The parameterized Steiner problem and the singular Plateau problem via energy

Authors:
Chikako Mese and Sumio Yamada

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

Published electronically:
March 1, 2006

MathSciNet review:
2216250

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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.

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Additional Information

**Chikako Mese**

Affiliation:
Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218

Email:
cmese@math.jhu.edu

**Sumio Yamada**

Affiliation:
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan

Email:
yamada@math.tohoku.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-06-04089-X

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.

Article copyright:
© Copyright 2006
American Mathematical Society