Gaussian curvature of a holomorphic curve in complex 2-space

Question

Let $M\subset\mathbb C^2$ be a Riemann surface that is a holomorphic submanifold of complex 2-space. As such it inherits a Riemannian metric from $\mathbb C^2\approx\mathbb R^4$.

Each point of $M$ has a tangent "line" — a real 2-plane — which when translated to the origin of $\mathbb C^2$ becomes a complex line in $\mathbb C^2$, i.e., a point of $\mathbb{CP}^1\approx\mathbf S^2$, the unit 2-sphere.

This defines a mapping $M \to \mathbf S^2$. Now view this as a mapping between real Riemannian surfaces.

Is it true that the Jacobian determinant of this mapping at any $p\in M$ is the Gaussian curvature of $M$ at $p$?

(If not, what is it?)

And are there generalizations to holomorphic curves in $\mathbb C^n$?

Any references to the literature will be appreciated.

Answer 1

There is a classic paper by Phillip Griffiths, On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 775–814, that answers your question and quite a bit more. The formulae you want occur in Section 4 (Holomorphic curves) or can easily be derived from the formulae there. This paper is available online from the IAS.

Just so you'll have the answer with the correct normalizations: Give $\mathbb{CP}^{n-1}$, regarded as the space of lines through the origin in $\mathbb{C}^n$, its $\mathrm{U}(n)$-invariant metric $h_{n-1}$, normalized so that a linear $\mathbb{CP}^1\subset\mathbb{CP}^{n-1}$ has Gauss curvature $+1$. Let $\gamma:M\to \mathbb{CP}^{n-1}$ be the Gauss map. Then $$ \gamma^*(h_{n-1}) = -2K\,g_M\,, $$ where $g_M$ is the metric induced on $M\subset\mathbb{C}^n$ and $K\le 0$ is the Gauss curvature of $g_M$. In particular, note that the Gauss map $\gamma$ is (weakly) conformal.

Remark: Note the factor of $2$, which is not present for the Gauss map of a minimal surface in $\mathbb{R}^3$. Also, note that some sources prefer to normalize $h_{n-1}$ so that the area of a linear $\mathbb{CP}^1\subset\mathbb{CP}^{n-1}$ is $\pi$ rather than $4\pi$. In this case, the factor of $2$ in the formula above gets replaced by a factor of $\tfrac12$.

Answer 2

In any dimension, you can take a Darboux frame $A_i$ about $p$ (this is an easy effective computation) and a local coordinate about $p$ say $z$, Then in a neighborhood about the the point, $C$ is given by $p+zA+z^2B+...$. Now the question becomes computing the curvature of a graph of a function.