New inequalities for eigenvalues of the Dirichlet Laplacian on the hyperbolic space

1. Problem Statement

The paper addresses the problem of establishing universal inequalities for the eigenvalues of the Dirichlet Laplacian on a bounded domain $\Omega$ within the hyperbolic space $\mathbb{H}^n(-1)$.

• The Eigenvalue Problem: The study focuses on the eigenvalues $\lambda_i$ satisfying:
  $$\begin{cases} \Delta u = -\lambda u & \text{in } \Omega, \\ u|_{\partial \Omega} = 0, \end{cases}$$
  where $0 < \lambda_1 < \lambda_2 \le \lambda_3 \le \dots$ are the successive eigenvalues.
• The Context: While universal inequalities (those independent of specific geometric quantities of $\Omega$ other than dimension) are well-established for Euclidean space ($\mathbb{R}^n$) and the Sphere ($S^n$), the case for Hyperbolic space is more complex.
• The Specific Challenge: Cheng and Yang (2009) established an inequality for $\mathbb{H}^n(-1)$ with a coefficient of 4 on the right-hand side. However, based on Weyl's asymptotic formula and recursion relations, it is conjectured that the optimal coefficient should be $4/n$.
• Cheng's Conjecture: The central open problem is whether the following inequality holds for eigenvalues in $\mathbb{H}^n(-1)$:
  $$\sum_{i=1}^k (\lambda_{k+1} - \lambda_i)^2 \le \frac{4}{n} \sum_{i=1}^k (\lambda_{k+1} - \lambda_i) \left( \lambda_i - \frac{(n-1)^2}{4} \right).$$
  Previous attempts (e.g., by Luo and Zheng, 2026) achieved a coefficient of $\frac{4}{n} \frac{\rho_{max}}{\rho_{min}}$, where $\rho$ relates to the conformal factor, but this factor is not always 1.

2. Methodology

The author employs the Cheng-Yang inequality framework, which utilizes a general inequality for eigenvalues involving test functions.

• Conformal Geometry Setup: The hyperbolic space is modeled using the Poincaré upper half-space model:
  $$\mathbb{H}^n(-1) = \left( \mathbb{R}^n_+, \frac{dx_1^2 + \dots + dx_n^2}{x_n^2} \right).$$
  The metric is conformally flat, $g = e^{2h}g_0$, where $g_0$ is the Euclidean metric and $e^{2h} = x_n^{-2}$.
• Key Tools:
  1. Cheng-Yang General Inequality: For any function $f \in C^3(M) \cap C^2(\partial M)$:
     $$\sum_{i=1}^k (\lambda_{k+1} - \lambda_i)^2 \|u_i \nabla f\|^2 \le \sum_{i=1}^k (\lambda_{k+1} - \lambda_i) \|2\nabla f \cdot \nabla u_i + u_i \Delta f\|^2.$$
  2. Conformal Transformation Formulas: The author derives specific expressions for the gradient and Laplacian under the conformal change $g = x_n^{-2} g_0$:
     $$\nabla F \cdot \nabla G = x_n^2 \nabla_0 F \cdot \nabla_0 G, \quad \Delta F = x_n^2 (\Delta_0 F + (n-2)\nabla_0 h \cdot \nabla_0 F).$$
• Strategy: The proof involves selecting specific test functions $f$ to minimize the error terms and maximize the coefficient on the right-hand side of the inequality.

3. Key Contributions and Results

A. Improvement of Previous Bounds (Theorem 1.2)

The author introduces new test functions to improve upon the recent result by Luo and Zheng.

• Test Functions:
  • $f_p = x_p$ for $p = 1, \dots, n-1$.
  • $f_n = \ln x_n$.
• Result: The paper proves the inequality:
  $$\sum_{i=1}^k (\lambda_{k+1} - \lambda_i)^2 \le \frac{\rho_{max}}{\rho_{min}} \frac{4}{n} \sum_{i=1}^k (\lambda_{k+1} - \lambda_i) \left( \lambda_i - \frac{(n-1)^2}{4} \right),$$
  where $\rho_{max} = \max_{\Omega} x_n^{-2}$ and $\rho_{min} = \min_{\Omega} x_n^{-2}$.
• Significance: This improves the constant term inside the parenthesis from $\frac{n^2-2n-4}{4}$ (in Luo-Zheng) to the optimal $\frac{(n-1)^2}{4}$, matching the conjectured form, though it still retains the geometric factor $\frac{\rho_{max}}{\rho_{min}}$.

B. Verification of Cheng's Conjecture (Corollary 1.3)

For domains where the ratio $\frac{\rho_{max}}{\rho_{min}}$ is close to 1 (i.e., domains where $x_n$ does not vary significantly), the conjecture is verified up to a small loss $\epsilon$.

• Condition: If $\frac{\rho_{max}}{\rho_{min}} \le 1 + \epsilon$.
• Result:
  $$\sum_{i=1}^k (\lambda_{k+1} - \lambda_i)^2 \le \frac{4(1+\epsilon)}{n} \sum_{i=1}^k (\lambda_{k+1} - \lambda_i) \left( \lambda_i - \frac{(n-1)^2}{4} \right).$$

C. Verification for a Second Class of Domains (Theorem 1.4)

The author proves the conjecture for a different class of domains by choosing a more sophisticated set of test functions involving inverse hyperbolic sine.

• Test Functions:
  • $f_p = \text{arcsinh}\left(\frac{x_p}{x_n}\right)$ for $p = 1, \dots, n-1$.
  • $f_n = \ln x_n$.
• Condition: The domain $\Omega$ must satisfy a geometric constraint related to the "flatness" in the horizontal directions relative to the vertical direction:
  $$\max_{x \in \Omega} \sum_{p=1}^{n-1} \frac{(x_p/x_n)^2}{1 + (x_p/x_n)^2} \le \frac{\epsilon^2}{1+\epsilon}.$$
• Result: Under this condition (and specific bounds on $\epsilon$ depending on dimension $n$), the inequality holds with the coefficient $4(1+\epsilon)/n$:
  $$\sum_{i=1}^k (\lambda_{k+1} - \lambda_i)^2 \le \frac{4(1+\epsilon)}{n} \sum_{i=1}^k (\lambda_{k+1} - \lambda_i) \left( \lambda_i - \frac{(n-1)^2}{4} \right).$$

4. Significance

1. Advancement of Spectral Geometry: The paper makes significant progress toward resolving Cheng's Conjecture regarding the sharp universal inequality for the Dirichlet Laplacian in hyperbolic space.
2. Optimization of Constants: It successfully replaces the suboptimal constant term $\frac{n^2-2n-4}{4}$ found in previous literature with the conjectured optimal term $\frac{(n-1)^2}{4}$.
3. Methodological Innovation: By utilizing arcsinh based test functions alongside standard logarithmic and linear functions, the author demonstrates a refined approach to handling the conformal factor in hyperbolic geometry, allowing for the verification of the conjecture for specific geometric configurations of domains.
4. Bridging Theory and Conjecture: While the full conjecture (coefficient exactly $4/n$ for all domains) remains open, this work verifies it "up to loss of $\epsilon$" for two distinct and physically meaningful classes of bounded domains, narrowing the gap between known results and the theoretical optimum.