Convex Programs for Minimal-Area Problems
Abstract The minimal-area problem that defines string diagrams in closed string field theory asks for the metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least $$2\pi $$ 2 π . This is an extremal length problem in conf...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Springer Berlin Heidelberg
2021
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| Online Access: | https://hdl.handle.net/1721.1/131449 |
| _version_ | 1811075304003731456 |
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| author | Headrick, Matthew Zwiebach, Barton |
| author_facet | Headrick, Matthew Zwiebach, Barton |
| author_sort | Headrick, Matthew |
| collection | MIT |
| description | Abstract
The minimal-area problem that defines string diagrams in closed string field theory asks for the metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least
$$2\pi $$
2
π
. This is an extremal length problem in conformal geometry as well as a problem in systolic geometry. We consider the analogous minimal-area problem for homology classes of curves and, with the aid of calibrations and the max flow-min cut theorem, formulate it as a local convex program. We derive an equivalent dual program involving maximization of a concave functional. These two programs give new insights into the form of the minimal-area metric and are amenable to numerical solution. We explain how the homology problem can be modified to provide the solution to the original homotopy problem. |
| first_indexed | 2024-09-23T10:03:55Z |
| format | Article |
| id | mit-1721.1/131449 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T10:03:55Z |
| publishDate | 2021 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1314492021-09-21T03:52:53Z Convex Programs for Minimal-Area Problems Headrick, Matthew Zwiebach, Barton Abstract The minimal-area problem that defines string diagrams in closed string field theory asks for the metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least $$2\pi $$ 2 π . This is an extremal length problem in conformal geometry as well as a problem in systolic geometry. We consider the analogous minimal-area problem for homology classes of curves and, with the aid of calibrations and the max flow-min cut theorem, formulate it as a local convex program. We derive an equivalent dual program involving maximization of a concave functional. These two programs give new insights into the form of the minimal-area metric and are amenable to numerical solution. We explain how the homology problem can be modified to provide the solution to the original homotopy problem. 2021-09-20T17:17:07Z 2021-09-20T17:17:07Z 2020-03-31 2020-09-24T20:52:52Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131449 en https://doi.org/10.1007/s00220-020-03732-1 Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ Springer-Verlag GmbH Germany, part of Springer Nature application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Headrick, Matthew Zwiebach, Barton Convex Programs for Minimal-Area Problems |
| title | Convex Programs for Minimal-Area Problems |
| title_full | Convex Programs for Minimal-Area Problems |
| title_fullStr | Convex Programs for Minimal-Area Problems |
| title_full_unstemmed | Convex Programs for Minimal-Area Problems |
| title_short | Convex Programs for Minimal-Area Problems |
| title_sort | convex programs for minimal area problems |
| url | https://hdl.handle.net/1721.1/131449 |
| work_keys_str_mv | AT headrickmatthew convexprogramsforminimalareaproblems AT zwiebachbarton convexprogramsforminimalareaproblems |