The title quote is from p.221 of the 2010 book,
*The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions*
by Shing-Tung Yau and Steve Nadis. "Nash's theorem" here refers to the Nash embedding theorem
(discussed in an earlier MO question: "[Nash embedding theorem for 2D manifolds](https://mathoverflow.net/questions/37708/)").
I would appreciate any pointer to literature that explains where and why embedding fails.
This is well-known in the right circles, but I am having difficulty locating sources. Thanks!



"The complex version of Nash's theorem is not true"
The failure is actually more profound than you might guess at first glance: 


There are conformal metrics on the Poincare disk that cannot (even locally) be isometrically induced by embedding in $\mathbb{C}^n$ by any holomorphic mapping. For example, there is no complex curve in $\mathbb{C}^n$ for which the induced metric has either curvature that is positive somewhere or constant negative curvature.


You can get around the positivity problem by looking for complex curves in $\mathbb{P}^n$ (with the Fubini-Study metric, say), but even there, there are no complex curves with constant negative curvature.


More generally, for any Kahler metric $g$ on an $n$-dimensional complex manifold $M$, there always exist many metrics on the Poincare disk that cannot be isometrically induced on the disk via a holomorphic embedding into $M$. 


This should not be surprising if you are willing to be a little heuristic: Holomorphic mappings of a disk into an $n$-dimensional complex manifold essentially depend on choosing $n$ holomorphic functions of one complex variable and each such holomorphic function essentially depends on choosing two (analytic) real functions of a single real variable. However, the conformal metrics on the disk depend essentially on one (positive) smooth function of two variables, which is too much `generality' for any finite number of functions of a single variable to provide. 


