- AutorIn
- Renan Assimos Martins
- Titel
- The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-361482
- Datum der Einreichung
- 08.03.2019
- Datum der Verteidigung
- 12.07.2019
- Abstract (EN)
- We develop a general method to construct subsets of complete Riemannian manifolds that cannot contain images of non-constant harmonic maps from compact manifolds. We apply our method to the special case where the harmonic map is the Gauss map of a minimal submanifold and the complete manifold is a Grassmannian. With the help of a result by Allard [Allard, W. K. (1972). On the first variation of a varifold. Annals of mathematics, 417-491.], we can study the graph case and have an approach to prove Bernstein-type theorems. This enables us to extend Moser’s Bernstein theorem [Moser, J. (1961). On Harnack's theorem for elliptic differential equations. Communications on Pure and Applied Mathematics, 14(3), 577-591.] to codimension two, i.e., a minimal p-submanifold in $R^{p+2}$, which is the graph of a smooth function defined on the entire $R^p$ with bounded slope, must be a p-plane.
- Freie Schlagwörter (EN)
- Bernstein theorem, minimal graph, harmonic map, Grassmannian, Gauss map, maximum principle.
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- publizierte Version / Verlagsversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-361482
- Veröffentlichungsdatum Qucosa
- 14.11.2019
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch