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 $(\mathrm{\Sigma },g)$ 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 the geometric origin and the solvability of a degenerate Monge–Ampere equation, Proc. Symp. Pure Math. 54 (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 $\mathrm{\Sigma }$ with positive curvature. To complement this result, we prove that asymptotic directions always exist locally about a point $x_0\in \mathrm{\Sigma }$ in either of the following cases (where $K$ is the Gaussian curvature on $\mathrm{\Sigma }$): (a), $K(x_0)>0$; (b) $K(x_0)<0$; or ©, $K$ changes sign cleanly at $x_0$, i.e., $K(x_0)=0$ and $\mathrm{\nabla }K(x_0)\mathrm{\ne }0$. 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–Ampère equation – which is of the elliptic, hyperbolic, and mixed types in cases (a), (b), and ©, respectively – locally equivalent to the Bao–Ratiu equations.