Hilbert volume in metric spaces. Part 1

• Autorzy: Misha Gromov
• Języki publikacji: EN

Abstrakty

EN

We introduce a notion of Hilbertian n-volume in metric spaces with Besicovitch-type inequalities built-in into the definitions. The present Part 1 of the article is, for the most part, dedicated to the reformulation of known results in our terms with proofs being reduced to (almost) pure tautologies. If there is any novelty in the paper, this is in forging certain terminology which, ultimately, may turn useful in an Alexandrov kind of approach to singular spaces with positive scalar curvature [Gromov M., Plateau-hedra, Scalar Curvature and Dirac Billiards, in preparation].

Słowa kluczowe

EN

Riemannian volume; Lipschitz maps; Besicovitch inequality; John’s ellipsoid

Kategorie tematyczne

• 53C22: Geodesics
• 53C23: Global geometric and topological methods (\`a la Gromov); differential geometric analysis on metric spaces
• 53C24: Rigidity results

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 2
• Strony: 371-400

Daty

wydano

2012-04-01

online

2012-01-18

Twórcy

autor

Misha Gromov

Bibliografia

• [1] Afsari B., Riemannian L p center of mass: existence, uniqueness, and convexity, Proc. Amer. Math. Soc., 2011, 139(2), 655–673 http://dx.doi.org/10.1090/S0002-9939-2010-10541-5
• [2] Almgren F.J., Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math., 1968, 87(2), 321–391 http://dx.doi.org/10.2307/1970587
• [3] Ambrosio L., Kirchheim B., Currents in metric spaces, Acta Math., 2000, 185(1), 1–80 http://dx.doi.org/10.1007/BF02392711
• [4] Babenko I.K., Asymptotic volume of tori and the geometry of convex bodies, Mat. Zametki, 1988, 44(2), 177–190 (in Russian)
• [5] Breuillard E., Geometry of locally compact groups of polynomial growth and shape of large balls, preprint available at http://arxiv.org/abs/0704.0095
• [6] Burago D.Yu., Periodic metrics, In: Representation Theory and Dynamical Systems, Adv. Soviet Math., 9, American Mathematical Society, Providence, 1992, 205–210
• [7] Burago D., Ivanov S., Riemannian tori without conjugate points are flat, Geom. Funct. Anal., 1994, 4(3), 259–269 http://dx.doi.org/10.1007/BF01896241
• [8] Burago D., Ivanov S., On asymptotic volume of tori, Geom. Funct. Anal., 1995, 5(5), 800–808 http://dx.doi.org/10.1007/BF01897051
• [9] Federer H., Geometric Measure Theory, Grundlehren Math. Wiss., 153, Springer, New York, 1969
• [10] Gromov M., Filling Riemannian manifolds, J. Differential Geom., 1983, 18(1), 1–147
• [11] Gromov M., Partial Differential Relations, Ergeb. Math. Grenzgeb., 9, Springer, Berlin, 1986
• [12] Gromov M., Metric Structures for Riemannian and Non-Riemannian Spaces, Progr. Math., Birkhäuser, Boston, 1999
• [13] Gromov M., Topological invariants of dynamical systems and spaces of holomorphic maps. I, Math. Phys. Anal. Geom., 1999, 2(4), 323–415 http://dx.doi.org/10.1023/A:1009841100168
• [14] Gromov M., Spaces and questions, Geom. Funct. Anal., 2000, Special Volume (I), 118–161
• [15] Gromov M., Manifolds: Where do we come from? What are we? Where are we going, preprint available at http://www.ihes.fr/_gromov/PDF/manifolds-Poincare.pdf
• [16] Gromov M., Super stable Kählerian horseshoe?, preprint available at http://www.ihes.fr/_gromov/PDF/horse-shoejan6-2011.pdf
• [17] Gromov M., Plateau-hedra, scalar curvature and Dirac billiards, in preparation
• [18] Grove K., Karcher H., How to conjugate C 1-close group actions, Math. Z., 1973, 132(1), 11–20 http://dx.doi.org/10.1007/BF01214029
• [19] Karcher H., Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math., 1977, 30(5), 509–541 http://dx.doi.org/10.1002/cpa.3160300502
• [20] Krat S.A., On pairs of metrics invariant under a cocompact action of a group, Electron. Res. Announc. Amer. Math. Soc., 2001, 7, 79–86 http://dx.doi.org/10.1090/S1079-6762-01-00097-X
• [21] Lang U., Schroeder V., Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal., 1997, 7(3), 535–560 http://dx.doi.org/10.1007/s000390050018
• [22] Pansu P., Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergodic Theory Dynam. Systems, 1983, 3(3), 415–445 http://dx.doi.org/10.1017/S0143385700002054
• [23] Sormani C., Wenger S., Weak convergence of currents and cancellation, Calc. Var. Partial Differential Equations, 2010, 38(1–2), 183–206 http://dx.doi.org/10.1007/s00526-009-0282-x
• [24] Wenger S., Compactness for manifolds and integral currents with bounded diameter and volume, Calc. Var. Partial Differential Equations, 2011, 40(3–4), 423–448 http://dx.doi.org/10.1007/s00526-010-0346-y

Identyfikatory

DOI

10.2478/s11533-011-0143-7