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<p>We are concerned with the existence of asymptotic directions for the group of volume-preserving diffeomorphisms of a closed 2-dimensional surface (Σ, <i>g</i>) within the full diffeomorphism group, described by the Bao-Ratiu equations, a second-order PDE system introduced by Bao et al. (1993). It is known by Palmer (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 <i>x</i><sub>0</sub> ∈ Σ in either of the following cases (where <i>K</i> is the Gaussian curvature on Σ): (a) <i>K</i>(<i>x</i><sub>0</sub>) &gt; 0; (b) <i>K</i>(<i>x</i><sub>0</sub>) &lt; 0; or (c) <i>K</i> changes sign cleanly at <i>x</i><sub>0</sub>, i.e., <i>K</i>(<i>x</i><sub>0</sub>) = 0 and ∇<i>K</i>(<i>x</i><sub>0</sub>) ≠ 0. The key ingredient of the proof is the analysis following Han (2005) of a degenerate Monge-Ampère equation, which is of the elliptic, hyperbolic, and mixed types in the cases (a)–(c), respectively, and is locally equivalent to the Bao-Ratiu equations.</p>

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

  • 期刊论文

  • 发表日期：

    2025年9月23日

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  • Siran Li,
  • Xiangxiang Su

摘要

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 second-order PDE system introduced by Bao et al. (1993). It is known by Palmer (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 (2005) of a degenerate Monge-Ampère equation, which is of the elliptic, hyperbolic, and mixed types in the cases (a)–(c), respectively, and is locally equivalent to the Bao-Ratiu equations.