Some properties of minimal surfaces in singular spaces

Mese, Chikako

doi:10.1090/S0002-9947-00-02481-8

Some properties of minimal surfaces in singular spaces
HTML articles powered by AMS MathViewer

by Chikako Mese

Trans. Amer. Math. Soc. 352 (2000), 3957-3969

DOI: https://doi.org/10.1090/S0002-9947-00-02481-8

Published electronically: May 22, 2000

PDF | Request permission

Abstract:

This paper involves the generalization of minimal surface theory to spaces with singularities. Let $X$ be an NPC space, i.e. a metric space of non-positive curvature. We define a (parametric) minimal surface in $X$ as a conformal energy minimizing map. Using this definition, many properties of classical minimal surfaces can also be observed for minimal surfaces in this general setting. In particular, we will prove the boundary monotonicity property and the isoperimetric inequality for minimal surfaces in $X$.

References

A. D. Aleksandrov, V. N. Berestovskiĭ, and I. G. Nikolaev, Generalized Riemannian spaces, Uspekhi Mat. Nauk 41 (1986), no. 3(249), 3–44, 240 (Russian). MR 854238
Kevin Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. (2) 135 (1992), no. 1, 165–182. MR 1147961, DOI 10.2307/2946567
Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 165–246. MR 1215595
Robert Hardt and Fang-Hua Lin, Harmonic maps into round cones and singularities of nematic liquid crystals, Math. Z. 213 (1993), no. 4, 575–593. MR 1231879, DOI 10.1007/BF03025739
W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, London Mathematical Society Monographs, No. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0460672
Jürgen Jost, Equilibrium maps between metric spaces, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 173–204. MR 1385525, DOI 10.1007/BF01191341
Jürgen Jost, Convex functionals and generalized harmonic maps into spaces of nonpositive curvature, Comment. Math. Helv. 70 (1995), no. 4, 659–673. MR 1360608, DOI 10.1007/BF02566027
Nicholas J. Korevaar and Richard M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), no. 3-4, 561–659. MR 1266480, DOI 10.4310/CAG.1993.v1.n4.a4
Nicholas J. Korevaar and Richard M. Schoen, Global existence theorems for harmonic maps to non-locally compact spaces, Comm. Anal. Geom. 5 (1997), no. 2, 333–387. MR 1483983, DOI 10.4310/CAG.1997.v5.n2.a4
C. Mese. The Curvature of Minimal Surfaces in Singular Spaces. to appear in Communications of Analysis and Geometry.
C. Mese. Minimal Surfaces and Conformal Mappings into Singular Spaces. Ph.D. thesis. Stanford University, 1996.
J. H. Michael and L. M. Simon, Sobolev and mean-value inequalities on generalized submanifolds of $R^{n}$, Comm. Pure Appl. Math. 26 (1973), 361–379. MR 344978, DOI 10.1002/cpa.3160260305
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
I. G. Nikolaev, Solution of the Plateau problem in spaces of curvature at most $K$, Sibirsk. Mat. Zh. 20 (1979), no. 2, 345–353, 459 (Russian). MR 530499
Y.G. Reshetnyak. Geometry IV Non-regular Riemann Geometry. Translated by E. Primrose. Encyclopedia of Mathematical Science, 70. Springer-Berlag, Berlin, 1993.
Ju. G. Rešetnjak, Non-expansive maps in a space of curvature no greater than $K$, Sibirsk. Mat. Ž. 9 (1968), 918–927 (Russian). MR 0244922
Tomasz Serbinowski, Boundary regularity of harmonic maps to nonpositively curved metric spaces, Comm. Anal. Geom. 2 (1994), no. 1, 139–153. MR 1312682, DOI 10.4310/CAG.1994.v2.n1.a8
T. Serbinowski. Harmonic Maps into Metric Spaces with Curvature Bounded Above. Ph.D. thesis. University of Utah, 1995.
Michael Wolf, Harmonic maps from surfaces to $\mathbf R$-trees, Math. Z. 218 (1995), no. 4, 577–593. MR 1326987, DOI 10.1007/BF02571924
Michael Wolf, On realizing measured foliations via quadratic differentials of harmonic maps to $\mathbf R$-trees, J. Anal. Math. 68 (1996), 107–120. MR 1403253, DOI 10.1007/BF02790206