To be more specific: Let $Q\subset\mathbb{C}$ be a Lipschitz bounded domain, and $V$ is a compact complex manifold without boundary. Consider the set of holomorphic maps $f:Q\rightarrow V$, and $f\in H^s(Q,V)$, with $s$ sufficiently large(say $s>3$). The notation $H^s(Q,V)$ means the derivatives up to order $s$ are in $L^2$(under local coordinates of $V$) Now the question is whether this set forms a complex Hilbert manifold? How do we define the coordinate chart?

Actually all I need is to identify the neighborhood of some give holomorphic map $f_0$ with an open subset of a Hilbert space(without asking for smoothness of coordinate transformation)

  • $\begingroup$ The real analogue of the question is easy, and coordinate chart can be constructed via the exponential map. $\endgroup$ – Jingrui Cheng May 31 '16 at 23:14
  • $\begingroup$ You will probably define charts (if you can) in terms of tubular neighborhoods of $f \partial Q$ and a finite chart on $V$, the harmonic maximum principle et.c. You will probably use that $\partial Q$ is both compact and piecewise rectifiable (and it is only on neighborhoods of $\partial Q$ that $s$ is really important). I sincerely hope the $Q$ you have in mind has $\partial Q$ of finitely many components? $\endgroup$ – Jesse C. McKeown Jun 1 '16 at 0:46
  • $\begingroup$ Just think of $Q$ to be simple, say $Q$ is a square or polygon. I don`t see how to use the tubular neighborhoods of $f\partial Q$. It does not necessarily behaves nicely under the image so I am not sure if tubular neighborhoods exists. Also does the use of tubular neighborhood compatible with being holomorphic? $\endgroup$ – Jingrui Cheng Jun 1 '16 at 0:58
  • $\begingroup$ so... have you answered your question yet? $\endgroup$ – Jesse C. McKeown Jun 2 '16 at 1:53
  • $\begingroup$ anyways, of course tubular neighborhoods do not behave well w.r.t. holomorphic maps! that's not the point. The point is to use holomorphic rigidity to reduce the size of your question from something 2-d to something ... somewhat less. E.g., I'm sure you can work out that a map $Q\to V$ is determined by its germ at any point, which can be specified by a power series w.r.t. some chart in $V$, but also that not all power series will work (and there are more details); but relating series to $H^s$ may be hard, so consider perhaps extension problems for boundary perturbations instead. $\endgroup$ – Jesse C. McKeown Jun 2 '16 at 7:31

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