Mixing
Geometric meaning of the $L^p$-norm of $|A|$
$begingroup$
Let be $M$ an oriented smooth surface on $mathbb{R}^3$, $|A|$ the norm of the second fundamental form of $M$ and $dV$ the element of area. I would like to know if there is a geometric meaning for the quantity $|| |A| ||_p^p = int_M |A|^p dV$? I saw this appears in "A compactness theorem for surfaces with $Lp$-bounded second fundamental form" by Joel Langer and this seems an important quantity since the integrand $|A| = sqrt{4H^2 - 2K}$ and the quantities $H$ and $K$ are invariant under change of the basis. I looked for an explanations about the $L^p$-norm of $|A|$ in Riemannian Geometry's textbooks and in Differential Geometry's textbooks, but I couldn't find nothing.
More generally, if $M$ is an oriented smooth $n$-manifold immersed on $mathbb{R}^{n+1}$, $|A|$ the norm of the second fundamental form of $M$ and $dV$ the element of volume, then there is an geometric meaning for the quantity $|| |A| ||_p^p = int_M |A|^p dV$? Again, it seems that there is an intepretation since $|A|^2 - frac{H^2}{n} = sum_{i < j} (lambda_i - lambda_j)^2$, where $lambda_i$ are the principal curvatures of $M$.
Thanks in advance!
differential-geometry riemannian-geometry
asked Jan 2 at 18:03
GeorgeGeorge
801515
$endgroup$
add a comment |
$begingroup$
Let be $M$ an oriented smooth surface on $mathbb{R}^3$, $|A|$ the norm of the second fundamental form of $M$ and $dV$ the element of area. I would like to know if there is a geometric meaning for the quantity $|| |A| ||_p^p = int_M |A|^p dV$? I saw this appears in "A compactness theorem for surfaces with $Lp$-bounded second fundamental form" by Joel Langer and this seems an important quantity since the integrand $|A| = sqrt{4H^2 - 2K}$ and the quantities $H$ and $K$ are invariant under change of the basis. I looked for an explanations about the $L^p$-norm of $|A|$ in Riemannian Geometry's textbooks and in Differential Geometry's textbooks, but I couldn't find nothing.
More generally, if $M$ is an oriented smooth $n$-manifold immersed on $mathbb{R}^{n+1}$, $|A|$ the norm of the second fundamental form of $M$ and $dV$ the element of volume, then there is an geometric meaning for the quantity $|| |A| ||_p^p = int_M |A|^p dV$? Again, it seems that there is an intepretation since $|A|^2 - frac{H^2}{n} = sum_{i < j} (lambda_i - lambda_j)^2$, where $lambda_i$ are the principal curvatures of $M$.
Thanks in advance!
differential-geometry riemannian-geometry
asked Jan 2 at 18:03
GeorgeGeorge
801515
$endgroup$
add a comment |
$begingroup$
Let be $M$ an oriented smooth surface on $mathbb{R}^3$, $|A|$ the norm of the second fundamental form of $M$ and $dV$ the element of area. I would like to know if there is a geometric meaning for the quantity $|| |A| ||_p^p = int_M |A|^p dV$? I saw this appears in "A compactness theorem for surfaces with $Lp$-bounded second fundamental form" by Joel Langer and this seems an important quantity since the integrand $|A| = sqrt{4H^2 - 2K}$ and the quantities $H$ and $K$ are invariant under change of the basis. I looked for an explanations about the $L^p$-norm of $|A|$ in Riemannian Geometry's textbooks and in Differential Geometry's textbooks, but I couldn't find nothing.
More generally, if $M$ is an oriented smooth $n$-manifold immersed on $mathbb{R}^{n+1}$, $|A|$ the norm of the second fundamental form of $M$ and $dV$ the element of volume, then there is an geometric meaning for the quantity $|| |A| ||_p^p = int_M |A|^p dV$? Again, it seems that there is an intepretation since $|A|^2 - frac{H^2}{n} = sum_{i < j} (lambda_i - lambda_j)^2$, where $lambda_i$ are the principal curvatures of $M$.
Thanks in advance!
differential-geometry riemannian-geometry
asked Jan 2 at 18:03
GeorgeGeorge
801515
$endgroup$
Let be $M$ an oriented smooth surface on $mathbb{R}^3$, $|A|$ the norm of the second fundamental form of $M$ and $dV$ the element of area. I would like to know if there is a geometric meaning for the quantity $|| |A| ||_p^p = int_M |A|^p dV$? I saw this appears in "A compactness theorem for surfaces with $Lp$-bounded second fundamental form" by Joel Langer and this seems an important quantity since the integrand $|A| = sqrt{4H^2 - 2K}$ and the quantities $H$ and $K$ are invariant under change of the basis. I looked for an explanations about the $L^p$-norm of $|A|$ in Riemannian Geometry's textbooks and in Differential Geometry's textbooks, but I couldn't find nothing.
More generally, if $M$ is an oriented smooth $n$-manifold immersed on $mathbb{R}^{n+1}$, $|A|$ the norm of the second fundamental form of $M$ and $dV$ the element of volume, then there is an geometric meaning for the quantity $|| |A| ||_p^p = int_M |A|^p dV$? Again, it seems that there is an intepretation since $|A|^2 - frac{H^2}{n} = sum_{i < j} (lambda_i - lambda_j)^2$, where $lambda_i$ are the principal curvatures of $M$.
Thanks in advance!
differential-geometry riemannian-geometry
differential-geometry riemannian-geometry
asked Jan 2 at 18:03
GeorgeGeorge
801515
asked Jan 2 at 18:03
GeorgeGeorge
801515
asked Jan 2 at 18:03
GeorgeGeorge
801515
asked Jan 2 at 18:03
GeorgeGeorge
801515
asked Jan 2 at 18:03
GeorgeGeorge
801515
801515
add a comment |
add a comment |
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