# Hartog's Theorem and Entire Functions

I'm interested in multivariable Complex Analysis, and I have two questions:

My first question is as follows: after reading about Hartog's Extension Theorem I started wondering about the following problem - assume $$B_r (0)$$ is an open ball of a positive radius $$r>0$$ in $$\mathbb{C}^n$$, and let $$f:B_r(0)\to\mathbb{C}$$ be holomorphic in some neighborhood of the closed ball. Can $$f$$ be extended to an entire function in $$\mathbb{C}^n$$?

On the one hand, my reasoning says that yes, it is possible - after all, we can "invert" the ball and observe $$f(\frac1z), z\in (B_r(0)^c)$$ the complement of the ball, with $$\frac1z$$ denoting a holomorphic rotation of the sphere $$\widehat{\mathbb{C}^n}$$ interchanging $$0,\infty$$. This would make $$f(\frac1z)$$ holomorphic in the complement of some compact set of $$\mathbb{C}^n$$, thus by Hartog's Theorem it can be extended to $$\mathbb{C}^n$$, hence so does $$f$$.

Edit - by $$\frac1z$$ I mean the mapping $$z=(v,w)\rightarrow{(\bar{v}/||v||^2,\bar{w}/||w||^2)}$$ in, say, $$\mathbb{C}^2$$.It seems to me that it should be holomorphic as it is holomorphic in every component separately. Am I missing something about it?

On the other hand, I also know that any convex set is a domain of holomorphy. Therefore, given a holomorphic function defined on $$B_r (0)$$ that is not extendable to $$\mathbb{C}^n$$, I could always choose $$r_1. $$f$$ would of course be holomorphic on some neighborhood of $$B_{r_1}(0)$$, and by previous argument could be extended uniquely and holomorphically to $$\mathbb{C}^n$$, which is impossible as $$B_r (0)$$ is a domain of holomorphy.

Am I missing something here? I mean, I probably am, but I can't see what exactly.

My second question is, is there a theory for the dynamics of multivariable holomorphic functions? In particular, do the nice properties of the Julia and Fatou sets carry over to several complex variables? I haven't found much about it, and I'd like to read more...

Thanks :)

• What does $1/z$ mean if $n>1?$
– zhw.
May 1 '20 at 2:51
• Hi, I edited my question and added the definition. May 1 '20 at 11:20

Your "inversion" of the mapping is not holomorphic. There is no biholomorphic map between a punctured ball and the complement of the ball. It is precisely Hartogs's theorem that says that there is no analogue of $$\frac{1}{z}$$ in several variables, and your procedure is the essentially one way to prove that no such mapping exists.