I'm reading A survey on classical minimal surface theory, by William H. Meeks and Joaquín Pérez. In the early beginning, they start giving eight definitions of minimal surfaces. The last of them is
Definition 2.1.8 A surface $M\subset \Bbb R^3$ is minimal if and only if its stereographically projected Gauß map $g:M\to \Bbb{C}\cup \{\infty\}$ is meromorphic with respect to the underlying Riemann surface structure.
The authors try to justify this definition as follows: a previous definition of minimal surfaces says that a surface $M\subset \Bbb R^3$ is minimal if it has indentically zero mean curvature. So, if $N:M\to \Bbb S^2\subset \Bbb R^3$ is the usual Gauß map, we know that $-dN_p:T_pM\to T_p\Bbb S^2\cong T_pM$ is a linear symmetric operator $\forall p\in M$ and, therefore, choosing an orthonormal basis for $T_pM$, $$-dN_p=\begin{pmatrix} a&b\\b&c\end{pmatrix},$$ a symmetric matrix. Since $$H=\text{arithmetic average of principal curvatures} = \frac{1}{2}\mathrm{trace} (-dN_p),$$ we must have $c=-a$, in order to get a minimal surface. Then they say that, identifying $\Bbb S^2$ with the Riemann sphere $\Bbb C\cup \{\infty\}$ together with the Cauchy-Riemann equations, this would show that the Gauß map $g:M\to \Bbb C\cup\{\infty\}$ must satisfy the definition above.
My doubt is: the matrix $$-dN_p=\begin{pmatrix} a&b\\ b&-a\end{pmatrix}$$ does not satisfy Cauchy-Riemann equations. For this, it should have the form $$\begin{pmatrix} a&-b\\ b&a\end{pmatrix}.$$
However, if I invert the orientation of $\Bbb C\cup\{\infty\}$ and "pretend" I did nothing, the matrix gets that form and Cauchy-Riemann equations are indeed satisfied.
Furthermore, other sources (like this Handbook of Differential Geometry) says in page 228 that such Gauß maps of minimal surfaces are indeed anti-holomorphic.
Which definition is correct?
