Asymptotic Plateau problem for $3$-convex hypersurface in $\mathbb{H}^5$

1. Problem Statement

The paper addresses the Asymptotic Plateau Problem in hyperbolic space $\mathbb{H}^{n+1}$ (specifically for dimension $n=4$, i.e., $\mathbb{H}^5$). The goal is to prove the existence of a smooth, complete hypersurface $\Sigma$ that satisfies a prescribed curvature equation and has a prescribed asymptotic boundary $\Gamma$ at infinity.

• The Equation: The hypersurface must satisfy:
  $$\prod_{i=1}^n (H - \kappa_i) = ((n-1)\sigma)^n$$
  where $H = \sum \kappa_i$ is the mean curvature, $\kappa_i$ are the principal curvatures, and $\sigma \in (0, 1)$ is a constant. This equation defines $(n-1)$-convex hypersurfaces. For $n=4$, this corresponds to 3-convex hypersurfaces.
• The Boundary Condition: The asymptotic boundary $\partial_\infty \Sigma = \Gamma$ is a disjoint collection of smooth closed embedded $(n-1)$-dimensional submanifolds in $\partial_\infty \mathbb{H}^{n+1} \cong \mathbb{R}^n$. It is assumed that $\Gamma$ has non-negative Euclidean mean curvature.
• The Challenge: Previous work by Guan and Spruck [15] established existence for general curvature functions but required a restrictive condition $\sigma > \sigma_0$ (where $\sigma_0 \approx 0.37$) to obtain uniform global curvature estimates. The singularity of the equation at the boundary $\Gamma$ makes deriving estimates independent of the approximation parameter $\epsilon$ extremely difficult. The main objective of this paper is to remove the assumption $\sigma > \sigma_0$ and prove existence for any $\sigma \in (0, 1)$.

2. Methodology

The proof follows the continuity method and the approximation scheme used in previous literature, but introduces novel techniques to handle the specific algebraic structure of the 3-convex curvature function in dimension 4.

A. Approximation and A Priori Estimates

The authors consider an approximate Dirichlet problem where the boundary condition is shifted by $\epsilon$ (i.e., $u = \epsilon$ on $\Gamma$). To pass to the limit $\epsilon \to 0$, one requires uniform global curvature estimates (specifically, a bound on the principal curvatures $|\kappa[\Sigma]| \leq C$ independent of $\epsilon$).

• All estimates up to the second order were previously established in [15] without the $\sigma > \sigma_0$ restriction.
• The core difficulty lies in the second-order estimate (the $C^2$ estimate), which requires analyzing the concavity of the curvature function.

B. The Test Function

The authors employ a maximum principle argument using a test function of the form:
$$M_0 = \sup_{X \in \Sigma} \frac{H}{(\nu_{n+1})^\beta}$$
where $H$ is the mean curvature, $\nu_{n+1}$ is the vertical component of the unit normal, and $\beta > 1$ is a parameter to be optimized.

C. Lagrange Multiplier Method for Concavity

A significant innovation of this paper is the application of the Lagrange multiplier method to compute the extreme value of the concavity of the function $f(\kappa)$.

• In the standard approach, one must estimate third-order terms arising from the linearization of the curvature equation.
• The authors formulate the estimation of the quadratic form involving third-order terms as a constrained optimization problem.
• They solve a system of linear equations derived from the Lagrange multipliers to find the exact minimum of the concavity term under the constraints imposed by the equation and the test function.
• This allows them to handle the difficult third-order terms delicately, balancing them against other terms in the hyperbolic setting.

D. Computational Algebra (Mathematica)

Due to the high dimensionality ($n=4$) and the complexity of the curvature function, the resulting algebraic expressions are extremely complicated.

• The authors use Mathematica to perform symbolic computations, specifically to:
  1. Derive the exact expressions for the critical points of the Lagrange multiplier system.
  2. Compute the exact value of the concavity quantity for each "layer" $i$ (corresponding to the principal curvature $\kappa_i$).
  3. Verify the non-negativity of high-degree homogeneous polynomials that arise in the estimation process.
• The paper explicitly notes that for $n=4$, the estimation requires dividing the problem into multiple cases based on the relative sizes of the principal curvatures (e.g., $\kappa_3 \geq -\phi \kappa_1$ vs. $\kappa_3 \leq -\phi \kappa_1$), a complexity not present in the $n=3$ case.

3. Key Contributions

1. Removal of the $\sigma > \sigma_0$ Restriction: The paper proves the existence of solutions for the asymptotic Plateau problem for any $\sigma \in (0, 1)$, resolving an open problem left by Guan and Spruck [15] for this specific class of curvature functions.
2. Lagrange Multiplier Technique: It introduces a systematic method using Lagrange multipliers to compute the extreme values of concavity for fully nonlinear operators. This technique is shown to be applicable to other operators (like $\sigma_2$ and $\sigma_2/\sigma_1$) and potentially recovers known inequalities (like Shankar-Yuan's almost Jacobi inequality) in a new form.
3. Detailed Analysis for $n=4$: The paper provides a rigorous, layer-by-layer analysis for dimension 4. It demonstrates that the $n=4$ case is structurally richer and more complex than $n=3$, requiring specific parameter choices and case distinctions that were not necessary in lower dimensions.
4. Explicit Parameter Selection: The authors provide explicit values for the parameters used in the proof (e.g., $\beta \approx 2.999$, $\theta \approx 0.499$, $\phi = 0.7$) that satisfy all necessary inequalities derived from the algebraic analysis.

4. Main Result

Theorem 1.9: Let $\Gamma = \partial \Omega \times \{0\} \subset \mathbb{R}^5$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^4$ with non-negative Euclidean mean curvature. For any constant $\sigma \in (0, 1)$, there exists a complete hypersurface $\Sigma$ in $\mathbb{H}^5$ satisfying the prescribed curvature equation:
$$\prod_{i=1}^4 (H - \kappa_i) = (3\sigma)^4$$
with asymptotic boundary $\Gamma$. Furthermore, the principal curvatures of $\Sigma$ are uniformly bounded:
$$|\kappa[\Sigma]| \leq C$$
where $C$ depends only on $\sigma$ and the geometry of $\Gamma$.

5. Significance

• Geometric Analysis: This work advances the understanding of fully nonlinear elliptic equations in hyperbolic space, specifically for curvature functions that are not elementary symmetric polynomials but rather products of shifted curvatures.
• Methodological Advancement: The successful application of the Lagrange multiplier method to compute concavity extremes in a geometric PDE context offers a new tool for tackling similar problems where standard concavity estimates fail or are insufficient.
• Completeness of Theory: By removing the $\sigma > \sigma_0$ barrier, the paper completes the existence theory for this specific class of 3-convex hypersurfaces in $\mathbb{H}^5$, aligning the results with the broader theory of asymptotic Plateau problems for general curvature functions.
• Computational Rigor: The paper highlights the role of computer algebra systems in modern geometric analysis, demonstrating how symbolic computation can be used to verify complex algebraic inequalities that are intractable by hand, thereby enabling proofs in higher dimensions.