The parameterized Steiner problem and the singular Plateau problem via energy
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- by Chikako Mese and Sumio Yamada PDF
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Abstract:
The Steiner problem is the problem of finding the shortest network connecting a given set of points. By the singular Plateau Problem, we will mean the problem of finding an area-minimizing surface (or a set of surfaces adjoined so that it is homeomorphic to a 2-complex) spanning a graph. In this paper, we study the parametric versions of the Steiner problem and the singular Plateau problem by a variational method using a modified energy functional for maps. The main results are that the solutions of our one- and two-dimensional variational problems yield length and area minimizing maps respectively, i.e. we provide new methods to solve the Steiner and singular Plateau problems by the use of energy functionals. Furthermore, we show that these solutions satisfy a natural balancing condition along its singular sets. The key issue involved in the two-dimensional problem is the understanding of the moduli space of conformal structures on a 2-complex.References
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Additional Information
- Chikako Mese
- Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
- MR Author ID: 641800
- Email: cmese@math.jhu.edu
- Sumio Yamada
- Affiliation: Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan
- MR Author ID: 641808
- Email: yamada@math.tohoku.ac.jp
- Received by editor(s): February 10, 2004
- Published electronically: March 1, 2006
- Additional Notes: The research for the first author was partially supported by grant NSF DMS-0450083 and for the second author by DMS-0071862.
- © Copyright 2006 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 2875-2895
- MSC (2000): Primary 58E12, 53A10
- DOI: https://doi.org/10.1090/S0002-9947-06-04089-X
- MathSciNet review: 2216250