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The $Pi$-operator, also known as Ahlfors-Beurling transform, plays an important role in solving the existence of locally quasiconformal solutions of Beltrami equations. In this paper, we first construct the $Pi$-operator on a general Clifford-Hilbert module. This $Pi$-operator is also an $L^2$ isometry. Further, it can also be used for solving certain Beltrami equations when the Hilbert space is the $L^2$ space of a measure space. Then, we show that this technique can be applied to construct the classical $Pi$-operator in the complex plane and some other examples on some conformally flat manifolds, which are constructed by $U/Gamma$, where $U$ is a simply connected subdomain of either $mathbb{R}^{n}$ or $mathbb{S}^{n}$, and $Gamma$ is a Kleinian group acting discontinuously on $U$. The $Pi$-operators on those manifolds also preserve the isometry property in certain $L^2$ spaces, and their $L^p$ norms are bounded by the $L^p$ norms of the $Pi$-operators on $mathbb{R}^{n}$ or $mathbb{S}^{n}$, depending on where $U$ lies. The applications of the $Pi$-operator to solutions of the Beltrami equations on those conformally flat manifolds are also discussed. At the end, we investigate the $Pi$-operator theory in the upper-half space with the hyperbolic metric.
We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex
In this paper we shall establish some Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. In particular, we show that for $a in (0, 1)$ constants are the only $C^1$ up to the boundary posi
By using the method of Loewner chains, we establish some sufficient conditions for the analyticity and univalency of functions defined by an integral operator. Also, we refine the result to a quasiconformal extension criterion with the help of Beckerss method.
The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood of any point where the gradient of the potential function is non-null. In opposition, if the
We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product.