All Questions
Tagged with complex-geometry hyperbolic-geometry
30 questions
0
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48
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When inclusion between two Kobayshi hyperbolic manifolds is distance decreasing?
Suppose that $X$ and $Y$ are two Kobayshi hyperbolic complex-analytic manifolds such that $X \subset Y$. It is known $d_Y(x_1, x_2) \leq d_X(x_1, x_2)$ for all $x_1, x_2 \in X$. In other words, the ...
- 41
0
votes
1
answer
98
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Number of regions created by r hyper-planes in n-dimensional space [closed]
I found this formula for calculating maximum number of regions created by r hyper-planes in n-dimensional space (n<=r)
...
- 50.8k
13
votes
2
answers
484
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How the hyperbolic metric changes when we add a puncture?
Suppose we have a surface $S$ of a finite genus, without boundary with a finite number of punctures. Suppose that this surface comes equipped with a hyperbolic metric of curvature $-1$.
Question 1: If ...
- 438
0
votes
1
answer
142
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Comparative between Kobayashi hyperbolicity and hyperbolicity in the sense of Koszul
In literature, there are several notions of hyperbolicity. My question is whether, for closed locally flat or affine manifolds, the notion of hyperbolicity in the sense of Kobayashi is equivalent to ...
- 1,379
4
votes
2
answers
444
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Mostow rigidity for complex hyperbolic manifolds
A Riemannian manifold $(X,g)$ is hyperbolic if the sectional curvatures are constant and negative. A theorem of Mostow says that these manifolds are determined by their fundamental group.
Theorem (...
- 586
5
votes
1
answer
431
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Isometric immersion of $\mathbb H^2$ into $\mathbb R^\infty$ built by Bieberbach
I'm analyzing the following isometric immersion of $(\mathbb H^2,g_D)$ in $(\ell^2,g_\infty)$ given by $f(x,y)=(x_1,x_2,\dots,x_{2m-1},x_{2m},\dots)$ with
\begin{align}\label{5.1}
x_{2m-1}=\color{...
- 143
1
vote
0
answers
70
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Classification of principal monodromy elements
Let $(X,0)$ be a germ of normal analytic space with an isolated singularity at $0$, and let $Y:=X\backslash\{0\}$. Suppose $Y$ has a complex-hyperbolic metric which is complete at $0$. Burns-Mazzeo ...
- 1,121
21
votes
1
answer
753
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Complexifications of hyperbolic manifolds
I'm wondering when a compact hyperbolic $n$-manifold ($n \geq 3$) can embed in a complex hyperbolic $n$-manifold as a real algebraic subvariety so that it is a component of the fixed point set of ...
- 68.9k
7
votes
3
answers
655
views
Hyperbolic 3-manifolds inside algebraic varieties
I have a hyperbolic 3-manifold $M$ that I'd like to see sit inside a complex algebraic variety $V$ as beautifully and snugly as possible. I don't want to specify attributes of "beauty" (e.g.,...
- 12.3k
3
votes
2
answers
1k
views
Explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface
I am finding the explicit description of genus $2$ surface as the upper half plane modulo group of Deck Transformation. I did't find it anywhere. Also, I found a similar question here. But, there is ...
CommunityBot
- 1
56
votes
6
answers
6k
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What do Weierstrass points look like?
As somebody who mostly works with smooth, real manifolds, I've always been a little uncomfortable with Weierstrass points. Smooth manifolds are totally homogeneous, but in the complex category you ...
- 63.9k
7
votes
4
answers
454
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Lattices of PU(n,1) with large abelianization
I am interested in properly discontinuous cocompact subgroups of the group $PU(n,1)$ of automorphisms of the complex hyperbolic space $H^n_{\mathbb{C}}$, says for $n=2,3$. Is there such a lattice $G$ ...
- 68.9k
9
votes
2
answers
456
views
Riemannian submersions from complex hyperbolic space into the hyperbolic space
Is there a (canonical) Riemannian submersion from the complex hyperbolic space $\mathbb C\mathbb H^n$ into the hyperbolic space $\mathbb H^n$?
In the affirmative case, what can we say about the ...
