A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem

1. Problem Statement

The paper addresses the Mixed Christoffel Problem, a generalization of the classical Christoffel problem in convex geometry.

• Context: Given $n-1$ fixed convex bodies $\Omega_1, \dots, \Omega_{n-1}$ in $\mathbb{R}^{n+1}$ and a prescribed Borel measure $\mu$ on the unit sphere $S^n$, the goal is to find a convex body $\Omega$ such that its mixed area measure with the given bodies equals $\mu$:
  $$S(\Omega_1, \dots, \Omega_{n-1}, \Omega; \cdot) = \mu(\cdot).$$
• Geometric Formulation: In the smooth setting ($C^{2,+}$ convex bodies), this problem reduces to finding a geometric solution (a support function $u$ of a convex body) to a linear second-order elliptic partial differential equation (PDE) on the sphere:
  $$\sum_{\alpha, \beta=1}^n a_{\alpha\beta}(x) (u_{\alpha\beta}(x) + \delta_{\alpha\beta}u(x)) = f(x),$$
  where:
  • $u$ is the support function of the unknown body $\Omega$.
  • $u_{\alpha\beta}$ are the second covariant derivatives of $u$.
  • $a_{\alpha\beta}$ are the entries of the mixed co-factor matrix derived from the inverse Weingarten forms of the fixed bodies $\Omega_1, \dots, \Omega_{n-1}$.
  • $f$ is the density of the measure $\mu$.
• The Core Challenge: Standard elliptic theory guarantees the existence of a solution $u$ to the PDE. However, for $u$ to represent a valid convex body, the associated matrix $W = (u_{ij} + u\delta_{ij})$ (the inverse Weingarten form) must be positive definite everywhere on $S^n$. The paper seeks sufficient conditions on the data ($\Omega_i$ and $f$) that ensure this geometric property holds.

2. Methodology

The authors employ the Constant Rank Theorem method, a powerful technique in geometric analysis used to prove that the rank of a tensor remains constant (and often maximal) under specific convexity conditions.

• The Constant Rank Strategy:

  1. Assume a solution $u$ exists such that $W \geq 0$ (positive semi-definite).
  2. Define a test function $\phi$ related to the elementary symmetric functions of the eigenvalues of $W$. Specifically, if the rank of $W$ drops to $l$ at some point, $\phi$ is constructed to vanish at that point.
  3. Derive a differential inequality for $\phi$ of the form $L\phi \leq C(\phi + |\nabla \phi|)$, where $L$ is a linear elliptic operator.
  4. Apply the Strong Minimum Principle: If $\phi$ achieves a minimum of 0, it must be identically zero. This implies the rank of $W$ is constant.
  5. Use topological arguments (Minkowski identities) to show that if the rank is constant and the domain is the whole sphere, the rank must be maximal ($n$), implying $W > 0$.

• Handling Dimensions:

  • Case $n=2$ (3D Space): The authors derive a simplified condition involving the convexity of a specific tensor extension.
  • Case $n \geq 3$: The analysis is more complex due to the non-linearity of the condition. The authors utilize a homotopy argument, connecting the given tensor $A$ to the identity matrix $I_n$ via a path $A_t = (1-t)I_n + tA$.

• Key Technical Tool: The proof relies heavily on the Newton-McLaurin inequalities and a specific auxiliary lemma (Lemma 3.5) regarding the minimization of a functional over the set of positive semi-definite matrices with a fixed trace constraint. This lemma ensures that the minimum of the relevant functional is attained at a rank-1 matrix, allowing the authors to verify the necessary inequality condition.

3. Key Contributions and Results

A. The Constant Rank Theorem (Theorem 1.1)

The paper establishes a general constant rank theorem for the linear equation:
$$\sum a_{\alpha\beta}(u_{\alpha\beta} + u\delta_{\alpha\beta}) = f.$$
Condition: If the data satisfies a specific convexity condition (involving the Hessian of $a_{qq}/f$ and the tensor $A$), then any solution $u$ with $W \geq 0$ must have $W$ of constant rank. If the domain is the entire sphere $S^n$, then $W > 0$.

B. Sufficient Conditions for Existence

The authors translate the abstract constant rank condition into concrete geometric conditions on the input data:

1. For $n \geq 3$ (Theorem 1.3):
   A solution exists if:

   • The density $f$ satisfies the standard compatibility condition (orthogonality to linear functions).
   • The function $1/f$ (extended homogeneously) is convex (a condition known from the classical Christoffel problem).
   • Crucially: A specific matrix inequality (Condition 18) holds for the homotopy path $A_t = (1-t)I_n + tA$ for all $t \in [0,1]$. This condition ensures the convexity requirement is maintained as one deforms from the standard Christoffel problem to the mixed problem.

2. For $n = 2$ (Theorems 1.4 & 1.5):
   The condition simplifies significantly. A solution exists if:

   • $f$ satisfies the standard convexity condition (Condition 8).
   • The tensor $W_0/f$ (where $W_0$ is the inverse Weingarten form of the fixed body $\Omega_0$) has a 1-homogeneous extension that is convex in $\mathbb{R}^3$.
   • Alternatively, a pointwise inequality involving the fourth-order derivatives of the support function of $\Omega_0$ (Condition 19) holds.

C. Application to Mixed Minkowski Problem

As a corollary, the paper provides sufficient conditions for the existence of solutions to the Mixed Minkowski Problem (where the prescribed measure is a mixed area measure involving the unknown body).

4. Significance and Impact

• Generalization: The work extends the seminal results of Guan and Ma (2003) on the classical Christoffel problem to the mixed setting, where the geometry is influenced by multiple fixed convex bodies.
• Geometric Regularity: It bridges the gap between the analytic existence of PDE solutions and the geometric requirement of convexity. Without such constant rank theorems, one cannot guarantee that a PDE solution corresponds to a physical convex body.
• Dimensional Distinction: The paper highlights a fundamental difference between 3D space ($n=2$) and higher dimensions ($n \geq 3$). In 3D, the condition is a direct convexity of a tensor extension. In higher dimensions, the non-linearity of the mixed discriminant requires a more restrictive homotopy condition to ensure the convexity is preserved along the deformation path.
• Methodological Advancement: The proof introduces refined estimates for the test function $\phi$ in the high-dimensional case, utilizing specific properties of symmetric functions and matrix calculus that could be applicable to other Hessian-type equations on manifolds.

In summary, this paper provides a rigorous framework for solving the Mixed Christoffel problem by establishing that under specific convexity constraints on the data, the solution to the associated linear elliptic PDE is guaranteed to be the support function of a strictly convex body.