Calabi-Yau Conjecture for Minimal Hypersurfaces in $\mathbb{R}^4$ with bounded geometry

Technical Summary: Calabi–Yau Conjecture for Minimal Hypersurfaces in $\mathbb{R}^4$ with Bounded Geometry

Problem Statement
The paper addresses the Calabi–Yau conjectures for complete minimal hypersurfaces $\Sigma^n \subset \mathbb{R}^{n+1}$ with $n \ge 2$. Specifically, it investigates whether completeness necessitates that such a hypersurface is unbounded, and more strongly, whether it must be a proper map (i.e., the preimage of any compact set is compact). While this question was resolved for embedded minimal surfaces in $\mathbb{R}^3$ (proving properness under conditions of finite topology or positive injectivity radius), the higher-dimensional case ($n \ge 3$) remains largely open due to the failure of tools like the half-space theorem and the absence of analytic representations such as the Weierstrass formula.

The authors focus on the case $n=3$, considering connected, complete, embedded minimal hypersurfaces $\Sigma^3 \subset \mathbb{R}^4$. The central problem is to determine if the combination of bounded second fundamental form ($\sup_\Sigma |A_\Sigma| < \infty$) and finite second Betti number ($b_2(\Sigma; \mathbb{Z}_2) < \infty$) forces the hypersurface to be proper.

Methodology and Proof Strategy
The proof proceeds by contradiction, assuming that $\Sigma$ is complete, embedded, satisfies the geometric and topological bounds, but is not proper.

1. Reduction to a Lamination in a Half-Space/Slab:
   Following arguments from [MIR05] and recent results on the Stable Bernstein conjecture ([CL24], [CL23], [CLMS26], [CMR24], [Maz24]), the authors establish that if $\Sigma$ is not proper, it can be rigidly moved (and dilated if necessary) to become a minimal lamination contained in an open half-space $U = \{x_4 > 0\}$ or a slab $U = \{0 < x_4 < H\}$. In this configuration, the boundary hyperplane $\{x_4 = 0\}$ is contained in the limit set of $\Sigma$.

2. Analysis of the Height Function:
   Let $h = x_4|_\Sigma$ be the height function. Under the assumption of non-properness, $h$ is a positive, non-constant harmonic function on $\Sigma$ with $\inf_\Sigma h = 0$.

   • Curvature Control: The bounded curvature assumption ($|A_\Sigma| \le \Lambda$) implies a gradient estimate $|\nabla_\Sigma h|^2 \le 2\Lambda h$. This ensures that for small heights, the vertical projection is uniformly non-degenerate.
   • Graph Structure: Using Lemma 1.4 from [MIR05], the authors show that connected components of sub-level sets $\{h < \tau\}$ are global graphs over domains in $\mathbb{R}^3$ with gradient bounds $|Du| < 1$.

3. Topological Constraints and Non-Compact Components:
   Leveraging the finiteness of the second Betti number $b_2(\Sigma; \mathbb{Z}_2)$, the authors utilize a homological argument (Lemma 2.1) to show that the level sets $\{h = c\}$ can only have a finite number of compact connected components.

   • Since $\{x_4 = 0\}$ is a limit leaf, there are infinitely many distinct components of $\{h < \tau\}$.
   • By the finiteness of compact components, the authors identify a specific connected component $V \subset \{h < \tau\}$ whose boundary contains a non-compact connected component $S \subset \{h = \tau\}$ and which intersects a lower level set $\{h = t\}$ in a non-compact connected component $\Gamma$.

4. Construction of a Global Lipschitz Function:
   The component $V$ is a graph $y \mapsto u(y)$ over a domain $D \subset \mathbb{R}^3$. The authors define a function $g: \mathbb{R}^3 \to \mathbb{R}$ by setting $g(y) = \tau - u(y)$ on $D$ and $0$ elsewhere.

   • Lipschitz Property: The gradient bound $|Du| < 1$ ensures $g$ is globally 1-Lipschitz.
   • Integral Estimate: Using the minimality of $\Sigma$ and integration by parts with a cutoff function, the authors derive a weighted $L^2$ bound on the gradient:
     $$\int_{\mathbb{R}^3} \frac{|\nabla g(y)|^2}{1 + |y|} \, dy < \infty.$$

5. Deriving the Contradiction:
   The sets $S$ and $\Gamma$ project to unbounded, connected sets $S_0$ and $\Gamma_0$ in $\mathbb{R}^3$. The function $g$ takes the value $0$ on $S_0$ and a distinct constant value $a = \tau - t$ on $\Gamma_0$.

   • Because $S_0$ and $\Gamma_0$ are unbounded and connected, they intersect every sufficiently large sphere $\partial B_r(0)$.
   • The Lipschitz condition implies that on these spheres, $g$ must transition from near $0$ to near $a$ within small geodesic caps.
   • Applying a logarithmic lower bound estimate for the Dirichlet energy of such functions on spheres (citing [AM26b, Lemma 5.8]), the authors show that the integral of $|\nabla g|^2$ over the exterior of a large ball diverges logarithmically:
     $$\int_{R_0}^\infty \frac{dr}{(1+r)\log(Cr/\rho_0)} = \infty.$$
     This divergence contradicts the previously established finiteness of the weighted integral, thereby proving that the initial assumption (that $\Sigma$ is not proper) is false.

Key Contributions and Results

• Main Theorem: The paper proves that any connected, complete, embedded minimal hypersurface $\Sigma^3 \subset \mathbb{R}^4$ with bounded second fundamental form and finite second Betti number ($b_2(\Sigma; \mathbb{Z}_2) < \infty$) is proper.
• Resolution of a Specific Case: This provides the first affirmative answer to the Calabi–Yau conjecture for embedded minimal hypersurfaces in $\mathbb{R}^4$ under specific geometric and topological constraints.
• New Proof for $\mathbb{R}^3$: The methodology yields a new proof of the properness of complete embedded minimal surfaces in $\mathbb{R}^3$ with bounded curvature, independent of Xavier's half-space theorem (which was used by Rosenberg [Ros01]). This is achieved by observing that for $\Sigma^2 \times \mathbb{R}$, the second Betti number vanishes.
• Necessity of Assumptions: The authors clarify the subtlety of the bounded curvature assumption. They note that without it, the theorem would imply that all complete embedded minimal surfaces in $\mathbb{R}^3$ are proper, a statement potentially contradicted by folklore examples (Meeks). Thus, the bounded curvature condition is likely essential unless stronger topological constraints (e.g., finite first Betti number) are imposed.

Significance and Scope
The paper claims to provide a "first answer" in the direction of understanding properness for minimal hypersurfaces in higher dimensions ($n \ge 3$), where standard tools fail. The authors explicitly state that their result applies to standard families such as the 3D catenoid, examples asymptotic to Simons cones, and products of bounded curvature minimal surfaces in $\mathbb{R}^3$ with $\mathbb{R}$.

The work does not claim to resolve the conjecture for all complete embedded minimal hypersurfaces in $\mathbb{R}^4$, nor does it claim to remove the bounded curvature or finite Betti number hypotheses. Instead, it establishes a rigorous link between bounded geometry, finite topology (specifically $b_2$), and properness, highlighting that the failure of properness in higher dimensions (as seen in examples by the first two authors with unbounded curvature and infinite topology) is intimately tied to the violation of these specific conditions.