Mean curvature flow in higher codimension: introduction and survey. (English) Zbl 1247.53004
Bär, Christian (ed.) et al., Global differential geometry. Berlin: Springer (ISBN 978-3-642-22841-4/hbk; 978-3-642-22842-1/ebook). Springer Proceedings in Mathematics 17, 231-274 (2012).
This is a useful survey on fundamental results on the mean curvature flow of closed submanifolds with higher codimension in a general Riemannian ambient space.
A detailed introduction is given that makes the survey recommendable for beginners on the subject, with all necessary equations explicitly derived.
Special cases such as codimension one, graphs, and Lagrangian submanifolds are considered throughout the survey. In the last section particular aspects are treated, some of them considered by the author in previous works.
Short-time existence and uniqueness of the mean curvature flow is explained by linearization, with some remarks concerning regularity and the non-compact complete case.
The problem of long-time existence and the formation and classification of singularities is addressed, with some proofs, by applying the parabolic maximum principle. When the ambient space is Euclidean, the rescaling techniques to study the two types of singularities are described, recalling for instance Huisken’s monotonicity formula. Corresponding self-similar shrinking solutions as well as eternal solutions are considered, with some partial classifications, including references to the higher-codimension cases obtained by the author.
In the last section, classes of submanifolds are distinguished by some properties preserved under the mean curvature flow, for instance convexity, graphs, Lagrangian submanifolds and \(\delta\)-pinched second fundamental form. In the latter case, for hypersurfaces and Lagrangian submanifolds, the author derives some rigidity results and classification.
As the author states in section 1, this survey is restricted to the case of Riemannian ambient space, but it could be interesting to consider in the same way the mean curvature flow of space-like submanifolds in a pseudo-Riemannian manifold as well, and to explain the main differences in the two cases.
For the entire collection see [Zbl 1230.53005].
A detailed introduction is given that makes the survey recommendable for beginners on the subject, with all necessary equations explicitly derived.
Special cases such as codimension one, graphs, and Lagrangian submanifolds are considered throughout the survey. In the last section particular aspects are treated, some of them considered by the author in previous works.
Short-time existence and uniqueness of the mean curvature flow is explained by linearization, with some remarks concerning regularity and the non-compact complete case.
The problem of long-time existence and the formation and classification of singularities is addressed, with some proofs, by applying the parabolic maximum principle. When the ambient space is Euclidean, the rescaling techniques to study the two types of singularities are described, recalling for instance Huisken’s monotonicity formula. Corresponding self-similar shrinking solutions as well as eternal solutions are considered, with some partial classifications, including references to the higher-codimension cases obtained by the author.
In the last section, classes of submanifolds are distinguished by some properties preserved under the mean curvature flow, for instance convexity, graphs, Lagrangian submanifolds and \(\delta\)-pinched second fundamental form. In the latter case, for hypersurfaces and Lagrangian submanifolds, the author derives some rigidity results and classification.
As the author states in section 1, this survey is restricted to the case of Riemannian ambient space, but it could be interesting to consider in the same way the mean curvature flow of space-like submanifolds in a pseudo-Riemannian manifold as well, and to explain the main differences in the two cases.
For the entire collection see [Zbl 1230.53005].
Reviewer: Isabel Salavessa (Lisboa)
MSC:
| 53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53C20 | Global Riemannian geometry, including pinching |
| 53C38 | Calibrations and calibrated geometries |
| 53D12 | Lagrangian submanifolds; Maslov index |
| 53C24 | Rigidity results |
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