundle ladder.

The authors conclude that the success of the method in dimension 3 for harmonic functions is a direct consequence of the two-dimensional spectral rigidity of the cross-section at infinity, which is necessarily an Alexandrov sphere with curvature $\ge 1$.