2
$\begingroup$

Let $B=\lbrace \|z\|<1\rbrace$ be a unit ball in $\mathbb{C}^n, n\geq 2.$ Let $f_1,\cdots, f_n, f$ be holomorphic functions on $B.$ Now, consider the following first order, linear PDE: $$f_1\partial_{z_1}(y)+\cdots +f_n\partial_{z_n}(y)+fy=0.$$ Here is my question. Is there some non-degeneracy condition of functions $f_1,\cdots, f_n$ that implies existence of a nontrivial solution $y$ which is holomorphic on $B?$ From what I understand about the Cauchy-Kovalevskaya theorem, if $f_1,\cdots, f_n$ have no common zero, then for any $w\in B$ there exists a nontrivial holomorphic solution (in fact infinitely many) defined near $w.$ But, I don't see how to get a global solution from it.Thank you very much in advance.

$\endgroup$
0
2
$\begingroup$

If the holomorphic vector field $X=\sum_i f_i\partial_{z_i}$ has no zeroes and there is a complex hypersurface nowhere tangent to it, and each point lies on a unique trajectory of that vector field through that hypersurface, then the differential equation becomes a linear ordinary differential equation for a holomorphic parameterization of each trajectory, so has global solvability. So you need to find your nowhere tangent hypersurface, and prove, for each point, both existence and uniqueness of a trajectory through that hypersurface and through that point.

$\endgroup$
1
  • $\begingroup$ I see, thanks! But I don't quite see apriori how doable that is. $\endgroup$
    – John Z.
    Sep 7 at 16:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.