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See, for example, L. Ronkin, Introduction to the theory of entire functions of several variables, AMS 1974, MR0346175 Ch. I section 3, and Ch. IV, section 4. Other books on entire functions of severa …

Your condition $\dim M>2$ does not save the situation: you can have many counterexamples with $M=M'\times C^n$ where $\dim M'=1$ and your functions are independent of the second variable. And in dimen …

Invariant metrics and pseudometrics are used to study holomorphic maps between complex manifolds. The model is the use of the hyperbolic metric in dimension 1. For example, a manifold is called hyperb …

On my opinion, there are two very general reasons why analytic functions are important. Solutions of many (almost all) important differential (and functional) equations are analytic. For example, …

(Edit). I think the answer is no. Let $\Omega$ be a region in the plane, and let me use positive superharmonic functions $F=-f,U=-u,V=-v$. Then your question is whether you can write a positive super …

Your first question is unclear. The answer to the second question is positive: you can have a plurisubharmonic function such that $f(x_k)=-\infty$ on some sequence $x_k\to 0$. For this function $\lims …

The usual Cauchy-Hadamard formula has a generalization to several variables. The numbers $r_1,\ldots,r_n$ are called conjugate radii of convergence if the series converges in the open polydisk $B(r_1, …

If you mean that your function $w$ vanishes on the critical values and nowhere else, then the answer is no. Take $(y_1,y_2)=(x_1^2,x_1x_2)$. The Jacobian determinant is zero on the line $x_1=0$ but t …

You have to state a specific problem to get a reasonable answer. The proof of the statements you cite is trivial. A function is called analytic at $x_0$ if in some neighborhood of $x_0$ it is represen …

Set $f_t(z)=\phi(t,z)$. Notice that entire functions $z\mapsto f_t(z)$ all commute with each other. I. N. Baker proved that if $f$ is a non-affine entire function with a repelling fixed point then the …

Here is a simple proof for complex-analytic case. If restrictions of $f$ on all complex lines are analytic, then $f$ is analytic. This reduces the problem to the case $n=1$. Now $f^2$ is analytic so n …