All Questions
Tagged with mg.metric-geometry cv.complex-variables
11 questions with no upvoted or accepted answers
9
votes
0
answers
212
views
A geometric characterization of quasicircles
I'm reading an article by complex analysists.
A Jordan curve $J$ in the extended complex plane $\hat{\mathbb{C}}=\mathbb{C} \cup \{\infty\}$ is called a quasicircle if there is a quasiconformal map ...
- 721
5
votes
0
answers
275
views
Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?
Recent questions showed that roots of a random polynomial tend to lie on the
unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...
- 151k
3
votes
0
answers
112
views
Modulus of image of a curve family in a rectangle
I don't expect to get a positive answer to this question but I may as well try.
Let $R$ be the rectangle in $\mathbb{C}$ given by $\{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\}$ for some $l,h>0$. ...
- 445
3
votes
0
answers
54
views
Extremal metric for image of a curve family
Let $U\subset \mathbb{C}$ be a domain and $\Gamma$ some family of curves in $U$ with $\textrm{mod}(\Gamma)<\infty$ and such that $\rho$ is an extremal metric for the modulus. Suppose we are given a ...
- 445
1
vote
0
answers
73
views
Vanishing components of Kähler metric
Let $(X, \omega) $ be a $n$-dimensional complex Kähler manifold such that $\omega^{n-1}=d\alpha $.
Does $\partial\alpha^{n-1,n-2} =0$ (resp. $\bar\partial\alpha^{n-2,n-1} =0$)
Where $\alpha^{n-1,n-2}$ ...
- 43
1
vote
0
answers
62
views
A question on a paper of B. S. Henriksen
I have been reading the article "A peak set of Hausdorff dimension $2n-1$ for the algebra $A(\mathcal{D})$ in the boundary of a domain $\mathcal{D}$ with $C^\infty$-boundary in $\mathbb{C}^n$&...
1
vote
0
answers
48
views
Supremum norm of certain quantity II
Can anyone solve the maximization problem...$\max_{|z_i|=1}\Big|\sum_{i,j=1}^nz_iz_j+\sum_{i,j=1}^n|z_i-z_j|\Big|$?
- 455
1
vote
0
answers
138
views
Finding a metric on a topological space with prescribed isometry group
Let $X$ be a (sufficiently nice) topological space and let $\mathcal{F}$ be a group of homeomorphisms of $X$. Assume that $\mathcal{F}$ is also closed under point-wise convergence. I would like to ...
- 1,169
1
vote
0
answers
56
views
Starlike curve tangent condition
Assume that $\gamma$ is a starlike Jordan curve in the complex plane w.r.t. 0. Let $\alpha\in (0,\pi/2]$. For each $z\in\gamma$, $z\neq x$, we let $\alpha(z,x)$
denote the acute angle which the ...
- 11
0
votes
0
answers
79
views
Geometry of inner products between the unit vector and several given vectors
Let $\mathcal{S}$ denote the set of all unit complex-valued $d$-dimensional vectors, i.e.,
$$
\mathcal{S} \triangleq \left\{ \mathbf{s}\in \mathbb{C}^{d} \mid \mathbf{s}^{\mathrm{H}}\mathbf{s}=1 \...
- 550
0
votes
0
answers
32
views
Reference Request: Carnot Groups over Complexes
Is there a theory of complex (analytic) Carnot groups and Caratheodory metrics?
- 5,405