On local solubility of Bao--Ratiu equations on surfaces related to the geometry of diffeomorphism group
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908794689159168 |
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| author | Li, Siran Su, Xiangxiang |
| author_facet | Li, Siran Su, Xiangxiang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15095 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On local solubility of Bao--Ratiu equations on surfaces related to the geometry of diffeomorphism group Li, Siran Su, Xiangxiang Differential Geometry Analysis of PDEs 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. |
| title | On local solubility of Bao--Ratiu equations on surfaces related to the geometry of diffeomorphism group |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2407.15095 |