Stationary surfaces with boundaries (Q2164355)

The authors study surfaces with boundary in \(\mathbb{R}^3\) (in general non empty), which appear as critical points of functionals depending on curvature quantities. More precisely, let \(\mathbf{r}:\Sigma \rightarrow \mathbb{R}^3\) be an isometric immersion of the oriented surface \(\Sigma\) into the Euclidean 3-space and a functional \(\mathcal{W}(\mathbf{r})\), which is the integral of a symmetric polynomial function in the principal curvatures: \[\mathcal{W}(\mathbf{r}):=\int_{\Sigma} \tilde{F}(\kappa_1,\kappa_2)d\mu,\] or \[\mathcal{W}(\mathbf{r}):=\int_{\Sigma} F(H,K)d\mu,\] where \(H=\kappa_1+\kappa_2\) is twice the mean curvature and \(K=\kappa_1\kappa_2\) is the Gauss curvature. This includes the cases of the surface area (\(F=1\)) and the conformal Willmore functional (\(F=H^2-4K\)). The computation of the first variation of \(\mathcal{W}\) allows one to define a stress tensor, whose divergence is related to the deviation from \(\mathcal{W}\)-criticality. In case of scaling-invariant \(\mathcal{W}\)-functionals a characterization of free-boundary \(\mathcal{W}\)-surfaces with rotational symmetry is given. If the functional is not scaling-invariant, certain boundary-to-interior consequences are discussed. Finally, some applications to the conformal Willmore energy and the \(p\)-Willmore energy of surfaces are presented.