On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space

Technical Summary: Weak Inverse Mean Curvature Flow and Minkowski-Type Inequalities in Hyperbolic Space

Problem Statement
The paper addresses the long-time behavior and regularity of the Inverse Mean Curvature Flow (IMCF) in hyperbolic space $\mathbb{H}^n$ ($3 \le n \le 7$). While IMCF is a powerful tool for proving geometric inequalities (such as Minkowski and Penrose inequalities), its application in hyperbolic space is obstructed by the lack of a "blow-down lemma" that exists in Euclidean space. In $\mathbb{R}^n$, weak solutions to IMCF are known to asymptotically approach round spheres, a property crucial for establishing monotonicity of geometric functionals. In $\mathbb{H}^n$, smooth solutions may not become round as $t \to \infty$, and the asymptotic topology of weak solutions was previously unclear. Furthermore, existing Minkowski inequalities in $\mathbb{H}^n$ were restricted to smooth, star-shaped, or mean-convex domains, leaving the case of arbitrary smooth domains (which may not be star-shaped initially) unresolved.

Methodology
The author employs a combination of geometric analysis techniques, primarily centered on an Alexandrov reflection method adapted to the hyperbolic setting.

1. Alexandrov Reflection via Sphere Inversion:
   Unlike the Euclidean moving plane method which uses reflections across hyperplanes, the author utilizes inversions across spheres in the Poincaré ball model of $\mathbb{H}^n$. By establishing a comparison principle for weak solutions under these inversions, the paper demonstrates that if the initial domain $\Omega_0$ is contained within a ball of radius $r_+$, the evolving weak flow surfaces $\Sigma_t$ eventually escape a specific region and become star-shaped.

2. Regularity via Approximation:
   The proof leverages the fact that for $3 \le n \le 7$, weak IMCF surfaces are $C^{1,\alpha}$ hypersurfaces with bounded weak mean curvature. The author invokes a theorem by Li and Wei [25], which states that a $C^1$ star-shaped domain with non-negative weak mean curvature can serve as initial data for a classical (smooth) IMCF. This bridges the gap between the weak solution (which may have jumps or singularities initially) and a smooth solution for large times.

3. Monotonicity of Geometric Functionals:
   To derive Minkowski inequalities, the paper analyzes the evolution of specific functionals under the weak flow.

   • Heintze-Karcher Inequality: The author extends the Heintze-Karcher inequality to $C^1$ hypersurfaces with bounded weak mean curvature by approximating them via Mean Curvature Flow (MCF).
   • Bulk and Surface Estimates: The growth of the bulk term ($\int_{\Omega_t} f d\Omega$) and the surface integral term ($\int_{\Sigma_t} fH d\sigma$) are estimated separately. The bulk term growth is handled via the co-area formula for Lipschitz functions, while the surface term is analyzed using elliptically regularized approximations ($u_\epsilon$) and techniques from Freire-Schwartz and Wei.

Key Contributions and Results

1. Regularity Theorem (Theorem 1.1):
   The paper proves that for any bounded initial domain $\Omega_0 \subset \mathbb{H}^n$ ($3 \le n \le 7$), the proper weak IMCF solution $\{\Omega_t\}$ becomes star-shaped and smooth after an explicit time $T$:
   $$T = (n-1) \log \left( \frac{\sinh(r_+)}{\sinh(r_-)} \right)$$
   where $r_+$ and $r_-$ are the geodesic out-radius and in-radius of $\Omega_0$. Beyond time $T$, the flow is free of "jumps" and constitutes a classical solution. This resolves the uncertainty regarding the asymptotic topology of weak IMCF in hyperbolic space.

2. Optimal Minkowski Inequalities (Theorem 1.2):
   Using the regularity result, the author derives optimal Minkowski inequalities for arbitrary smooth domains in $\mathbb{H}^n$ ($3 \le n \le 7$). These inequalities generalize previous results by Brendle-Hung-Wang and De Lima-Girao, which required the domain to be star-shaped or mean-convex. The inequalities involve the strictly minimizing hull $\Omega_0^+$ of the initial domain:
   $$\frac{1}{(n-1)w_{n-1}} \int_{\Sigma_0} |fH| d\sigma \ge \left( \frac{|\Sigma_0^+|}{w_{n-1}} \right)^{\frac{n-2}{n-1}} + \frac{n}{w_{n-1}} \int_{\Omega_0^+} f d\Omega$$
   Equality holds if and only if the boundary is a geodesic sphere (a slice of $\mathbb{H}^n$).

3. Rigidity of Ancient Solutions (Theorem 1.4):
   The gradient estimates derived for the star-shaped flow imply a rigidity result: the only proper weak solution to IMCF on $\mathbb{H}^n \setminus \{0\}$ is the expanding sphere solution. Consequently, the expanding sphere is the unique embedded ancient smooth solution converging to a point as $t \to -\infty$.

4. Penrose Inequality for Balanced Graphs (Corollary 1.3):
   The results extend the Riemannian Penrose Inequality to balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains in $\mathbb{H}^n$. The paper removes the previous requirement that the underlying domain be star-shaped, requiring only that it be outer-minimizing.

Significance
The paper claims that its primary significance lies in overcoming the obstruction posed by the lack of a blow-down lemma in hyperbolic space. By proving that weak IMCF solutions eventually become smooth and star-shaped, the author enables the application of monotonicity arguments (previously restricted to smooth or star-shaped cases) to arbitrary smooth domains. This allows for the extension of fundamental geometric inequalities (Minkowski and Penrose) to a much broader class of initial data in hyperbolic geometry, specifically for dimensions $3 \le n \le 7$. The work also provides a complete classification of proper weak IMCF on $\mathbb{H}^n \setminus \{0\}$, characterizing expanding spheres as the unique solution of this type.