St. Petersburg Mathematical Journal

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1547-7371(e) ISSN 1061-0022(p)

Volumes and areas of Lipschitz metrics

Author(s): S. V. Ivanov
Translated by: the author
Original publication: Algebra i Analiz, tom 20 (2008), nomer 3.
Journal: St. Petersburg Math. J. 20 (2009), 381-405.
MSC (2000): Primary 53B40
Posted: April 7, 2009
MathSciNet review: 2454453
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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:

1.: 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 0225243 (37:837)
2.: J. C. Álvarez-Paiva and G. Berck, What is wrong with the Hausdorff measure in Finsler spaces, Preprint, 2004, arXiv:math.DG/0408413.
3.: J. C. Álvarez-Paiva and A. C. Thompson, Volumes on normed and Finsler spaces, A Sampler of Riemann-Finsler Geometry (D. Bao, R. Bryant, S.-S. Chern, and Z. Shen, eds.), Math. Sci. Res. Inst. Publ., No. 50, Cambridge Univ. Press, Cambridge, 2004, pp. 1-48. MR 2132656 (2006c:53079)
4.: Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, Amer. Math. Soc. Colloq. Publ., vol. 48, Amer. Math. Soc., Providence, RI, 2000. MR 1727673 (2001b:46001)
5.: A. Besicovitch, On two problems of Loewner, J. London Math. Soc. 27 (1952), 141-144. MR 0047126 (13:831d)
6.: D. Burago and S. Ivanov, Isometric embeddings of Finsler manifolds, Algebra i Analiz 5 (1993), no. 1, 179-192; English transl., St. Petersburg Math. J. 5 (1994), no. 1, 159-169. MR 1220494 (94e:53073)
7.: -, On asymptotic volume of tori, Geom. Funct. Anal. 5 (1995), 800-808. MR 1354290 (96h:53041)
8.: -, On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume, Ann. of Math. (2) 156 (2002), 891-914. MR 1954238 (2003k:53088)
9.: -, Gaussian images of surfaces and ellipticity of surface area functionals, Geom. Funct. Anal. 14 (2004), 469-490. MR 2100668 (2006b:53099)
10.: -, Boundary rigidity and filling volume minimality of metrics close to a flat one (to appear).
11.: D. Yu. Burago, Yu. D. Burago, and S. V. Ivanov, A course in metric geometry, Inst. Komp'yut. Issled., Moscow-Izhevsk, 2004; English transl., Grad. Stud. in Math., vol. 33, Amer. Math. Soc., Providence, RI, 2001. MR 1835418 (2002e:53053)
12.: H. Busemann, Intrinsic area, Ann. of Math. (2) 48 (1947), 234-267. MR 0020626 (8:573a)
13.: -, A theorem on convex bodies of the Brunn-Minkowski type, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 27-31. MR 0028046 (10:395c)
14.: H. Busemann, G. Ewald, and G. C. Shephard, Convex bodies and convexity on Grassmann cones. I-IV, Math. Ann. 151 (1963), 1-41. MR 0157286 (28:522a)
15.: G. De Cecco and G. Palmieri, LIP manifolds: From metric to Finslerian structure, Math. Z. 218 (1995), 223-237. MR 1318157 (96a:53090)
16.: H. Federer, Geometric measure theory, Grundlehren Math. Wiss., Bd. 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325 (41:1976)
17.: M. Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), 1-147. MR 0697984 (85h:53029)
18.: R. D. Holmes and A. C. Thompson, n-dimensional area and content in Minkowski spaces, Pacific J. Math. 85 (1979), 77-110. MR 0571628 (81k:52023)
19.: S. Ivanov, On two-dimensional minimal fillings, Algebra i Analiz 13 (2001), no. 1, 26-38; English transl., St. Petersburg Math. J. 13 (2002), no. 1, 17-25. MR 1819361 (2002b:58016)
20.: F. John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Intersci. Publ., Inc., New York, NY, 1948, pp. 187-204. MR 0030135 (10:719b)
21.: R. Michel, Sur la rigidité imposée par la longuer des géodésiques, Invent. Math. 65 (1981/82), 71-83. MR 0636880 (83d:58021)
22.: P. Pu, Some inequalities in certain nonorientable Riemannian manifolds, Pacific J. Math. 2 (1952), 55-71. MR 0048886 (14:87e)
23.: A. C. Thompson, Minkowski geometry, Encyclopedia Math. Appl., vol. 63, Cambridge Univ. Press, Cambridge, 1996. MR 1406315 (97f:52001)
24.: R. Schneider, On the Busemann area in Minkowski spaces, Beiträge Algebra Geom. 42 (2001), 263-273. MR 1824764 (2002k:52006)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|