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 eq...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 anyΣwith positive curvature.To complement this result,we prove that asymptotic directions always exist locally about a point x_(0)∈Σin either of the following cases(where K is the Gaussian curvature onΣ):(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∇K(x_(0))≠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.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.12201399,12331008,and 12411530065)Young Elite Scientists Sponsorship Program by China Association of Science and Technology(Grant No.2023QNRC001)+3 种基金the National Key Research&Development Programs(Grant Nos.2023YFA1010900 and 2024YFA1014900)Shanghai Rising-Star Program(Grant No.24QA2703600)the Shanghai Frontier Science Center of Modern Analysissupported by the National Key Research&Development Programs(Grant Nos.2023YFA1010900 and 2024YFA1014900).
文摘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 anyΣwith positive curvature.To complement this result,we prove that asymptotic directions always exist locally about a point x_(0)∈Σin either of the following cases(where K is the Gaussian curvature onΣ):(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∇K(x_(0))≠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.
