Extremal volume of $partial P^n$ with fixed volume of $P_n$
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It is well-known that sphere has the least area of all surfaces bounding a fixed volume. I want to prove the generalized result: let $P^n$ be a compact submanifold (with fixed $n$-volume) of a Reimann manifold $M^n$ and let $N^{n-1}=partial P^n$ be its boundary. $N^{n-1}$ has extremal $n-1$-volume, iff its mean curvative is constant.
differential-geometry optimization manifolds
asked Dec 26 '18 at 19:13
Michael FreimannMichael Freimann
300113
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add a comment |
$begingroup$
It is well-known that sphere has the least area of all surfaces bounding a fixed volume. I want to prove the generalized result: let $P^n$ be a compact submanifold (with fixed $n$-volume) of a Reimann manifold $M^n$ and let $N^{n-1}=partial P^n$ be its boundary. $N^{n-1}$ has extremal $n-1$-volume, iff its mean curvative is constant.
differential-geometry optimization manifolds
asked Dec 26 '18 at 19:13
Michael FreimannMichael Freimann
300113
$endgroup$
add a comment |
$begingroup$
It is well-known that sphere has the least area of all surfaces bounding a fixed volume. I want to prove the generalized result: let $P^n$ be a compact submanifold (with fixed $n$-volume) of a Reimann manifold $M^n$ and let $N^{n-1}=partial P^n$ be its boundary. $N^{n-1}$ has extremal $n-1$-volume, iff its mean curvative is constant.
differential-geometry optimization manifolds
asked Dec 26 '18 at 19:13
Michael FreimannMichael Freimann
300113
$endgroup$
It is well-known that sphere has the least area of all surfaces bounding a fixed volume. I want to prove the generalized result: let $P^n$ be a compact submanifold (with fixed $n$-volume) of a Reimann manifold $M^n$ and let $N^{n-1}=partial P^n$ be its boundary. $N^{n-1}$ has extremal $n-1$-volume, iff its mean curvative is constant.
differential-geometry optimization manifolds
differential-geometry optimization manifolds
asked Dec 26 '18 at 19:13
Michael FreimannMichael Freimann
300113
asked Dec 26 '18 at 19:13
Michael FreimannMichael Freimann
300113
asked Dec 26 '18 at 19:13
Michael FreimannMichael Freimann
300113
asked Dec 26 '18 at 19:13
Michael FreimannMichael Freimann
300113
asked Dec 26 '18 at 19:13
Michael FreimannMichael Freimann
300113
300113
add a comment |
add a comment |
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