Singularity removability at branch points for Willmore surfaces (Q387715)

Let \(\vec{\Phi}\) be an immersion of a closed surface \(\Sigma\) into \(\mathbb{R}^m\). The Willmore energy functional is \(W(\vec{\Phi}):=\int_\Sigma|\vec{H}|^2d\text{\,vol}_g\). Here \(\vec{H}\) is the mean curvature vector of the immersion and \(g=\vec{\Phi}^*_{g_{\mathbb{R}^m}}\) is the pullback by \(\vec{\Phi}\) of the flat canonical metric \(g_{\mathbb{R}^m}\) of \(\mathbb{R}^m\).NEWLINENEWLINEA Willmore surface is a critical point of the functional \(W(\vec{\Phi})\). Investigation of the closure of the space of Willmore surfaces leads to study Willmore surfaces with a finite set of singular points. An important problem is to define when the singularities are removable. The model surface is the plane unit disk \(D^2\) with the origin as a singular point, so we can investigate maps \(\vec{\Phi}:D^2\setminus \{0\}\to \mathbb{R}^m\) satisfying: NEWLINENEWLINENEWLINENEWLINE (i) \(\vec{\Phi}\in C^0(D^2)\cap C^\infty(D^2\setminus \{0\})\); NEWLINENEWLINENEWLINENEWLINE (ii) \(\mathcal{H}^2(\vec{\Phi}(D^2))<+\infty\);NEWLINENEWLINENEWLINENEWLINE (iii) \(\int_{D^2}|\vec{\mathbb{I}}|^2d\text{vol}_g<+\infty\); NEWLINENEWLINENEWLINENEWLINE here \(\mathcal{H}^2\) is the Hausdorff \(2\)-measure and \(\vec{\mathbb{I}}\) is the second fundamental form of the immersion. NEWLINENEWLINENEWLINENEWLINE The mean curvature vector \(\vec{H}\) satisfies a differential equation; the authors rewrite it in a divergent form. Let us define a vector \(\vec{\gamma}_0\), named the first residue, as follows: NEWLINENEWLINE\[NEWLINE\vec{\gamma}_0:=(1/4\pi)\int_{\partial D^2}\vec{\nu}\cdot\left(\nabla \vec{H}-3\pi_{\vec{n}}(\nabla \vec{H})+*(\nabla^\perp \vec{n}\wedge \vec{H})\right)NEWLINE\]NEWLINENEWLINEhere \(\vec{\nu}\) is the unit onward normal vector to \(\partial D^2\), \(\nabla=(\partial_{x_1},\partial_{x_2})\), \(\nabla^\perp=(-\partial_{x_2},\partial_{x_1})\) are differential operators with respect to flat coordinates \(x_1\), \(x_2\) on the unit disk, \(*\) is the Hodge operator, \(\vec{n}\) is the Gauss map, and \(\pi_{\vec{n}}\) is the orthogonal projection from \(\mathbb{R}^m\) onto the \((m-2)\)-plane given by \(\vec{n}\). NEWLINENEWLINENEWLINENEWLINE It is known that there exists a positive integer \(\theta_0\) such that \(|\vec{\Phi}|\simeq |x^{\theta_0}|\) and \(|\nabla \vec{\Phi}|\simeq |x^{\theta_0-1}|\) near the origin. The investigation of the behavior of \(\vec{H}\) near the point \(x=0\) leads to the representation \(\vec{H}+\vec{\gamma}_0\log|x|=\Re(\vec{E}(x)-\vec{T}(x))\) (Proposition 1.6); here \(\vec{T}\) satisfies \(\partial_x\vec{T}=O(|\vec{H}|\,|\nabla \vec{n}|)\) and \(\vec{T}=O(|x|^{2+\theta_0-\epsilon})\) for all \(\epsilon>0\), and \(\vec{E}=(E_1,\ldots,E_m)\) is antiholomorphic with possibly a polar singularity at the origin of order at most \((\theta_0-1)\). The second residue associated with \(\vec{\Phi}\) is the vector \(\vec{\gamma}=(\gamma_1,\ldots,\gamma_m)\) where NEWLINENEWLINE\[NEWLINE\gamma_j:=(1/(2\pi i)) \int_{\partial D^2}d\,\log E_j.NEWLINE\]NEWLINENEWLINENEWLINENEWLINEThe main result of the paper is Theorem 1.9 which states that if \(\theta_0=1\) and \(\vec{\gamma}_0=\vec{0}\), then the immersion \(\Phi\) is smooth across the branch point; if \(\theta_0>1\), then the conditions \(\vec{\gamma}_0=\vec{0}\) and \(\vec{\gamma}=\vec{0}\) imply the smoothness of \(\Phi\) across the branch point.