Weak immersions of surfaces with \(L^2\)-bounded second fundamental form. (English) Zbl 1359.53002
Bray, Hubert L. (ed.) et al., Geometric analysis. Lecture notes from the graduate minicourse of the 2013 IAS/Park City Mathematics Institute session on geometric analysis, Park City, UT, USA, 2013. Providence, RI: American Mathematical Society (AMS); Princeton, NJ: Institute for Advanced Study (IAS) (ISBN 978-1-4704-2313-1/hbk; 978-1-4704-2881-5/ebook). IAS/Park City Mathematics Series 22, 303-384 (2016).
From the text: In the present series of five lectures, we develop some fundamental tools in the analytic study of the Willmore Lagrangian motivated by the following set of questions:
i) Does there exist a minimizer of the Willmore functional among all smooth immersions of a fixed surface \(\Sigma^2\)? If there is, can one estimate the energy and special properties of such a minimizer?
ii) Does there exist a minimizer of the Willmore functional among a more restricted class of immersions, for example, considering only the conformal immersions relative to a fixed conformal class \(c\) on \(\Sigma\), or among all immersions of \(\Sigma\) into \(\mathbb{R}^3\) which enclose a domain of given volume and which have a fixed area?
iii) How stable is the Willmore equation? In other words, does a sequence of “approximately Willmore” surfaces, e.g., Palais-Smale sequences for the Willmore energy, necessarily converge to a Willmore surface?
iv) Can one procedure Willmore surfaces using min-max arguments? More specifically can one apply fundamental variational principles such as Ekeland’s variational principles or the mountain pass lemma to the Willmore functional?
In 2010, the author introduced the notion of weak immersions in order to provide a suitable framework in which general variations of the Willmore Lagrangian [the author, J. Reine Angew. Math. 695, 41–98 (2014; Zbl 1304.49095)] are well posed. The goal of this mini-course is to present fundamental properties of this approach, which turns out to be a fundamental tool for addressing all of the questions i)…iv). To illustrate this, we explain in the last lecture how to use these properties to give a new proof of Simon’s existence result. More completely, we address in this mini-course the following questions:
a) Does a weak immersion define a smooth conformal structure?
b) What happens to a sequence of weak immersions of a given surface \(\Sigma^2\) for which the Willmore energy is uniformly bounded? Does it convergence in some sense to a weak immersion?
c) Is there a weak formulation of the Willmore equation which is compatible with the notion of weak immersion?
d) Are weak immersions which are solutions to the Willmore equation necessarily smooth?
For the entire collection see [Zbl 1343.53002].
i) Does there exist a minimizer of the Willmore functional among all smooth immersions of a fixed surface \(\Sigma^2\)? If there is, can one estimate the energy and special properties of such a minimizer?
ii) Does there exist a minimizer of the Willmore functional among a more restricted class of immersions, for example, considering only the conformal immersions relative to a fixed conformal class \(c\) on \(\Sigma\), or among all immersions of \(\Sigma\) into \(\mathbb{R}^3\) which enclose a domain of given volume and which have a fixed area?
iii) How stable is the Willmore equation? In other words, does a sequence of “approximately Willmore” surfaces, e.g., Palais-Smale sequences for the Willmore energy, necessarily converge to a Willmore surface?
iv) Can one procedure Willmore surfaces using min-max arguments? More specifically can one apply fundamental variational principles such as Ekeland’s variational principles or the mountain pass lemma to the Willmore functional?
In 2010, the author introduced the notion of weak immersions in order to provide a suitable framework in which general variations of the Willmore Lagrangian [the author, J. Reine Angew. Math. 695, 41–98 (2014; Zbl 1304.49095)] are well posed. The goal of this mini-course is to present fundamental properties of this approach, which turns out to be a fundamental tool for addressing all of the questions i)…iv). To illustrate this, we explain in the last lecture how to use these properties to give a new proof of Simon’s existence result. More completely, we address in this mini-course the following questions:
a) Does a weak immersion define a smooth conformal structure?
b) What happens to a sequence of weak immersions of a given surface \(\Sigma^2\) for which the Willmore energy is uniformly bounded? Does it convergence in some sense to a weak immersion?
c) Is there a weak formulation of the Willmore equation which is compatible with the notion of weak immersion?
d) Are weak immersions which are solutions to the Willmore equation necessarily smooth?
For the entire collection see [Zbl 1343.53002].
MSC:
| 53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |
| 53A05 | Surfaces in Euclidean and related spaces |
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 53A30 | Conformal differential geometry (MSC2010) |
