Are complete minimal submanifolds closed?
$begingroup$
Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset?
What about the case in which the ambient manifold is an euclidean space?
riemannian-geometry smooth-manifolds minimal-surfaces
asked 4 hours ago
ValentinoValentino
1275
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add a comment |
$begingroup$
Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset?
What about the case in which the ambient manifold is an euclidean space?
riemannian-geometry smooth-manifolds minimal-surfaces
asked 4 hours ago
ValentinoValentino
1275
$endgroup$
add a comment |
$begingroup$
Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset?
What about the case in which the ambient manifold is an euclidean space?
riemannian-geometry smooth-manifolds minimal-surfaces
asked 4 hours ago
ValentinoValentino
1275
$endgroup$
Is it true that any complete minimal submanifold of some Riemannian manifold is closed as a subset?
What about the case in which the ambient manifold is an euclidean space?
riemannian-geometry smooth-manifolds minimal-surfaces
riemannian-geometry smooth-manifolds minimal-surfaces
asked 4 hours ago
ValentinoValentino
1275
asked 4 hours ago
ValentinoValentino
1275
asked 4 hours ago
ValentinoValentino
1275
asked 4 hours ago
ValentinoValentino
1275
asked 4 hours ago
ValentinoValentino
1275
1275
add a comment |
add a comment |
1 Answer
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It depends on what you mean by submanifold (at least for surfaces in three dimensional ambient spaces).
Nadirashvilli constructed an example of a complete minimal immersion into the unit ball in $mathbb{R}^3$. In particular, this immersion is not proper and so the image is not a closed subset of $mathbb{R}^3$. In contrast, Colding-Minicozzi showed that any complete embedded minimal surface in $mathbb{R}^3$ (i.e. a two dimensional submanifold of $mathbb{R}^3$ in the usual sense) that has finite topology must be properly embedded and hence be a closed subset. This has been extended in various ways by Meeks-Perez-Ros. There are still a number of difficult open problems about exactly how general this phenomena is -- e.g. is it true for surfaces of finite genus? Any complete embedded minimal surface?
answered 4 hours ago
RBega2RBega2
52629
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
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$begingroup$
It depends on what you mean by submanifold (at least for surfaces in three dimensional ambient spaces).
Nadirashvilli constructed an example of a complete minimal immersion into the unit ball in $mathbb{R}^3$. In particular, this immersion is not proper and so the image is not a closed subset of $mathbb{R}^3$. In contrast, Colding-Minicozzi showed that any complete embedded minimal surface in $mathbb{R}^3$ (i.e. a two dimensional submanifold of $mathbb{R}^3$ in the usual sense) that has finite topology must be properly embedded and hence be a closed subset. This has been extended in various ways by Meeks-Perez-Ros. There are still a number of difficult open problems about exactly how general this phenomena is -- e.g. is it true for surfaces of finite genus? Any complete embedded minimal surface?
answered 4 hours ago
RBega2RBega2
52629
$endgroup$
add a comment |
$begingroup$
It depends on what you mean by submanifold (at least for surfaces in three dimensional ambient spaces).
Nadirashvilli constructed an example of a complete minimal immersion into the unit ball in $mathbb{R}^3$. In particular, this immersion is not proper and so the image is not a closed subset of $mathbb{R}^3$. In contrast, Colding-Minicozzi showed that any complete embedded minimal surface in $mathbb{R}^3$ (i.e. a two dimensional submanifold of $mathbb{R}^3$ in the usual sense) that has finite topology must be properly embedded and hence be a closed subset. This has been extended in various ways by Meeks-Perez-Ros. There are still a number of difficult open problems about exactly how general this phenomena is -- e.g. is it true for surfaces of finite genus? Any complete embedded minimal surface?
answered 4 hours ago
RBega2RBega2
52629
$endgroup$
add a comment |
$begingroup$
It depends on what you mean by submanifold (at least for surfaces in three dimensional ambient spaces).
Nadirashvilli constructed an example of a complete minimal immersion into the unit ball in $mathbb{R}^3$. In particular, this immersion is not proper and so the image is not a closed subset of $mathbb{R}^3$. In contrast, Colding-Minicozzi showed that any complete embedded minimal surface in $mathbb{R}^3$ (i.e. a two dimensional submanifold of $mathbb{R}^3$ in the usual sense) that has finite topology must be properly embedded and hence be a closed subset. This has been extended in various ways by Meeks-Perez-Ros. There are still a number of difficult open problems about exactly how general this phenomena is -- e.g. is it true for surfaces of finite genus? Any complete embedded minimal surface?
answered 4 hours ago
RBega2RBega2
52629
$endgroup$
It depends on what you mean by submanifold (at least for surfaces in three dimensional ambient spaces).
Nadirashvilli constructed an example of a complete minimal immersion into the unit ball in $mathbb{R}^3$. In particular, this immersion is not proper and so the image is not a closed subset of $mathbb{R}^3$. In contrast, Colding-Minicozzi showed that any complete embedded minimal surface in $mathbb{R}^3$ (i.e. a two dimensional submanifold of $mathbb{R}^3$ in the usual sense) that has finite topology must be properly embedded and hence be a closed subset. This has been extended in various ways by Meeks-Perez-Ros. There are still a number of difficult open problems about exactly how general this phenomena is -- e.g. is it true for surfaces of finite genus? Any complete embedded minimal surface?
answered 4 hours ago
RBega2RBega2
52629
edited 2 hours ago
answered 4 hours ago
RBega2RBega2
52629
answered 4 hours ago
RBega2RBega2
52629
answered 4 hours ago
RBega2RBega2
52629
52629
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