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 (, 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 with positive curvature. To complement this result, we prove that asymptotic directions always exist locally about a point x₀ in either of the following cases (where K is the Gaussian curvature on): (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\`ere equation -- which is of the elliptic, hyperbolic, and mixed types in cases (a), (b), and (c), respectively -- locally equivalent to the Bao--Ratiu equations.