An Ahlfors derivative for conformal immersions (Q2256847)

The author firstly defines the Ahlfors derivative \(\mathcal{A}\) of a conformal immersion \(F:(M^m,g)\to (P^n,h)\) that is the symmetric covariant tensor field of order two in \(M\) defined by \[ \mathcal{A}F=\mathcal{H}\varphi+\frac 12F^\ast(k_h h)+F^\ast\left(\nu^P_{F(M)}\right)-\frac 12 k_g g, \] where \(\| DF\| =e^\varphi\), the operator \(\mathcal{H}\) for a smooth function \(u:M\to \mathbb R\) is defined by \[ \mathcal{H}u=\operatorname{Hess}u-du\otimes du+\frac12\| du\|^2 g, \] \(k_g(x)=1/m(m-1)\operatorname{Scal}g(x)\), and the tensor \(\nu=\nu^P_M\) is defined by \[ \nu(X,Y)=h(\mathrm{I}\!\mathrm{I}(X,Y),\mu)-\frac 12 h(X,Y)\|\mu\|^2, \quad X,Y\in T_xM, \] in which \(\mu=\mu^P_M\) is the mean curvature vector of a submanifold \(M\subseteq (P,h)\). Then the author studies its relation to other operators, such as the Schwarzian derivative of a locally injective holomorphic function in a domain in \(\mathbb C\) and others. He provides a conceptual foundation, and establishes some basic properties. Finally, as an application, he gives some injectivity criterion for conformal immersions into \(\mathbb R^n\).