Is the “immersed proper” hypothesis necessary in Half-space Theorem?
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I'm using the following version of Half-space Theorem:
$textbf{Theorem}$(Half-space) A connected, immersed proper, nonplanar minimal surface $M$ in $mathbb{R}^3$is not contained in a halfspace.
I supposedly proved this theorem using the closed and complete hypothesis instead of immersed proper.
Is the "immersed proper" hypothesis necessary?
The original statement is found in the paper "The strong halfspace theorem for minimal surfaces" by D. Hoffman and W. H. Meeks, III, 1990.
In the statement they use the hypothesis of immersed proper and allow the surface to be "possibly branched".
Does the fact that the surface is "possibly branched" need the proper immersed hypothesis?
Can someone help me understand this better?
Follow the link in the paper below
http://www.math.jhu.edu/~js/Math748/hoffman-meeks.halfspace.pdf
analysis differential-geometry riemannian-geometry surfaces minimal-surfaces
asked Nov 27 at 1:15
Takashi
1946
add a comment |
up vote
3
down vote
favorite
I'm using the following version of Half-space Theorem:
$textbf{Theorem}$(Half-space) A connected, immersed proper, nonplanar minimal surface $M$ in $mathbb{R}^3$is not contained in a halfspace.
I supposedly proved this theorem using the closed and complete hypothesis instead of immersed proper.
Is the "immersed proper" hypothesis necessary?
The original statement is found in the paper "The strong halfspace theorem for minimal surfaces" by D. Hoffman and W. H. Meeks, III, 1990.
In the statement they use the hypothesis of immersed proper and allow the surface to be "possibly branched".
Does the fact that the surface is "possibly branched" need the proper immersed hypothesis?
Can someone help me understand this better?
Follow the link in the paper below
http://www.math.jhu.edu/~js/Math748/hoffman-meeks.halfspace.pdf
analysis differential-geometry riemannian-geometry surfaces minimal-surfaces
asked Nov 27 at 1:15
Takashi
1946
If you didn't require the properness assumption you could simply take a closed half-space and delete the part of any minimal surface intersecting that. What you are left with is a nonproper minimal surface lying in the open half-space we didn't delete.
– Mike Miller
Nov 27 at 16:31
@MikeMiller Is the immersed proper hypothesis to avoid self intersection? If you take that hypothesis, is there any counterexample that Theorem is false?
– Takashi
Nov 30 at 17:46
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I'm using the following version of Half-space Theorem:
$textbf{Theorem}$(Half-space) A connected, immersed proper, nonplanar minimal surface $M$ in $mathbb{R}^3$is not contained in a halfspace.
I supposedly proved this theorem using the closed and complete hypothesis instead of immersed proper.
Is the "immersed proper" hypothesis necessary?
The original statement is found in the paper "The strong halfspace theorem for minimal surfaces" by D. Hoffman and W. H. Meeks, III, 1990.
In the statement they use the hypothesis of immersed proper and allow the surface to be "possibly branched".
Does the fact that the surface is "possibly branched" need the proper immersed hypothesis?
Can someone help me understand this better?
Follow the link in the paper below
http://www.math.jhu.edu/~js/Math748/hoffman-meeks.halfspace.pdf
analysis differential-geometry riemannian-geometry surfaces minimal-surfaces
asked Nov 27 at 1:15
Takashi
1946
I'm using the following version of Half-space Theorem:
$textbf{Theorem}$(Half-space) A connected, immersed proper, nonplanar minimal surface $M$ in $mathbb{R}^3$is not contained in a halfspace.
I supposedly proved this theorem using the closed and complete hypothesis instead of immersed proper.
Is the "immersed proper" hypothesis necessary?
The original statement is found in the paper "The strong halfspace theorem for minimal surfaces" by D. Hoffman and W. H. Meeks, III, 1990.
In the statement they use the hypothesis of immersed proper and allow the surface to be "possibly branched".
Does the fact that the surface is "possibly branched" need the proper immersed hypothesis?
Can someone help me understand this better?
Follow the link in the paper below
http://www.math.jhu.edu/~js/Math748/hoffman-meeks.halfspace.pdf
analysis differential-geometry riemannian-geometry surfaces minimal-surfaces
analysis differential-geometry riemannian-geometry surfaces minimal-surfaces
asked Nov 27 at 1:15
Takashi
1946
asked Nov 27 at 1:15
Takashi
1946
edited Nov 27 at 13:15
asked Nov 27 at 1:15
Takashi
1946
asked Nov 27 at 1:15
Takashi
1946
asked Nov 27 at 1:15
Takashi
1946
1946
If you didn't require the properness assumption you could simply take a closed half-space and delete the part of any minimal surface intersecting that. What you are left with is a nonproper minimal surface lying in the open half-space we didn't delete.
– Mike Miller
Nov 27 at 16:31
@MikeMiller Is the immersed proper hypothesis to avoid self intersection? If you take that hypothesis, is there any counterexample that Theorem is false?
– Takashi
Nov 30 at 17:46
add a comment |
If you didn't require the properness assumption you could simply take a closed half-space and delete the part of any minimal surface intersecting that. What you are left with is a nonproper minimal surface lying in the open half-space we didn't delete.
– Mike Miller
Nov 27 at 16:31
@MikeMiller Is the immersed proper hypothesis to avoid self intersection? If you take that hypothesis, is there any counterexample that Theorem is false?
– Takashi
Nov 30 at 17:46
If you didn't require the properness assumption you could simply take a closed half-space and delete the part of any minimal surface intersecting that. What you are left with is a nonproper minimal surface lying in the open half-space we didn't delete.
– Mike Miller
Nov 27 at 16:31
If you didn't require the properness assumption you could simply take a closed half-space and delete the part of any minimal surface intersecting that. What you are left with is a nonproper minimal surface lying in the open half-space we didn't delete.
– Mike Miller
Nov 27 at 16:31
@MikeMiller Is the immersed proper hypothesis to avoid self intersection? If you take that hypothesis, is there any counterexample that Theorem is false?
– Takashi
Nov 30 at 17:46
@MikeMiller Is the immersed proper hypothesis to avoid self intersection? If you take that hypothesis, is there any counterexample that Theorem is false?
– Takashi
Nov 30 at 17:46
add a comment |
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If you didn't require the properness assumption you could simply take a closed half-space and delete the part of any minimal surface intersecting that. What you are left with is a nonproper minimal surface lying in the open half-space we didn't delete.
– Mike Miller
Nov 27 at 16:31
@MikeMiller Is the immersed proper hypothesis to avoid self intersection? If you take that hypothesis, is there any counterexample that Theorem is false?
– Takashi
Nov 30 at 17:46