7
$\begingroup$

Let $D\subset\mathbb{C}$ be the open unit disk. Suppose $f,g,F,G:D\rightarrow\mathbb{C}$ are analytic functions linked by $$\vert f(z)\vert^2+\vert g(z)\vert^2=\vert F(z)\vert^2+\vert G(z)\vert^2; \qquad \forall z\in D.$$

Question 1. If $f\neq \alpha g$ and $g\neq\beta f$ for any $\alpha, \beta\in\mathbb{C}$ then is the same true for $F$ and $G$?

Caveat. Not true for real analytic functions: take $f=\sin x, \,g=\cos x, \, F=\frac1{\sqrt{2}}=G$.

EDIT. Encouraged by Christian's positive answer, let's upgrade the problem.

Question 2. Suppose $f, g, h$ are linearly independent (over $\mathbb{C}$) such that $$\vert f\vert^2+\vert g\vert^2+\vert h\vert^2=\vert F\vert^2+\vert G\vert^2+\vert H\vert^2.$$ Should $F, G, H$ be linearly independent, too?

$\endgroup$
2
$\begingroup$

In fact, more is true. If $f_j$ are linearly independent and $F_i$ are linearly independent, and $$\sum_{j=1}^n|f_j|^2=\sum_{i=1}^m|F_i|^2,$$ then $m=n$ and $F_i$ are obtained from $g_i$ by a unitary transformation. See, for example, https://arxiv.org/pdf/math/0007030.pdf, section 3. This is called the "Calabi rigidity".

| cite | improve this answer | |
$\endgroup$
8
$\begingroup$

Yes, for Question 1.

We can divide through by one of the functions on the left-hand side and also assume $F=G$, and then the question becomes: Can $1+|h(z)|^2$ be the square of the absolute value of a holomorphic function, for a non-constant holomorphic $h$?

For this to happen, we need $\log (1+|h(z)|^2)$ to be harmonic. If you now just work out the Laplacian of this function and use the Cauchy-Riemann equations for $h$, you'll find that the numerator of the expression you obtain from two applications of the quotient rule equals $4((h_1)_x^2+(h_2)_x^2)$, so $h$ has to be constant.

| cite | improve this answer | |
$\endgroup$

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.