Ricci scalars (Newman-Penrose formalism)

The DBAR problem is the problem of solving the differential equation

$\overline{\partial }f(z,\overline{z})=g(z)$

for the function $f(z,\overline{z})$, where $g(z)$ is assumed to be known and $z=x+iy$ is a complex number in a domain $R\subseteq \mathbb{ℂ}$. The operator $\overline{\partial }$ is called the DBAR operator

$\overline{\partial }=\frac{1}{2}(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y})$

The DBAR operator is nothing other than the complex conjugate of the operator

$\partial =\frac{\partial }{\partial z}=\frac{1}{2}(\frac{\partial }{\partial x}-i\frac{\partial }{\partial y})$

denoting the usual differentiation in the complex $z$-plane.

The DBAR problem is of key importance in the theory of integrable systems and generalizes the Riemann–Hilbert problem.

References

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