Immersions with Bounded Second Fundamental Form
- 370 Downloads
We first consider immersions on compact manifolds with uniform L p -bounds on the second fundamental form and uniformly bounded volume. We show compactness in arbitrary dimension and codimension, generalizing a classical result of J. Langer. In the second part, this result is used to deduce a localized version, being more convenient for many applications, such as convergence proofs for geometric flows.
KeywordsImmersions Compactness Second fundamental form Bounded curvature Compact and noncompact manifolds
Mathematics Subject Classification53C42 53C23 53B25
I would like to thank my advisor Ernst Kuwert for his support. Moreover, I would like to thank Manuel Breuning for proofreading my dissertation , where the results of this paper were established first.
- 2.Baker, C.: The mean curvature flow of submanifolds of high codimension. Ph.D. thesis (2010) Google Scholar
- 3.Baker, C.: A partial classification of Type 1 singularities of the mean curvature flow in high codimension. Preprint (2011). arXiv:1104.4592
- 6.Breuning, P.: Immersions with local Lipschitz representation. Ph.D. thesis, Freiburg (2011) Google Scholar
- 9.Cooper, A.A.: A compactness theorem for the second fundamental form. Preprint (2011). arXiv:1006.5697v4
- 13.Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces, second printing with corrections. Birkhäuser, Boston (2001) Google Scholar
- 19.Link, F.: Gradient Flow for the Willmore Functional in Riemannian Manifolds. Ph.D. thesis, Freiburg (2013). arXiv:1308.6055
- 20.Mondino, A.: Existence of integral m-varifolds minimizing ∫|A|p and ∫|H|p, p>m, in Riemannian manifolds. Preprint (2010). arXiv:1010.4514v1
- 22.Schlichting, A.: Mittlerer Krümmungsfluss fast konvexer Lipschitzflächen. Diploma thesis, Freiburg (2009) Google Scholar