# Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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### Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability

We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth ...
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### Algebraic independence of certain values implies algebraic independence of functions?

It is quite general and elementary question. Is it possible that some holomorphic functions $f_1,\cdots,f_m$ on a region $\Omega$ of $\mathbb C$ satisfies: Whenever $(f_1(z), \cdots, f_m (z))$ is ...
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### Simple Closed Hyperbolic Geodesics on Punctured Spheres

Thinking of $\mathbb {CP^1}$ as the sphere $S^2\subset\mathbb R^3$, we can define the notion of a circle on it to be a subset that is got by a hyperplane section of $S^2$ inside $\mathbb R^3$. This ...
Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint. It is not difficult to see that $$\int_{1<|z|&... 0answers 133 views ### Analytic Aspects of Rational Maps I would like some help finding references for a analytic treatment of rational maps between compact complex manifolds (that is holomorphic maps defined away from a codimension at least 2 subvariety). ... 0answers 90 views ### Analytic continuation of an NLS soliton The attractive nonlinear Schroedinger equation i \partial_t \psi = - \frac {1} {2} \Delta \psi - |\psi|^{p-1} \psi in the H^1-subcritical case 1 < p, \frac {d} {2} + \frac {2} {p-1} < 1 ... 3answers 224 views ### about the Hausdorff dimension of Removable singularities of PDE There are some interesting phenomenons about removable singularities (or extension problems). In the theory of functions of several complex variables, we know the classical Hartogs theorem: Let f ... 2answers 398 views ### Conformal mappings that preserve angles and areas but not perimeters? Conformal mappings from U to V, both subsets of \mathbb{C}, locally preserve angles. But, in general, such mappings neither preserve areas nor preserve perimeters. Q. Are there examples of ... 0answers 43 views ### What is the ring of functions on the open unit disc with polynomially bounded Maclaurin coefficients called? Let R be the the set of complex-valued (analytic) functions f on the open unit disc \mathrm D:=\{z\in\Bbb C:|z|<1\} for which there exist constants a_0, a_1, ... in \Bbb C and n in \... 4answers 185 views ### PDE with Laplacian and squared of the gradient Let u be a real function in \mathbb{R}^2. Does anybody know that the following PDE$$\Delta u+|\nabla u|^2=0 has any non-constant general solution or not? It would be appreciated if any one ...
Let $\mathfrak f(\tau)=e^{-\pi i/24}\frac{\eta\left(\frac{\tau+1}{2}\right)}{\eta(\tau)}=q^{-1/48}\prod_{n=1}^{\infty}\left(1+q^{n+1/2}\right)$ be the Weber modular function. The function $\mathfrak f$...