Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [Ch] J. Chen. On energy minimizing mappings between and into singular spaces. Duke Math. J. 79 (1995) 77-99. MR 1340295 (96f:58041)
  • [D] J. Douglas. Solution of the problem of Plateau. Trans. AMS 33 (1931) 263-321. MR 1501590
  • [DM] G. Daskalopoulos and C. Mese. Harmonic maps from 2-complexes and geometric actions on trees. preprint.
  • [EF] J. Eells and B. Fuglede. Harmonic maps between Riemannian polyhedra. Cambridge Tracts in Mathematics 142, Cambridge University Press, Cambridge, 2001. MR 1848068 (2002h:58017)
  • [F] B. Fuglede. Hölder continuity of harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature. Calc. Var. 16 (2003) 375-403. MR 1971035 (2004g:58016)
  • [GS] M. Gromov and R. Schoen. Harmonic maps into singular spaces and $ p$-adic superrigidity for lattices in groups of rank one. IHES Publ. Math. 76 (1992) 165-246. MR 1215595 (94e:58032)
  • [IT] A. Ivanov and A. Tuzhilin. Geometry of minimal networks and the one-dimensional Plateau problem. Russian Math. Surveys 47:2 (1992) 59-131. MR 1185285 (93i:05050)
  • [H] P. Hartman. On homotopic harmonic maps. Canadian J. Math. 19 (1967) 673-683. MR 0214004 (35:4856)
  • [J] J. Jost. Nonpositive Curvature: Geometric and Analytic Aspects. Lectures in Mathematics, ETH Zürich. Birkhaüser, Basel, 1997. MR 1451625 (98g:53070)
  • [KNS] D. Kinderlerer, L. Nirenberg and J. Spruck. Regularity in elliptic free boundary problems. I. J. Anal. Math. 34 (1978) 86-119. MR 0531272 (83d:35060)
  • [KS1] N. Korevaar and R. Schoen. Sobolev spaces and harmonic maps for metric space targets. Communications in Analysis and Geometry 1 (1993) 561-659. MR 1266480 (95b:58043)
  • [KS2] N. Korevaar and R. Schoen. Global existence theorem for harmonic maps to non-locally compact spaces. Communications in Analysis and Geometry 5 (1997) 333-387. MR 1483983 (99b:58061)
  • [KS3] N. Korevaar and R. Schoen. Global existence theorems for harmonic maps: finite rank spaces and an approach to rigidity for smooth actions. Preprint.
  • [LM] G. Lawlor and F. Morgan. Curvy slicing proves that triple junctions locally minimize area. J. Diff. Geom. 44 (1996) 514-528. MR 1431003 (98a:53012)
  • [La] B. Lawson. Lectures on minimal submanifolds, Vol. 1. Publish or Perish, 1980. MR 0576752 (82d:53035b)
  • [Le] O. Lehto. Univalent Function and Teichmüller Spaces. Springer-Verlag, New York, 1987. MR 0867407 (88f:30073)
  • [M] C. Mese. Regularity of harmonic maps from a flat complex. Progress in Non-Linear Differential Equations 133-148. Birkhauser Publications, Basel, 2004. MR 2076271 (2005e:58022)
  • [SY] R. Schoen and S.-T. Yau. Lectures on Harmonic Maps. International Press, Boston, 1997. MR 1474501 (98i:58072)
  • [T] J. Taylor. The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces. Ann. of Math. 103 (1976) 489-539. MR 0428181 (55:1208a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 58E12, 53A10

Retrieve articles in all journals with MSC (2000): 58E12, 53A10


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

American Mathematical Society