Regularity and a Liouville theorem for a class of boundary-degenerate second order equations

A classification of scalar-flat toric Kähler instantons in dimension 4

“…The functions ϕ 1 − φ1 and ϕ 2 − φ2 are zero on {y = 0}. Then the Liouville Theorem from [24] says ϕ i differs from the constructed solution φi by at worst a multiple of y 2 , completing the classification.…”

“…Theorem 5.2 (Liouville theorem, cf. Corollary 1.11 of [24]). Assume ϕ ∈ C 0 (H 2 )∩ C 2 (H 2 ) is non-negative and solves…”

“…To establish our classification we want to use off-the-shelf analytic results from [24], but before this becomes available, a good coordinate system is required against which the momentum functions can be measured. This is done by showing the volumetric normal function z : Σ 2 → H 2 is a global coordinate, in particular that the analytic function z = x + √ −1y is unramified and surjective.…”

“…In Section 5.1, we match momentum functions to any outline, in a boundary-matching process essentially the same as that from [3]. In Section 5.2 we create a slightly improved version of one of the Liouville theorems of [24]; this is the essential result that allows for the classification. In Sections 5.3, 5.4, and 5.5 we apply the Liouville theorem to classify all variations of the momentum functions found in Section 5.1.…”

“…To take care of the ϕ 2 variable, note that since ϕ 2 (0, y) = 0 and ϕ 2 ≥ 0 we can use the Liouville theorem of [24], recorded here as Theorem 5.2, to guarantee…”