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gaoxinge
gaoxinge
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Let us assume $H$$\text{H}$ is the mean curvature of a compact surface in $E^3$$\mathbb E^3$ and $g$ is its genus.
(1) When $g$ is arbitrary, we have $\int_{S^{2}}H^{2}dV=4\pi$ and $\int_{\Sigma}H^{2}dV\geq4\pi$.
(2)When $g=1$, we have $\int_{T^{2}}H^{2}dV\geq2\pi^{2}$ and $\int_{\Sigma}H^{2}dV\geq2\pi^{2}$.
(3)I want to ask: can these theorems be generalized to a compact surface with $g>1$ in $E^3$? And the statement below is right or not?
When $g(\Sigma_{1})>g(\Sigma_{2})$, then do we have $\int_{\Sigma_{1}}H^{2}dV\geq\int_{\Sigma_{2}}H^{2}dV$?

  • When $g$ is arbitrary, we have $\int_{\mathbb S^2}\text{H}^2dV=4\pi$ and $\int_{\Sigma}\text{H}^2dV\geq4\pi$.

  • When $g=1$, we have $\int_{\mathbb T^2}\text{H}^2dV\geq2\pi^{2}$ and $\int_{\Sigma}\text{H}^2dV\geq2\pi^{2}$.

  • I want to ask that these theorems can be generalized to a compact surface with $g>1$ in $\mathbb E^3$? And the statement below is right or not?
    $$\text{When}~~g(\Sigma_{1})>g(\Sigma_{2})\text{, then do we have} \int_{\Sigma_{1}}\text{H}^2dV\geq\int_{\Sigma_{2}}\text{H}^2dV\text{?}$$

Let us assume $H$ is the mean curvature of a compact surface in $E^3$ and $g$ is its genus.
(1) When $g$ is arbitrary, we have $\int_{S^{2}}H^{2}dV=4\pi$ and $\int_{\Sigma}H^{2}dV\geq4\pi$.
(2)When $g=1$, we have $\int_{T^{2}}H^{2}dV\geq2\pi^{2}$ and $\int_{\Sigma}H^{2}dV\geq2\pi^{2}$.
(3)I want to ask: can these theorems be generalized to a compact surface with $g>1$ in $E^3$? And the statement below is right or not?
When $g(\Sigma_{1})>g(\Sigma_{2})$, then do we have $\int_{\Sigma_{1}}H^{2}dV\geq\int_{\Sigma_{2}}H^{2}dV$?

Let us assume $\text{H}$ is the mean curvature of a compact surface in $\mathbb E^3$ and $g$ is its genus.

  • When $g$ is arbitrary, we have $\int_{\mathbb S^2}\text{H}^2dV=4\pi$ and $\int_{\Sigma}\text{H}^2dV\geq4\pi$.

  • When $g=1$, we have $\int_{\mathbb T^2}\text{H}^2dV\geq2\pi^{2}$ and $\int_{\Sigma}\text{H}^2dV\geq2\pi^{2}$.

  • I want to ask that these theorems can be generalized to a compact surface with $g>1$ in $\mathbb E^3$? And the statement below is right or not?
    $$\text{When}~~g(\Sigma_{1})>g(\Sigma_{2})\text{, then do we have} \int_{\Sigma_{1}}\text{H}^2dV\geq\int_{\Sigma_{2}}\text{H}^2dV\text{?}$$

replaced deprecated tag 'geometry'; added more tags; slight rewording
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Ricardo Andrade
Ricardo Andrade
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Let weus assume $H$ is the mean curvature of a compact surface in $E^3$ and $g$ is its genus.
(1)When When $g$ is arbitrary, we have $\int_{S^{2}}H^{2}dV=4\pi$ and $\int_{\Sigma}H^{2}dV\geq4\pi$.
(2)When $g=1$, we have $\int_{T^{2}}H^{2}dV\geq2\pi^{2}$ and $\int_{\Sigma}H^{2}dV\geq2\pi^{2}$.
(3)I want to ask: can these theorems be generalized to a compact surface whosewith $g>1$ in $E^3$? And the statement below is right or not?
0.0WhenWhen $g(\Sigma_{1})>g(\Sigma_{2})$, then do we have $\int_{\Sigma_{1}}H^{2}dV\geq\int_{\Sigma_{2}}H^{2}dV$.?

Let we assume $H$ is mean curvature of a compact surface in $E^3$ and $g$ is its genus.
(1)When $g$ is arbitrary, we have $\int_{S^{2}}H^{2}dV=4\pi$ and $\int_{\Sigma}H^{2}dV\geq4\pi$.
(2)When $g=1$, we have $\int_{T^{2}}H^{2}dV\geq2\pi^{2}$ and $\int_{\Sigma}H^{2}dV\geq2\pi^{2}$.
(3)I want to ask can these theorems be generalized to a compact surface whose $g>1$ in $E^3$? And the statement below is right or not?
0.0When $g(\Sigma_{1})>g(\Sigma_{2})$, then we have $\int_{\Sigma_{1}}H^{2}dV\geq\int_{\Sigma_{2}}H^{2}dV$.

Let us assume $H$ is the mean curvature of a compact surface in $E^3$ and $g$ is its genus.
(1) When $g$ is arbitrary, we have $\int_{S^{2}}H^{2}dV=4\pi$ and $\int_{\Sigma}H^{2}dV\geq4\pi$.
(2)When $g=1$, we have $\int_{T^{2}}H^{2}dV\geq2\pi^{2}$ and $\int_{\Sigma}H^{2}dV\geq2\pi^{2}$.
(3)I want to ask: can these theorems be generalized to a compact surface with $g>1$ in $E^3$? And the statement below is right or not?
When $g(\Sigma_{1})>g(\Sigma_{2})$, then do we have $\int_{\Sigma_{1}}H^{2}dV\geq\int_{\Sigma_{2}}H^{2}dV$?

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gaoxinge
gaoxinge
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Integral of Square of Mean Curvature

Let we assume $H$ is mean curvature of a compact surface in $E^3$ and $g$ is its genus.
(1)When $g$ is arbitrary, we have $\int_{S^{2}}H^{2}dV=4\pi$ and $\int_{\Sigma}H^{2}dV\geq4\pi$.
(2)When $g=1$, we have $\int_{T^{2}}H^{2}dV\geq2\pi^{2}$ and $\int_{\Sigma}H^{2}dV\geq2\pi^{2}$.
(3)I want to ask can these theorems be generalized to a compact surface whose $g>1$ in $E^3$? And the statement below is right or not?
0.0When $g(\Sigma_{1})>g(\Sigma_{2})$, then we have $\int_{\Sigma_{1}}H^{2}dV\geq\int_{\Sigma_{2}}H^{2}dV$.