Let $M$ be a 3 dimensional manifold, $N$ a surface in $M$ and $A$ the second fundamental form on $N$, $H$ the mean curvature. $h$ is the metric induced on $N$.
I need to show that \begin{equation*} \int_{N_l}|A - \frac{H}{2}h|^2 = \frac{1}{2}\int_{N_l}(\lambda_1 - \lambda_2)^2. \end{equation*} where $\lambda_1, \lambda_2$ are the eigenvalues of $A$ with respect to $h$.
This is what I tied: In matrix form, we get \begin{equation*} A =\left( \begin{array}{cc} \lambda_1 & 0\\ 0 & \lambda_2 \\ \end{array} \right) \hspace{0.5cm} \text{and $H = \lambda_1 + \lambda_2.$} \end{equation*} We can express $h$ as the following (with an appropriate basis) \begin{equation*} h =\left( \begin{array}{cc} 1 & 0\\ 0 & 1 \\ \end{array} \right) \end{equation*} Thus, we have \begin{equation*} A-\frac{H}{2}h =\left( \begin{array}{cc} \frac{\lambda_1 -\lambda_2}{2} & 0\\ 0 & \frac{\lambda_2-\lambda_1}{2} \\ \end{array} \right), \end{equation*} and $|A-\frac{H}{2}h|^2 = \frac{1}{4}(\lambda_1 - \lambda_2)^2$. If the calculation is correct then I am wrong by a factor 1/2... What am I doing wrong ? Thanks!
I guess you are making, for some reason not clear to me, a mistake when calculating the squared norm of $A-\frac{H}{2}h$. This is just, using your matrix representation, $$\left(\frac{\lambda_1-\lambda_2}{2}\right)^2 + \left(\frac{\lambda_2-\lambda_1}{2}\right)^2 = 2\frac{1}{4}\left({\lambda_2-\lambda_1}\right)^2$$