Embeddedness of timelike maximal surfaces in (1+2)-Minkowski Space
We prove that if ϕ: R2→ R1 + 2 is a smooth, proper, timelike immersion with vanishing mean curvature, then necessarily ϕ is an embedding, and every compact subset of ϕ(R2) is a smooth graph. It follows that if one evolves any smooth, self-intersecting spaceli...
| Main Author: | |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Springer
2020
|
| _version_ | 1826257123067035648 |
|---|---|
| author | Paxton, EA |
| author_facet | Paxton, EA |
| author_sort | Paxton, EA |
| collection | OXFORD |
| description | We prove that if ϕ: R2→ R1 + 2 is a smooth, proper, timelike immersion with vanishing mean curvature, then necessarily ϕ is an embedding, and every compact subset of ϕ(R2) is a smooth graph. It follows that if one evolves any smooth, self-intersecting spacelike curve (or any planar spacelike curve whose unit tangent vector spans a closed semi-circle) so as to trace a timelike surface of vanishing mean curvature in R1 + 2, then the evolving surface will either fail to remain timelike, or it will fail to remain smooth. We show that, even allowing for null points, such a Cauchy evolution will be C2 inextendible beyond some singular time. In addition we study the continuity of the unit tangent for the evolution of a self-intersecting curve in isothermal gauge, which defines a well-known evolution beyond singular time. |
| first_indexed | 2024-03-06T18:13:08Z |
| format | Journal article |
| id | oxford-uuid:03b400e7-e32f-4c93-9746-d87c0fc912b3 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T18:13:08Z |
| publishDate | 2020 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:03b400e7-e32f-4c93-9746-d87c0fc912b32022-03-26T08:47:50ZEmbeddedness of timelike maximal surfaces in (1+2)-Minkowski SpaceJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:03b400e7-e32f-4c93-9746-d87c0fc912b3EnglishSymplectic ElementsSpringer2020Paxton, EAWe prove that if ϕ: R2→ R1 + 2 is a smooth, proper, timelike immersion with vanishing mean curvature, then necessarily ϕ is an embedding, and every compact subset of ϕ(R2) is a smooth graph. It follows that if one evolves any smooth, self-intersecting spacelike curve (or any planar spacelike curve whose unit tangent vector spans a closed semi-circle) so as to trace a timelike surface of vanishing mean curvature in R1 + 2, then the evolving surface will either fail to remain timelike, or it will fail to remain smooth. We show that, even allowing for null points, such a Cauchy evolution will be C2 inextendible beyond some singular time. In addition we study the continuity of the unit tangent for the evolution of a self-intersecting curve in isothermal gauge, which defines a well-known evolution beyond singular time. |
| spellingShingle | Paxton, EA Embeddedness of timelike maximal surfaces in (1+2)-Minkowski Space |
| title | Embeddedness of timelike maximal surfaces in (1+2)-Minkowski Space |
| title_full | Embeddedness of timelike maximal surfaces in (1+2)-Minkowski Space |
| title_fullStr | Embeddedness of timelike maximal surfaces in (1+2)-Minkowski Space |
| title_full_unstemmed | Embeddedness of timelike maximal surfaces in (1+2)-Minkowski Space |
| title_short | Embeddedness of timelike maximal surfaces in (1+2)-Minkowski Space |
| title_sort | embeddedness of timelike maximal surfaces in 1 2 minkowski space |
| work_keys_str_mv | AT paxtonea embeddednessoftimelikemaximalsurfacesin12minkowskispace |