MiS Preprint Repository
We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
From Isometric Embeddings to Turbulence
László SzékelyhidiAbstract
The following dichotomy concerning isometric embeddings of the sphere is well-known: whereas the only C2 isometric embedding of S2 into R3 is the standard embedding modulo rigid motion, there exists many C1 isometric embeddings which can "wrinkle" S2 into arbitrary small regions. The latter flexibility, known as the Nash-Kuiper theorem, involves an iteration scheme called convex integration which turned out to have surprisingly wide applicability. These lecture notes are meant as an analysts exposition of convex integration.
In particular the notes contain:
- an essentially complete proof of the rigidity and flexibility for isometric embeddings,
- a general discussion of the existence theory for first order partial differential inclusions and
- the recent application of the same methods to the incompressible Euler equations of fluid mechanics, extending the work of Scheffer and Snirelman.
- Received:
- 20.12.12
- Published:
- 08.01.13