Volumes and areas of Lipschitz metrics

Ivanov, S.

doi:10.1090/S1061-0022-09-01053-X

Volumes and areas of Lipschitz metrics
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by S. V. Ivanov
Translated by: the author

St. Petersburg Math. J. 20 (2009), 381-405

DOI: https://doi.org/10.1090/S1061-0022-09-01053-X

Published electronically: April 7, 2009

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Abstract:

Methods of estimating (Riemannian and Finsler) filling volumes by using nonexpanding maps to Banach spaces of $L^\infty$-type are developed and generalized. For every Finsler volume functional (such as the Busemann volume or the Holmes–Thompson volume), a natural extension is constructed from the class of Finsler metrics to all Lipschitz metrics, and the notion of area is defined for Lipschitz surfaces in a Banach space. A correspondence is established between minimal fillings and minimal surfaces in $L^\infty$-type spaces. A Finsler volume functional for which the Riemannian and the Finsler filling volumes are equal is introduced; it is proved that this functional is semielliptic.

References

F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. (2) 87 (1968), 321–391. MR 225243, DOI 10.2307/1970587
J. C. Álvarez-Paiva and G. Berck, What is wrong with the Hausdorff measure in Finsler spaces, Preprint, 2004, arXiv:math.DG/0408413.
J. C. Álvarez Paiva and A. C. Thompson, Volumes on normed and Finsler spaces, A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, Cambridge, 2004, pp. 1–48. MR 2132656, DOI 10.4171/prims/123
Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
A. S. Besicovitch, On two problems of Loewner, J. London Math. Soc. 27 (1952), 141–144. MR 47126, DOI 10.1112/jlms/s1-27.2.141
D. Burago and S. Ivanov, Isometric embeddings of Finsler manifolds, Algebra i Analiz 5 (1993), no. 1, 179–192 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 1, 159–169. MR 1220494
D. Burago and S. Ivanov, On asymptotic volume of tori, Geom. Funct. Anal. 5 (1995), no. 5, 800–808. MR 1354290, DOI 10.1007/BF01897051
D. Burago and S. Ivanov, On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume, Ann. of Math. (2) 156 (2002), no. 3, 891–914. MR 1954238, DOI 10.2307/3597285
D. Burago and S. Ivanov, Gaussian images of surfaces and ellipticity of surface area functionals, Geom. Funct. Anal. 14 (2004), no. 3, 469–490. MR 2100668, DOI 10.1007/s00039-004-0465-8
—, Boundary rigidity and filling volume minimality of metrics close to a flat one (to appear).
Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
Herbert Busemann, Intrinsic area, Ann. of Math. (2) 48 (1947), 234–267. MR 20626, DOI 10.2307/1969168
Herbert Busemann, A theorem on convex bodies of the Brunn-Minkowski type, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 27–31. MR 28046, DOI 10.1073/pnas.35.1.27
H. Busemann, G. Ewald, and G. C. Shephard, Convex bodies and convexity on Grassmann cones. I–IV, Math. Ann. 151 (1963), 1–41. MR 157286, DOI 10.1007/BF01343323
Giuseppe De Cecco and Giuliana Palmieri, LIP manifolds: from metric to Finslerian structure, Math. Z. 218 (1995), no. 2, 223–237. MR 1318157, DOI 10.1007/BF02571901
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
R. D. Holmes and A. C. Thompson, $n$-dimensional area and content in Minkowski spaces, Pacific J. Math. 85 (1979), no. 1, 77–110. MR 571628
S. V. Ivanov, On two-dimensional minimal fillings, Algebra i Analiz 13 (2001), no. 1, 26–38 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 1, 17–25. MR 1819361
Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, 1948, pp. 187–204. MR 0030135
René Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65 (1981/82), no. 1, 71–83 (French). MR 636880, DOI 10.1007/BF01389295
P. M. Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55–71. MR 48886
A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315, DOI 10.1017/CBO9781107325845
Rolf Schneider, On the Busemann area in Minkowski spaces, Beiträge Algebra Geom. 42 (2001), no. 1, 263–273. MR 1824764