On local solubility of Bao--Ratiu equations on surfaces related to the geometry of diffeomorphism group

We are concerned with the existence of asymptotic directions for the group of volume-preserving diffeomorphisms of a closed 2-dimensional surface (Σ,g) within the full diffeomorphism group, described by the Bao--Ratiu equations, a system of second-order PDEs introduced in [On a non-linear equation related to the geometry of the diffeomorphism group, Pacific J. Math. 158 (1993); On a non-linear equation related to the geometry of the diffeomorphism group, Pacific J. Math. 158 (1993)]. It is known [The Bao--Ratiu equations on surfaces, Proc. R. Soc. Lond. A 449 (1995)] that asymptotic directions cannot exist globally on any Σ with positive curvature. To complement this result, we prove that asymptotic directions always exist locally about a point x₀ ∈ Σ in either of the following cases (where K is the Gaussian curvature on Σ): (a), K(x₀)>0; (b) K(x₀)<0; or (c), K changes sign cleanly at x₀, i.e., K(x₀)=0 and ∇ K(x₀) ≠ 0. The key ingredient of the proof is the analysis following Han [On the isometric embedding of surfaces with Gauss curvature changing sign cleanly, Comm. Pure Appl. Math. 58 (2005)] of a degenerate Monge--Amp\`{e}re equation -- which is of the elliptic, hyperbolic, and mixed types in cases (a), (b), and (c), respectively -- locally equivalent to the Bao--Ratiu equations.