Nonlocal Isoperimetric Problem in Hyperbolic Space

• The paper demonstrates that for small volumes, minimizers exist uniquely as geodesic balls using advanced variational methods.
• It employs vertical scaling and horospherical slicing to navigate the challenges posed by hyperbolic geometry's lack of dilation symmetry.
• For large volumes with α<2, the splitting method shows that nonlocal repulsive effects cause the nonexistence of minimizing sets.

The nonlocal isoperimetric problem in hyperbolic space concerns the minimization of a variational functional that combines the geometric perimeter of a set with a nonlocal repulsive term. Specifically, the problem seeks to determine, for fixed volume, those sets in $n$-dimensional hyperbolic space $(\mathbb{H}^n,g)$ that minimize the sum of hyperbolic perimeter and a double integral encoding repulsion via a singular kernel depending on hyperbolic distance. A principal focus is the existence, uniqueness, and qualitative nature of minimizers depending on volume and the kernel exponent, paralleling classical theory in Euclidean space but requiring new techniques due to the geometric and analytic peculiarities of negative curvature (Li et al., 14 Dec 2025).

1. Variational Formulation and Functionals

Given a measurable set $\Omega \subset \mathbb{H}^n$ of finite hyperbolic volume $|\Omega|_g$, the nonlocal isoperimetric energy functional is defined as: $$E(\Omega) = P_{\mathbb{H}^n}(\Omega) + \gamma \iint_{\Omega\times\Omega} \frac{1}{d_g(x,y)^\alpha}\,dV_g(x)\,dV_g(y),$$ where:

• $P_{\mathbb{H}^n}(\Omega)$ is the De Giorgi perimeter in $\mathbb{H}^n$,
• $d_g(x,y)$ is the hyperbolic geodesic distance,
• $dV_g$ is the associated Riemannian volume,
• $\alpha \in (0, n)$ is the singular-kernel exponent,
• $\gamma > 0$ is the strength of the nonlocal term.

The well-posedness of the energy requires $\alpha \in (0, n)$ to ensure finiteness, with perimeter favoring regularity and compaction, and the nonlocal term penalizing mass concentration via long-range repulsion. For fixed mass $m>0$, the problem is to determine: $E(m) = \inf\{E(\Omega)\ :\ |\Omega|_g = m\}.$ This competition underpins the dichotomy of minimizer behavior as the volume parameter $m$ varies (Li et al., 14 Dec 2025).

2. Small-Volume Regime: Existence and Uniqueness

For small mass, precise existence and uniqueness results hold:

• There exists $m_0 = m_0(n, \alpha, \gamma) > 0$ such that for all $m \in (0, m_0]$, the infimum $E(m)$ is attained.
• Any minimizer $\Omega_m$ is, up to a hyperbolic isometry, a geodesic ball of volume $m$.

The proof proceeds via the calculus of variations:

• Minimizing sequences have uniformly bounded perimeter and nonlocal energy, hence (via compactness of sets of bounded perimeter in $\mathbb{H}^n$) admit convergent subsequences.
• The energy is lower semicontinuous under $L^1$-convergence, yielding existence.
• A major difficulty in $\mathbb{H}^n$ is the lack of true dilations. Instead, a vertical scaling map $\Phi_\lambda(x',x_n)=(x',\lambda x_n)$ in the upper-half-space model is employed to relate mass and radius of candidate sets.
• Any minimizer must be contained in a ball of radius $r \sim (1 + O(r^{1/2n}))\,r$ for small $r$, implying roundness for small mass.

Regularity theory ensures the minimizer’s boundary is $C^{1,1/2}$, and quantitative stability (a Fuglede-type inequality) shows that geodesic balls are uniquely minimizing (Li et al., 14 Dec 2025).

