Willmore energy of an ellipsoid

Question

Given an ellipsoid of equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ How can I calculate the Willmore energy of this surface knowing that its definition is: $$W=\frac{1}{4}\int_S(k_1-k_2)^2dA$$where $k_1$ and $k_2$ are the principal curvatures of the surface? Thanks

Answer 1

Curvature formulas for implicit surfaces $F(x,y,z)=0$ can be found in Curvature formulas for implicit curves and surfaces by Ron Goldman, with derivation. Let $H$ be the Hessian of $F$. Then the principal curvatures are $$k_{1,2}=\frac{-1}{|\nabla F|}\lambda_{1,2}\tag1$$ (the minus sign is arbitrary, of course) where $\lambda_{1,2}$ are the roots of the equation $$\det\begin{pmatrix} H-\lambda I & \nabla F^T \\ \nabla F & 0 \end{pmatrix}=0 \tag2$$ (see equation (4.4) of the paper). Here $\nabla F$ is understood as a row vector.

Although the matrix in (2) is $4\times 4$, the fact that $H$ is diagonal in your example should help.