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Decomposition of sub-harmonic Fixed norm problem for analytic functionsHi there, I have the following problems on my hand: Given arbitrary positive sub-harmonic function $l(x,y)$ on plane, for all holomorphic analytic functions: $f=a(x,y)+ib(x,y)$ on complex plane, consider the equation: $|f|^2=l$, i.e. $a^2+b^2=l$ 1) for what kind of $l(x,y)$, the equation has $C^2$ solution? 2) for what kind of $l(x,y)$, the number of solutions is finite, after quotient of $S^1$ action: $e^{i\theta}\cdot f$ ? Thanks! |
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Hi there, I have the following problems on my hand: Given arbitrary positive sub-harmonic function $l(x,y)$ on plane, for all holomorphic functionfunctions: $f=a(x,y)+ib(x,y)$ on complex plane, consider the equation: $|f|^2=l$, i.e. $a^2+b^2=l$ 1) for what kind of $l(x,y)$, the equation has $C^2$ solution? 2) for what kind of $l(x,y)$, the solution number of solutions is finite, after quotient of $S^1$ action: $e^{i\theta}\cdot f$ ? Thanks! |
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Decomposition of sub-harmonic functionsHi there, I have the following problems on my hand: Given arbitrary positive sub-harmonic function $l(x,y)$ on plane, for all holomorphic function: $f=a(x,y)+ib(x,y)$ on complex plane, consider the equation: $|f|^2=l$, i.e. $a^2+b^2=l$ 1) for what kind of $l(x,y)$, the equation has $C^2$ solution? 2) for what kind of $l(x,y)$, the solution is finite, after quotient of $S^1$ action: $e^{i\theta}\cdot f$ ? Thanks!
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