Poincaré metric on the Riemann sphere minus more than two points
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If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let us denote the hyperbolic metric on this surface by $d_h$, and the usual spherical metric on the Riemann sphere by $d$. Here is my question:
Can we find a constant $C>0$ such that for any two points $x$ and $y$ in this punctured sphere we have:
$$d(x,y)<C d_h(x,y).$$
complex-geometry riemann-surfaces complex-dynamics
edited Nov 26 at 18:04
Ivan Izmestiev
4,0631238
asked Nov 26 at 15:44
Amin Talebi
634
add a comment |
up vote
12
down vote
favorite
If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let us denote the hyperbolic metric on this surface by $d_h$, and the usual spherical metric on the Riemann sphere by $d$. Here is my question:
Can we find a constant $C>0$ such that for any two points $x$ and $y$ in this punctured sphere we have:
$$d(x,y)<C d_h(x,y).$$
complex-geometry riemann-surfaces complex-dynamics
edited Nov 26 at 18:04
Ivan Izmestiev
4,0631238
asked Nov 26 at 15:44
Amin Talebi
634
4
Welcome to Mathoverflow.
– Mahdi
Nov 26 at 18:06
@Mahdi Thank you very much.
– Amin Talebi
Nov 27 at 9:14
add a comment |
up vote
12
down vote
favorite
up vote
12
down vote
favorite
If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let us denote the hyperbolic metric on this surface by $d_h$, and the usual spherical metric on the Riemann sphere by $d$. Here is my question:
Can we find a constant $C>0$ such that for any two points $x$ and $y$ in this punctured sphere we have:
$$d(x,y)<C d_h(x,y).$$
complex-geometry riemann-surfaces complex-dynamics
edited Nov 26 at 18:04
Ivan Izmestiev
4,0631238
asked Nov 26 at 15:44
Amin Talebi
634
If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let us denote the hyperbolic metric on this surface by $d_h$, and the usual spherical metric on the Riemann sphere by $d$. Here is my question:
Can we find a constant $C>0$ such that for any two points $x$ and $y$ in this punctured sphere we have:
$$d(x,y)<C d_h(x,y).$$
complex-geometry riemann-surfaces complex-dynamics
complex-geometry riemann-surfaces complex-dynamics
edited Nov 26 at 18:04
Ivan Izmestiev
4,0631238
asked Nov 26 at 15:44
Amin Talebi
634
edited Nov 26 at 18:04
Ivan Izmestiev
4,0631238
asked Nov 26 at 15:44
Amin Talebi
634
edited Nov 26 at 18:04
Ivan Izmestiev
4,0631238
edited Nov 26 at 18:04
Ivan Izmestiev
4,0631238
edited Nov 26 at 18:04
Ivan Izmestiev
4,0631238
4,0631238
asked Nov 26 at 15:44
Amin Talebi
634
asked Nov 26 at 15:44
Amin Talebi
634
asked Nov 26 at 15:44
Amin Talebi
634
634
4
Welcome to Mathoverflow.
– Mahdi
Nov 26 at 18:06
@Mahdi Thank you very much.
– Amin Talebi
Nov 27 at 9:14
add a comment |
4
Welcome to Mathoverflow.
– Mahdi
Nov 26 at 18:06
@Mahdi Thank you very much.
– Amin Talebi
Nov 27 at 9:14
4
4
Welcome to Mathoverflow.
– Mahdi
Nov 26 at 18:06
Welcome to Mathoverflow.
– Mahdi
Nov 26 at 18:06
@Mahdi Thank you very much.
– Amin Talebi
Nov 27 at 9:14
@Mahdi Thank you very much.
– Amin Talebi
Nov 27 at 9:14
add a comment |
1 Answer
1
active
oldest
votes
up vote
16
down vote
accepted
Yes. The density of the Poincare metric with respect to the spherical metric is
a positive continuous function which tends to infinity at the punctures. Thus it
is bounded from below by some positive constant. The constant depends only
on the configuration of the punctures. For some special configurations of punctures, the exact constant has been explicitly found:
MR1428102
Bonk, Mario; Cherry, William,
Bounds on spherical derivatives for maps into regions with symmetries.
J. Anal. Math. 69 (1996), 249–274.
The authors of this paper say that for general punctures, the explicit determination of
the optimal constant is hopeless, and I agree with them.
answered Nov 26 at 16:11
Alexandre Eremenko
48.8k6136253
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
16
down vote
accepted
Yes. The density of the Poincare metric with respect to the spherical metric is
a positive continuous function which tends to infinity at the punctures. Thus it
is bounded from below by some positive constant. The constant depends only
on the configuration of the punctures. For some special configurations of punctures, the exact constant has been explicitly found:
MR1428102
Bonk, Mario; Cherry, William,
Bounds on spherical derivatives for maps into regions with symmetries.
J. Anal. Math. 69 (1996), 249–274.
The authors of this paper say that for general punctures, the explicit determination of
the optimal constant is hopeless, and I agree with them.
answered Nov 26 at 16:11
Alexandre Eremenko
48.8k6136253
add a comment |
up vote
16
down vote
accepted
Yes. The density of the Poincare metric with respect to the spherical metric is
a positive continuous function which tends to infinity at the punctures. Thus it
is bounded from below by some positive constant. The constant depends only
on the configuration of the punctures. For some special configurations of punctures, the exact constant has been explicitly found:
MR1428102
Bonk, Mario; Cherry, William,
Bounds on spherical derivatives for maps into regions with symmetries.
J. Anal. Math. 69 (1996), 249–274.
The authors of this paper say that for general punctures, the explicit determination of
the optimal constant is hopeless, and I agree with them.
answered Nov 26 at 16:11
Alexandre Eremenko
48.8k6136253
add a comment |
up vote
16
down vote
accepted
up vote
16
down vote
accepted
Yes. The density of the Poincare metric with respect to the spherical metric is
a positive continuous function which tends to infinity at the punctures. Thus it
is bounded from below by some positive constant. The constant depends only
on the configuration of the punctures. For some special configurations of punctures, the exact constant has been explicitly found:
MR1428102
Bonk, Mario; Cherry, William,
Bounds on spherical derivatives for maps into regions with symmetries.
J. Anal. Math. 69 (1996), 249–274.
The authors of this paper say that for general punctures, the explicit determination of
the optimal constant is hopeless, and I agree with them.
answered Nov 26 at 16:11
Alexandre Eremenko
48.8k6136253
Yes. The density of the Poincare metric with respect to the spherical metric is
a positive continuous function which tends to infinity at the punctures. Thus it
is bounded from below by some positive constant. The constant depends only
on the configuration of the punctures. For some special configurations of punctures, the exact constant has been explicitly found:
MR1428102
Bonk, Mario; Cherry, William,
Bounds on spherical derivatives for maps into regions with symmetries.
J. Anal. Math. 69 (1996), 249–274.
The authors of this paper say that for general punctures, the explicit determination of
the optimal constant is hopeless, and I agree with them.
answered Nov 26 at 16:11
Alexandre Eremenko
48.8k6136253
edited Nov 27 at 1:22
answered Nov 26 at 16:11
Alexandre Eremenko
48.8k6136253
answered Nov 26 at 16:11
Alexandre Eremenko
48.8k6136253
answered Nov 26 at 16:11
Alexandre Eremenko
48.8k6136253
48.8k6136253
add a comment |
add a comment |
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4
Welcome to Mathoverflow.
– Mahdi
Nov 26 at 18:06
@Mahdi Thank you very much.
– Amin Talebi
Nov 27 at 9:14