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.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:36148 |
| Date | 14 November 2019 |
| Creators | Assimos Martins, Renan |
| Contributors | Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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