Invariant differential operators associated with a conformal metric (Q2469316)

Peschl defined invariant higher-order derivatives of a holomorphic or meromorphic function on the unit disk. Here, the invariance is concerned with the hyperbolic metric of the source domain and the canonical metric of the target domain. Minda and Schippers extended Peschl's invariant derivatives to the case of general conformal metrics. In this paper, the authors introduce similar invariant derivatives for smooth functions on a Riemann surface and show a complete analogue of Faà di Bruno's formula for the composition of a smooth function with a holomorphic map with respect to the derivatives. The authors define the invariant ope\-ra\-tors \(\partial^n_\rho\) acting on the space \(C^\infty(V)\) of smooth (complex-valued) functions on a plane domain \(V\) with smooth conformal metric \(\rho=\rho(z)|dz|\) inductively by \[ \partial^1_\rho\varphi=\partial_\rho\varphi=\frac{1}{\rho(z)} \frac{\partial\varphi(z)}{\partial z},\quad \partial^{n+1}_\rho\varphi=(\partial_\rho\circ\partial_\rho^n)\varphi- n(\partial_\rho\log\rho)\cdot\partial^{n}_\rho\varphi,\quad n\geq1, \] for \(\varphi\in C^\infty(V)\) in a natural way. Since these operators are described as tensors of specific types, it is a routine task to see that they obey certain transformation rules. The invariant derivatives \(D^nf = D^n_{\sigma,\rho}f\) arise for a holomorphic map \(f: V\to W\) between Riemann surfaces with conformal metrics \(\rho\) and \(\sigma\), respectively, \[ D^1f=\frac{\sigma\circ f}{\rho}f', \quad D^{n+1}f = [\partial_\rho-n(\partial_\rho\log\rho) + (\partial_\sigma\log\sigma)\circ f\cdot D^1f]D^nf, \quad n\geq1. \] Basic properties of the (exponential) Bell polynomials \(A_{n,k}\) are summarized as well as a principle leading to Faà di Bruno-type formulas for a sort of differential operators. This principle plays a decisive role in the proof of Theorem 1.1. Let \(V\) and \(W\) be plane domains with smooth conformal metrics \(\rho\) and \(\sigma\), respectively, and let \(f : V \to W\) be holomorphic. Then, for every function \(\varphi \in C^\infty(W)\), the relation \[ \partial ^n_\rho(\varphi\circ f) =\sum_{k=1}^n\left(\partial ^k_\sigma\varphi\right)\circ f\cdot A_{n,k}\left(D^1f,\dots,D^{n-k+ 1}f\right) \] holds for each \(n\geq 1\). Toward this end, the authors introduce an auxiliary differential operator \(d_\rho\varphi=\rho^{-1}\partial_\rho\varphi=\rho^{-2}\partial\varphi\), \(\varphi\in C^\infty(V).\) A remarkable fact is that the \(n\)th iterate of this differential operator can describe the differential operators \(\partial^n_\rho\) and \(D^n_{\sigma,\rho}\) in simple ways, which makes the proof of Theorem 1.1 dramatically short. The defining recursive relations give apparently complicated expressions of \(D^nf\). As an application of Theorem 1.1, another expression of \(D^nf\) is derived in terms of \(f^{(n)}\) and the lower-order derivatives \(D^1f,\dots, D^{n-1}f\). Moreover, the authors give concrete forms for \(D^nf\) in terms of only the ordinary derivatives \(f',\dots, f^{(n)}\) and for \(f^{(n)}\) in terms of \(D^1,\dots, D^nf\). The consequences of the previous results for the canonical surfaces \(\mathbb C_\varepsilon\) for \(\varepsilon = +1,0, -1\) are explored. An interpretation of these derivatives in terms of intrinsic geometry and some applications is also given. Although some of them are known already, we believe that the proposed approach will give a further insight even into the classical invariant derivatives.