Weinstock inequality in hyperbolic space II

Technical Summary: Weinstock Inequality in Hyperbolic Space II

Problem Statement
This paper addresses the shape optimization problem for the first non-zero Steklov eigenvalue, $\sigma_1(\Omega)$, on bounded domains within hyperbolic space $\mathbb{H}^n$ for dimensions $n \ge 3$. Specifically, the authors investigate the validity of a perimeter-constrained Weinstock-type inequality. The central question, formulated as Conjecture 1 (Open Question 4.27 in [13]), asks whether, for a bounded convex domain $\Omega \subset \mathbb{H}^n$, the eigenvalue $\sigma_1(\Omega)$ is maximized by a geodesic ball $\Omega^*$ having the same boundary area ($|\partial \Omega^*| = |\partial \Omega|$). While the conjecture was known to hold for $n=2$ (via conformal mapping techniques by Fraser and Schoen [15]) and for the volume-constrained case in higher dimensions (Binoy and Santhanam [5]), the perimeter-constrained case for $n \ge 3$ remained open. The authors previously resolved this for $n \ge 4$ in [23]; this paper extends the result to the critical case $n=3$ and provides a unified proof for all $n \ge 3$ under the assumption of star-shaped mean convex boundaries.

Methodology
The proof strategy synthesizes several advanced geometric and analytic tools:

1. Inverse Mean Curvature Flow (IMCF): The authors utilize the IMCF, introduced by Gerhardt [21], to evolve the boundary of the domain $\Sigma_t$. This flow preserves the star-shaped and mean convex properties while expanding the surface area exponentially. The evolution equations for geometric quantities along this flow are used to establish monotonicity formulas.
2. Weighted Mass Transplantation: A core innovation of this work is the development of a "weighted mass transplantation" method. Standard mass transplantation (Weinberger [44]) compares integrals over a domain $\Omega$ and a geodesic ball $B(R)$ of equal volume. However, for lower dimensions ($n=3, 4$), the specific auxiliary functions required for the proof lack global monotonicity. To overcome this, the authors introduce a weighted variant (Theorem 4.8) inspired by the Bathtub principle, allowing for the comparison of integrals involving non-monotonic weight functions.
3. Auxiliary Functions and Geometric Analysis: The authors define a specific volume-area ratio function $g(r)$ associated with geodesic balls. They rigorously analyze the analytic properties of $g(r)$ and its derivatives, establishing that certain combinations of these functions (such as $\lambda^2 g'/g$) are monotonic. An auxiliary function $h$, defined via the inverse of the volume integral of a specific radial term, is constructed to bridge the gap between the boundary integrals and the interior energy.
4. Variational Characterization: The upper bound for $\sigma_1(\Omega)$ is derived using the Rayleigh quotient with carefully chosen test functions involving the center of mass and the function $g(r)$.

Key Contributions and Results

• Main Theorem (Theorem 1.2): The authors prove that for any smooth bounded domain $\Omega \subset \mathbb{H}^n$ ($n \ge 3$) with a star-shaped mean convex boundary, and for a geodesic ball $\Omega^*$ with $|\partial \Omega^*| = |\partial \Omega|$, the inequality
  $$\sigma_1(\Omega) \le \sigma_1(\Omega^*)$$
  holds. Equality occurs if and only if $\Omega$ is a geodesic ball. This result provides an affirmative answer to Conjecture 1 for convex domains (Corollary 1.4).
• Resolution of the $n=3$ Case: The paper specifically addresses the analytical difficulties arising in dimension $n=3$ (and $n=4$), where standard monotonicity arguments fail. By introducing the weighted mass transplantation method, the authors successfully extend the inequality to these lower dimensions, completing the proof for all $n \ge 3$.
• Extension to Higher Eigenvalues (Corollary 1.6): As a byproduct of their estimates, the authors establish an isoperimetric-type inequality for the harmonic mean of the first $n-1$ non-zero Steklov eigenvalues. They prove that for star-shaped mean convex domains,
  $$\sum_{i=1}^{n-1} \frac{1}{\sigma_i(\Omega)} \ge \sum_{i=1}^{n-1} \frac{1}{\sigma_i(\Omega^*)}$$
  with equality if and only if $\Omega$ is a geodesic ball. This extends Verma's isoperimetric inequality [42] to the hyperbolic setting.

Significance and Claims
The paper claims to "completely settle" the problem of the perimeter-constrained Weinstock inequality in hyperbolic space for dimensions $n \ge 3$. By bridging the gap for $n=3$ through the novel weighted mass transplantation technique, the authors provide a unified framework that confirms the rigidity of the geodesic ball as the unique maximizer of the first Steklov eigenvalue under fixed perimeter constraints in hyperbolic geometry. The work resolves a specific open question in spectral geometry (Open Question 4.27 in [13]) and demonstrates that, unlike in positively curved spaces where rigidity often fails, negatively curved spaces like $\mathbb{H}^n$ exhibit strong spectral rigidity for convex domains. The methodology, particularly the weighted mass transplantation, is presented as a tool to overcome analytical obstructions caused by the lack of global monotonicity in lower dimensions.