- 2,703
3
votes
0
answers
49
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Geodesics in norm balls
Recently, some problems that I work on require that I understand a bit of hyperbolic complex geometry. Assume that $B \subset \mathbb{C}^n$ is the unit ball of some norm $\|\cdot\|$ (not induced by an ...
- 431
6
votes
2
answers
495
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Riemann Theta Function On Hyperbolic Riemann Surfaces
The Riemann theta function for a genus $g$ closed Riemann surface with period matrix $\tau=[\tau_{ij}]$ is defined by
$$\theta(\{z_1,\cdots,z_g\}|\tau)=\Sigma_{n\in\mathbb{Z}^g}e^{\pi i(n\cdot\tau\...
- 96.4k
2
votes
0
answers
163
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Geometric and holomorphic structure of $\mathbb{C} \rtimes \mathbb{C} \setminus \{ 0 \}$
Put $G= \mathbb{C} \rtimes_{\phi} \mathbb{C} \setminus \{0\}$ where $\phi_{a} (z)= az$ for $a \in \mathbb{C} \setminus \{0\}$.
$G$ is a real $4$ dimensional Lie group; then it has a unique left ...
- 356
5
votes
1
answer
353
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Quantifying the monotonicity property of the hyperbolic metric
Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} &...
CommunityBot
- 1
1
vote
2
answers
425
views
( finite ) Blaschke product in higher dimensions ?
Hello, as we know, the (finite) Blaschke product $P$ in $\mathbb{C}$ or in $ \mathbb{R}^2 $ is defined by $\prod_{j=1}^{k} \frac{z-a_j}{1-\bar{a_j}z}, a_j \in \mathbb{D}$. I was wondering whether ...
1
vote
1
answer
618
views
Conformal equivalence of square and upper-plane, including part of the boundary
I was reading the book "Conformal Invariants" of L. Ahlfors, and seen (P.74) he says that "It is in fact obvious that the part of the teichmuller annulus (for R=1) in the upper plane is conformally ...
- 91.8k
1
vote
0
answers
245
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Schwarz–Ahlfors–Pick theorem for hyperbolic pair of pants
Let $\mathcal{P}$ be a hyperbolic pair of pants with geodesic boundary and $\Sigma_3$ be the hyperbolic thrice punctures sphere. I want to construct a conformal map $f:\mathcal{P}\rightarrow \Sigma_3$ ...
- 1,713
10
votes
1
answer
314
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Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?
Is there a closed subscheme $D$ in $\mathbb P^2_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map
$$ \phi: X(\mathbb C) \to \mathbb P^...
- 103
3
votes
1
answer
388
views
Kobayashi distance function on the upper half-space
I asked this question already in mathstackexchange but got no answer, so I ask it again here.
Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ ...
- 96.4k
5
votes
1
answer
307
views
Infinitesimal deformations of fake projective planes (or ball quotients)
This question is related to the answer I gave to this MO question. What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial.
By ...
CommunityBot
- 1
2
votes
0
answers
115
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Characterisation of convergence in Deligne-Mumford compactifiaction
1) Is there a (simple) way to characterise the convergence of complex curves toward a stable curve in term of hyperbolic metrics for the Deligne Mumford compactification of the moduli space of complex ...
- 2,696
8
votes
1
answer
573
views
Do elements of the fundamental group give rise to isometries
Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.
...
- 2,727
8
votes
3
answers
942
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Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$
Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite.
What ...
- 19.4k
6
votes
1
answer
479
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If rational points are like entire curves, then what do algebraic points correspond to
I read somewhere that if $X$ is a projective variety of general type over a number field $K$, then rational points are an analogue of entire curves $\mathbf{C}\to X^{an}$ (with $X^{an}$ the ...
- 19.2k
8
votes
3
answers
2k
views
Conformal Mappings for hyperbolic polygon
I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics.
The classical Schwarz Christoffel theorem does the job ...
- 91.8k
2
votes
2
answers
811
views
Surface of a Ideal Tetrahedron in Hyperbolic Space H3
The hyperbolic space $\mathbb{H}^3$, has a boundary $\mathbb{CP}^1$.
A ideal tetrahedron in $\mathbb{H}^3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb{CP}^1$.
The four ...
- 28.2k
3
votes
2
answers
618
views
Schwarz Lemma in terms of conformal surfaces or holomorphic curves?
Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting.
Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ...
- 85.6k