On the analysis of isometric immersions of Riemannian manifolds
<p>This thesis is a study of problems related to isometric immersions of Riemannian manifolds in Euclidean space. We address three main questions: the weak continuity of geometric invariants along sequences of immersions of Riemannian manifolds; the construction of isometric immersions by weak...
| Hoofdauteur: | |
|---|---|
| Andere auteurs: | |
| Formaat: | Thesis |
| Taal: | English |
| Gepubliceerd in: |
2020
|
| Onderwerpen: |
| _version_ | 1826296714846273536 |
|---|---|
| author | Giron, T |
| author2 | Chen, G |
| author_facet | Chen, G Giron, T |
| author_sort | Giron, T |
| collection | OXFORD |
| description | <p>This thesis is a study of problems related to isometric immersions of Riemannian manifolds in Euclidean space. We address three main questions: the weak continuity of geometric invariants along sequences of immersions of Riemannian manifolds; the construction of isometric immersions by weak compactness methods; and the validity and regularity of the Gauss equation.</p>
<p>First, we investigate the validity of Cartan's equations for W<sup>1,<em>p</em></sup> coframes on surfaces, for all 1 ≤ <em>p</em> ≤ ∞, and employ this to derive a version of the Gauß equation valid for W<sup>2,<em>p</em></sup> immersed surfaces in <strong>R</strong><sup>3</sup>. Under some additional regularity hypotheses, a distributional formulation of the Gauß equation on immersed surfaces in <strong>R</strong><sup>3</sup> is proved, and as a corollary, a new local regularity result is established for isometric immersions of positively curved surfaces.</p>
<p>Investigating the weak continuity properties of immersions of Riemannian manifolds, we first prove a general weak continuity result for curvatures of connections with $L^p$ bounds on principal bundles. As a corollary, it is proved that when $p>2$, curvatures are weakly continuous for the weak $W^{2,p}$ convergence of immersions. In some cases, we also recover the weak continuity of the general Gauss equation when $p=2$.</p>
<p>Finally, we give a general viscosity framework for constructing isometric immersions in prescribed target spaces under natural boundedness assumptions in $L^p$ spaces. Assuming this set-up, we prove a new weak compactness theorem for approximate solutions of the Gauss--Codazzi--Ricci equations.</p> |
| first_indexed | 2024-03-07T04:20:36Z |
| format | Thesis |
| id | oxford-uuid:cae2f41d-c5a1-4138-9dec-5e824d21044e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T04:20:36Z |
| publishDate | 2020 |
| record_format | dspace |
| spelling | oxford-uuid:cae2f41d-c5a1-4138-9dec-5e824d21044e2022-03-27T07:11:06ZOn the analysis of isometric immersions of Riemannian manifoldsThesishttp://purl.org/coar/resource_type/c_db06uuid:cae2f41d-c5a1-4138-9dec-5e824d21044eMathematicsEnglishHyrax Deposit2020Giron, TChen, GNguyen, L<p>This thesis is a study of problems related to isometric immersions of Riemannian manifolds in Euclidean space. We address three main questions: the weak continuity of geometric invariants along sequences of immersions of Riemannian manifolds; the construction of isometric immersions by weak compactness methods; and the validity and regularity of the Gauss equation.</p> <p>First, we investigate the validity of Cartan's equations for W<sup>1,<em>p</em></sup> coframes on surfaces, for all 1 ≤ <em>p</em> ≤ ∞, and employ this to derive a version of the Gauß equation valid for W<sup>2,<em>p</em></sup> immersed surfaces in <strong>R</strong><sup>3</sup>. Under some additional regularity hypotheses, a distributional formulation of the Gauß equation on immersed surfaces in <strong>R</strong><sup>3</sup> is proved, and as a corollary, a new local regularity result is established for isometric immersions of positively curved surfaces.</p> <p>Investigating the weak continuity properties of immersions of Riemannian manifolds, we first prove a general weak continuity result for curvatures of connections with $L^p$ bounds on principal bundles. As a corollary, it is proved that when $p>2$, curvatures are weakly continuous for the weak $W^{2,p}$ convergence of immersions. In some cases, we also recover the weak continuity of the general Gauss equation when $p=2$.</p> <p>Finally, we give a general viscosity framework for constructing isometric immersions in prescribed target spaces under natural boundedness assumptions in $L^p$ spaces. Assuming this set-up, we prove a new weak compactness theorem for approximate solutions of the Gauss--Codazzi--Ricci equations.</p> |
| spellingShingle | Mathematics Giron, T On the analysis of isometric immersions of Riemannian manifolds |
| title | On the analysis of isometric immersions of Riemannian manifolds |
| title_full | On the analysis of isometric immersions of Riemannian manifolds |
| title_fullStr | On the analysis of isometric immersions of Riemannian manifolds |
| title_full_unstemmed | On the analysis of isometric immersions of Riemannian manifolds |
| title_short | On the analysis of isometric immersions of Riemannian manifolds |
| title_sort | on the analysis of isometric immersions of riemannian manifolds |
| topic | Mathematics |
| work_keys_str_mv | AT giront ontheanalysisofisometricimmersionsofriemannianmanifolds |