Suppose $\Sigma\subset \mathbb{R}^{n+1}$ is a closed embedded hypersurface. We know that when $n=1$

$$ \int_{\Sigma} |H|^2 \geq \frac{4 \pi^2}{|\Sigma|} $$ by Gauss-Bonnet and that this is saturated on the round circle -- here $|\Sigma|$ is the length of $\Sigma$ and $H$ the mean curvature.

Likewise, if $n=2$ we have $$ \int_{\Sigma} |H|^2\geq 16\pi $$ which is also saturated on the round sphere (of course as we now know this can be improved for positive genus surfaces).

I'm wondering to what extent one can get sharp lower bounds in dimensions $n+1>3$. Specifically, something like $$ \int_{\Sigma} |H|^2 \geq C_n |\Sigma|^{n-2}. $$

Ideally, this bound would be sharp on the round sphere at least amongst convex competitors (I suspect otherwise it wouldn't be true).

Any references would be appreciated.

Suppose $\Sigma\subset \mathbb{R}^{n+1}$ is a closed embedded hypersurface. We know that when $n=1$

$$ \int_{\Sigma} |H|^2 \geq \frac{4 \pi^2}{|\Sigma|} $$ by Gauss-Bonnet and that this is saturated on the round circle -- here $|\Sigma|$ is the length of $\Sigma$ and $H$ the mean curvature.

Likewise, if $n=2$ we have $$ \int_{\Sigma} |H|^2\geq 16\pi $$ which is also saturated on the round sphere (of course as we now know this can be improved for positive genus surfaces).

I'm wondering to what extent one can get sharp lower bounds in dimensions $n+1>3$. Specifically, something like $$ \int_{\Sigma} |H|^2 \geq C_n |\Sigma|^{(n-2)/n}. $$

Ideally, this bound would be sharp on the round sphere at least amongst convex competitors (I suspect otherwise it wouldn't be true).

Any references would be appreciated.

Suppose $\Sigma\subset \mathbb{R}^{n+1}$ is a closed embedded hypersurface. We know that when $n=1$

$$ \int_{\Sigma} |H|^2 \geq \frac{4 \pi^2}{|\Sigma|} $$ by Gauss-Bonnet and that this is saturated on the round circle -- here $|\Sigma|$ is the length of $\Sigma$ and $H$ the mean curvature.

Likewise, if $n=2$ we have $$ \int_{\Sigma} |H|^2\geq 8\pi $$ which is also saturated on the round sphere (of course as we now know this can be improved for positive genus surfaces).

I'm wondering to what extent one can get sharp lower bounds in dimensions $n+1>3$. Specifically, something like $$ \int_{\Sigma} |H|^2 \geq C_n |\Sigma|^{n-2}. $$

Ideally, this bound would be sharp on the round sphere at least amongst convex competitors (I suspect otherwise it wouldn't be true).

Any references would be appreciated.

Suppose $\Sigma\subset \mathbb{R}^{n+1}$ is a closed embedded hypersurface. We know that when $n=1$

$$ \int_{\Sigma} |H|^2 \geq \frac{4 \pi^2}{|\Sigma|} $$ by Gauss-Bonnet and that this is saturated on the round circle -- here $|\Sigma|$ is the length of $\Sigma$ and $H$ the mean curvature.

Likewise, if $n=2$ we have $$ \int_{\Sigma} |H|^2\geq 16\pi $$ which is also saturated on the round sphere (of course as we now know this can be improved for positive genus surfaces).

I'm wondering to what extent one can get sharp lower bounds in dimensions $n+1>3$. Specifically, something like $$ \int_{\Sigma} |H|^2 \geq C_n |\Sigma|^{n-2}. $$

Ideally, this bound would be sharp on the round sphere at least amongst convex competitors (I suspect otherwise it wouldn't be true).

Any references would be appreciated.