Integer multiplicity current without boundary is boundary of another current
$begingroup$
Let us say we have an integer multiplicity current $Tin mathcal{D}_n(Omega)$, $Omegasubset mathbb{R}^m$, $n+1leq m$ and $partial T =0$. Do we always find another current $Rin mathcal{D}_{n+1}(Omega)$ such that $partial R=T$?
In case the support of $T$ is compact the answer to this question is yes by just taking a cone or using the result for the isoperimetric inequality by Federer and Fleming, which can be deduced by the Deformation Theorem.
So what happens if the support is not compact anymore? Are there counterexamples? I am especially interested in the case $m=n+1$.
If the answer is no, what if $T$ is a minimizing current?
Many thanks for your time and effort.
measure-theory geometric-measure-theory
asked Jan 25 at 11:11
humanStampedisthumanStampedist
2,228214
$endgroup$
add a comment |
$begingroup$
Let us say we have an integer multiplicity current $Tin mathcal{D}_n(Omega)$, $Omegasubset mathbb{R}^m$, $n+1leq m$ and $partial T =0$. Do we always find another current $Rin mathcal{D}_{n+1}(Omega)$ such that $partial R=T$?
In case the support of $T$ is compact the answer to this question is yes by just taking a cone or using the result for the isoperimetric inequality by Federer and Fleming, which can be deduced by the Deformation Theorem.
So what happens if the support is not compact anymore? Are there counterexamples? I am especially interested in the case $m=n+1$.
If the answer is no, what if $T$ is a minimizing current?
Many thanks for your time and effort.
measure-theory geometric-measure-theory
asked Jan 25 at 11:11
humanStampedisthumanStampedist
2,228214
$endgroup$
add a comment |
$begingroup$
Let us say we have an integer multiplicity current $Tin mathcal{D}_n(Omega)$, $Omegasubset mathbb{R}^m$, $n+1leq m$ and $partial T =0$. Do we always find another current $Rin mathcal{D}_{n+1}(Omega)$ such that $partial R=T$?
In case the support of $T$ is compact the answer to this question is yes by just taking a cone or using the result for the isoperimetric inequality by Federer and Fleming, which can be deduced by the Deformation Theorem.
So what happens if the support is not compact anymore? Are there counterexamples? I am especially interested in the case $m=n+1$.
If the answer is no, what if $T$ is a minimizing current?
Many thanks for your time and effort.
measure-theory geometric-measure-theory
asked Jan 25 at 11:11
humanStampedisthumanStampedist
2,228214
$endgroup$
Let us say we have an integer multiplicity current $Tin mathcal{D}_n(Omega)$, $Omegasubset mathbb{R}^m$, $n+1leq m$ and $partial T =0$. Do we always find another current $Rin mathcal{D}_{n+1}(Omega)$ such that $partial R=T$?
In case the support of $T$ is compact the answer to this question is yes by just taking a cone or using the result for the isoperimetric inequality by Federer and Fleming, which can be deduced by the Deformation Theorem.
So what happens if the support is not compact anymore? Are there counterexamples? I am especially interested in the case $m=n+1$.
If the answer is no, what if $T$ is a minimizing current?
Many thanks for your time and effort.
measure-theory geometric-measure-theory
measure-theory geometric-measure-theory
asked Jan 25 at 11:11
humanStampedisthumanStampedist
2,228214
asked Jan 25 at 11:11
humanStampedisthumanStampedist
2,228214
asked Jan 25 at 11:11
humanStampedisthumanStampedist
2,228214
asked Jan 25 at 11:11
humanStampedisthumanStampedist
2,228214
asked Jan 25 at 11:11
humanStampedisthumanStampedist
2,228214
2,228214
add a comment |
add a comment |
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