Appropriateness of boundary condition (2.3) for higher dimensions

Determine whether the boundary condition ∑_{i=1}^{n+1} ∂φ/∂x_i = 0 on ∂(R_+^{n+1})—equivalently, v⋆ = 0 on ∂P—for the real Monge–Ampère problems (2.1) and (2.2) is appropriate in dimensions n + 1 > 2, in the sense of serving as the correct second boundary condition for constructing the intended complete Calabi–Yau metrics and their asymptotics.

Background

In the inductive approach to constructing complete Calabi–Yau metrics on complements of simple normal crossing divisors, the boundary behavior of the associated real Monge–Ampère equation is prescribed by a second boundary condition. For n+1 = 2, prior work established a specific boundary condition that fits the inductive structure.

The authors generalize this boundary condition to higher dimensions via (2.3) but explicitly note uncertainty about its suitability when n+1 > 2. Their existence and asymptotic results provide evidence, yet a definitive determination remains open.