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Beltrami equations \(\overline{L}_t (g) = \mu ( \, \cdot \, , \, t) \, L_t (g)\) on \(S^3\) , where \(L_t\) , \(|t| < 1\) , are the Rossi operators (i.e., \(L_t\) spans the globally non-embeddable CR structure \({\mathscr {H}}(t)\) on \(S^3\) discovered by H. Rossi, [Ros]), are derived such that to describe quasiconformal mappings \(f : S^3 \rightarrow N \subset {\mathbb C}^2\) from the Rossi sphere \(\big ( S^3 \, , \, {\mathscr {H}}(t) \big )\) . Using the Greiner-Kohn-Stein solution (cf. [GKS]) to the Levi equation, and the Bargman representations of the Heisenberg group we solve the Beltrami equations for Sobolev type solutions \(g_t\) such that \(g_t - v \in W^{1,2}_F \big ( S^3 , \, \theta \big )\) with \(v \in \textrm{CR}^\infty (S^3 )\) .

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Beltrami Equations on Rossi Spheres

  • 章节

  • 发表日期：

    2025年

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  • Elisabetta Barletta,
  • Sorin Dragomir,
  • Mohammad Hasan Shahid,
  • Falleh R. Al-Solamy

摘要

Beltrami equations \(\overline{L}_t (g) = \mu ( \, \cdot \, , \, t) \, L_t (g)\) on \(S^3\) , where \(L_t\) , \(|t| < 1\) , are the Rossi operators (i.e., \(L_t\) spans the globally non-embeddable CR structure \({\mathscr {H}}(t)\) on \(S^3\) discovered by H. Rossi, [Ros]), are derived such that to describe quasiconformal mappings \(f : S^3 \rightarrow N \subset {\mathbb C}^2\) from the Rossi sphere \(\big ( S^3 \, , \, {\mathscr {H}}(t) \big )\) . Using the Greiner-Kohn-Stein solution (cf. [GKS]) to the Levi equation, and the Bargman representations of the Heisenberg group we solve the Beltrami equations for Sobolev type solutions \(g_t\) such that \(g_t - v \in W^{1,2}_F \big ( S^3 , \, \theta \big )\) with \(v \in \textrm{CR}^\infty (S^3 )\) .