Analysis aspects of Willmore surfaces
| dc.creator | Tristan, Riviere | |
| dc.date | 2006-12-18 | |
| dc.date.accessioned | 2026-07-07T07:35:44Z | |
| dc.date.available | 2026-07-07T07:35:44Z | |
| dc.description | We found a new formulation to the Euler-Lagrange equation of the Willmore functional for immersed surfaces in ${\R}^m$. This new formulation of Willmore equation appears to be of divergence form, moreover, the non-linearities are made of jacobians. Additionally to that, if $\bH$ denotes the mean curvature vector of the surface, this new form writes ${\mathcal L}\bH=0$ where ${\mathcal L}$ is a well defined locally invertible self-adjoint operator. These 3 facts have numerous consequences in the analysis of Willmore surfaces. One first consequence is that the long standing open problem to give a meaning to the Willmore Euler-Lagrange equation for immersions having only $L^2$ bounded second fundamental form is now solved. We then establish the regularity of weak $W^{2,p}-$Willmore surfaces for any $p$ for which the Gauss map is continuous : $p>2$. This is based on the proof of an $ε-$regularity result for weak Willmore surfaces. We establish then a weak compactness result for Willmore surfaces of energy less than $8π-δ$ for every $δ>0$. This theorem is based on a point removability result we prove for Wilmore surfaces in ${\R}^m$. This result extends to arbitrary codimension a result that E.Kuwert and R.Schaetzle established for surfaces in ${\R}^3$. Finally, we deduce from this point removability result the strong compactness, modulo the Möbius group action, of Willmore tori below the energy level $8π-δ$ in dimensions 3 and 4. The dimension 3 case was already solved in a previous work. | |
| dc.identifier | https://arxiv.org/abs/math/0612526 | |
| dc.identifier | http://arxiv.org/abs/math/0612526 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/120191 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Differential Geometry | |
| dc.subject | 53A05; 53C42; 58E15; 53A30; 35Jxx | |
| dc.title | Analysis aspects of Willmore surfaces | |
| dc.type | text |