Gradient flow for the Willmore functional
The Willmore energy of a closed, immersed surface \(f:\Sigma\to \mathbb{R}^n\) with induced area measure \(d\mu\) is given by \(W(f)=\int_\Sigma|A^0 |^2d\mu\), where \(A^0=A-\frac 12 g \otimes H\) is the tracefree part of the second fundamental form \(A\). It is not known whether there can develop singularities under the Willmore flow (related to the above energy) in finite time. One of the main results of the paper under review is to obtain a lower bound on the lifespan which depends only on the concentration of curvature for the intial surface. The main theorem is the following: Theorem: Let \(f_0:\Sigma \to\mathbb{R}^n\) be a smooth immersion. There are constants \(\varepsilon_0>0\), \(c<\infty\) depending only on \(n\), such that if \(\rho>0\) is chosen with \[ \int_{B_\rho(x)} |A_0 |^2 d\mu_0\leq \varepsilon\leq \varepsilon_0 \] for any \(x\in\mathbb{R}^n\), then the maximal time \(T\) of smooth existence for the Willmore flow with initial data \(f_0\) satisfies \(T\geq {1\over c}\rho^4\), and one has the estimate \[ \int_{B_\rho (x)}|A|^2 d\mu\leq c\varepsilon \] for \(0\leq t\leq {1\over c}\rho^4\).
scientific article; zbMATH DE number 1893848