i have the following problem. Let $M$ be a complex n-dim manifold and $X \subset M$ be a n-dim real analytic submanifold. Consider $d_{X}(z)$ be the squared distance from $z \in M$ to $X$. For $z$ sufficiently near $X$ this function is smooth. My quaestion is, if (with respect to complex coordinates $z_{1}, ..., z_{n}$) the function $\frac{\partial^{2}d_{X}}{\partial z_{i} \partial \overline{z}_{j}}$ is holomorphic? (assume that we have alredy choosen a hermitian metric to compute $d_{X}$).
is the differential of the distance function holomorphic?
2
$\begingroup$
complex-analysis
differential-geometry
complex-geometry
-
0I'm a total novice in $\mathbf{C}^n$, so it may be easy, but: Can we say something about those subsets $X$ for which the squared distance $d_X(z)$ can be written in the form $d_X(z)=f(z)\overline{f(z)}$ for some holomorphic function $f$. Works when $X$ is a point and $n=1$. Works in higher $n$, if we also allow vector valued functions $f$ (and want to write $d_X$ as the squared hermitian norm. Any other way to make this question more interesting? – 2011-08-13
1 Answers
4
No. Take $n=1$, $M=\mathbb C$ and $X=\{|z|=1\}$. Then $ d(z)=\left|z-\frac z{|z|}\right|^2=\left(z-\frac z{\sqrt{z\bar z}}\right) \left(\bar z-\frac {\bar z}{\sqrt{z\bar z}}\right)= z\bar z -2\sqrt{z\bar z}+1 $ and $ \frac{\partial^{2}d(z)}{\partial z \partial \bar{z}}= 1-\frac {1}{2\sqrt{z\bar z}}. $
-
2It makes the question look kind of sad if the first metric that comes in mind doesn't satisfy OP's wishes. I believe he should've worked a little bit more... – 2011-08-13