3. Mechanisms for Nonexistence at Large Volume

If $\alpha < 2$, there exists $M_0 = M_0(n, \alpha, \gamma)$ such that for any $m \geq M_0$, the infimum $E(m)$ is not attained:

• The splitting method is central: if a large-mass minimizer existed, a horosphere $\{x_n = t\}$ divides it into pieces that can be separated to infinity in $\mathbb{H}^n$.
• While perimeter loss from splitting is controlled by twice the horospheric measure, the nonlocal energy gains—becoming arbitrarily negative as the repulsive component drops—overwhelm, lowering total energy and contradicting minimality.
• The curvature and exponential volume growth of $\mathbb{H}^n$ are crucial in these arguments, requiring an interpolation lemma (Lemma 5.1 in (Li et al., 14 Dec 2025)) to control geometry after splitting.

This demonstrates the failure of existence for minimizing sets at large mass for the kernel regime $\alpha < 2$, echoing the Euclidean scenario but requiring adaptation to hyperbolic geometry.

4. Analytical Tools and Stability Inequalities

A variety of specialized mathematical tools are necessary:

• Vertical Scaling ($\Phi_\lambda$): Substitutes for dilations, enabling comparison between sets of different volume, despite distortion of distances.
• Quantitative Stability: A Fuglede-type inequality in hyperbolic space controls the difference in nonlocal energy between a geodesic ball and perturbed competitor, bounding the nonlocal deficit via the $W^{1,2}(S^{n-1})$ norm of radial deformations and hyperbolic geometric factors ($\sinh r$, $\cosh r$).
• Boundary Regularity: Quasi-minimizer regularity theory ensures the existence of smooth boundaries for small-mass minimizers.
• Horospherical Slicing: Essential in the nonexistence argument, using the horospherical foliation to realize the splitting construction.
• Isoperimetric Profile Concavity: The strict concavity of the function $\xi(v)$ linking hyperbolic ball perimeter and volume is critical, both in the splitting argument and absence of dilations when compared to Euclidean space.

These methods constitute a toolkit specifically tailored to the analytic and geometric context of negatively curved spaces (Li et al., 14 Dec 2025).

5. Relation to the Euclidean Nonlocal Isoperimetric Problem

Important distinctions and parallels arise between the Euclidean and hyperbolic settings:

• Scale Invariance: The Euclidean case enjoys full dilation symmetry, allowing direct rescaling of solutions; no analogous symmetry exists in $\mathbb{H}^n$, and vertical scaling distorts geometries.
• Volume Growth: Hyperbolic volume grows exponentially with radius, sharpening the perimeter to volume ratio for large sets and fundamentally altering energy scaling.
• Isoperimetric Profile: In $\mathbb{R}^n$, the isoperimetric function is linear, but in $\mathbb{H}^n$, its strict concavity affects minimizer structure and nonexistence for large masses.
• Stability Inequalities: While linearization and stability are classical in Euclidean settings, in $\mathbb{H}^n$ curvature modifies all quantitative estimates.

Notwithstanding these differences, the dichotomy between unique ball-type minimizers at low volume and absence of minimizers for large mass with $\alpha<2$ persists, with curvature and lack of scaling symmetry necessitating refined tools (Li et al., 14 Dec 2025).

6. Technical Landscape and Research Directions

The current resolution of the hyperbolic nonlocal isoperimetric problem delineates the interplay between geometry, analysis, and the competing energetic effects of perimeter and nonlocality:

• The critical range for the kernel exponent ($\alpha \in (0, n)$) aligns with integrability requirements of the nonlocal term.
• The role of hyperbolic isometries in the uniqueness of minimizers.
• The potential for further study lies in quantitative stability in higher regularity ($C^{1,\beta}$), bifurcations beyond the small-mass regime, and application of analogous functionals to other negatively curved or symmetric spaces.

A plausible implication is that the technical machinery developed—vertical rescalings, horospherical slicing, adapted deficit inequalities—is applicable to a broader class of variational problems in non-Euclidean manifolds, particularly where geometric analysis and singular interactions interface (Li et al., 14 Dec 2025).