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

TL;DR

This work establishes the local solvability of the Bao–Ratiu equations for asymptotic directions on a closed 2D surface by converting the problem to a degenerate Monge–Ampère equation for a scalar potential. The authors treat three curvature regimes—elliptic ($K>0$), hyperbolic ($K<0$), and mixed-type ($K$ changing sign cleanly)—using a Han-inspired perturbative approach and carefully constructed initial approximations. They obtain precise Sobolev regularity for the asymptotic direction $X$ in terms of the ambient $H^r$ metric: $X\in H^{r-3}$ for $K>0$, $X\in H^{r-4}$ for $K<0$, and $X\in H^{r-5}$ for sign-changing curvature, with corresponding restrictions on $r$. These results complement Palmer's global nonexistence findings and connect to isometric embedding theory through a transformational strategy that parallels the Darboux equation framework.

Abstract

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 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 $Σ$ with positive curvature. To complement this result, we prove that asymptotic directions always exist locally about a point $x_0 \in Σ$ in either of the following cases (where $K$ is the Gaussian curvature on $Σ$): (a), $K(x_0)>0$; (b) $K(x_0)<0$; or (c), $K$ changes sign cleanly at $x_0$, i.e., $K(x_0)=0$ and $\nabla K(x_0) \neq 0$. 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 (c), respectively -- locally equivalent to the Bao--Ratiu equations.

Theorems & Definitions (6)

• Theorem 2
• Theorem 3
• Remark 1.1
• Lemma 2.1
• proof : Proof of Lemma \ref{['lemma 2.1']}
• Lemma 3.1: Solubility for positively symmetric hyperbolic system