Are complete minimal submanifolds closed?












2












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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







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    2












    $begingroup$


    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







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      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







      share|cite|improve this question









      $endgroup$




      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






      share|cite|improve this question













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      asked 5 hours ago









      ValentinoValentino

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      1325






















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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?






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            1 Answer
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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?






            share|cite|improve this answer











            $endgroup$


















              5












              $begingroup$

              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?






              share|cite|improve this answer











              $endgroup$
















                5












                5








                5





                $begingroup$

                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?






                share|cite|improve this answer











                $endgroup$



                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?







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 3 hours ago

























                answered 5 hours ago









                RBega2RBega2

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                53629






























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