Properties of the Intrinsic Flat Distance

Portegies, J.; Sormani, C.

doi:10.1090/spmj/1504

Properties of the Intrinsic Flat Distance
HTML articles powered by AMS MathViewer

by J. Portegies and C. Sormani

St. Petersburg Math. J. 29 (2018), 475-528

DOI: https://doi.org/10.1090/spmj/1504

Published electronically: March 30, 2018

PDF | Request permission

Abstract:

In this paper written in honor of Yuri Burago, we explore a variety of properties of intrinsic flat convergence. We introduce the sliced filling volume and interval sliced filling volume and explore the relationship between these notions, the tetrahedral property and the disappearance of points under intrinsic flat convergence. We prove two new Gromov–Hausdorff and intrinsic flat compactness theorems including the Tetrahedral Compactness Theorem. Much of the work in this paper builds upon Ambrosio–Kirchheim’s Slicing Theorem combined with an adapted version of Gromov’s Filling Volume. We are grateful to have been invited to submit a paper in honor of Yuri Burago, in thanks not only for his beautiful book written jointly with Dimitri Burago and Sergei Ivanov but also for his many thoughtful communications with us and other young mathematicians over the years.

References

Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), no. 1, 1–80. MR 1794185, DOI 10.1007/BF02392711
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
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
Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom. 46 (1997), no. 3, 406–480. MR 1484888
Jeff Cheeger and Tobias H. Colding, On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom. 54 (2000), no. 1, 37–74. MR 1815411
Tobias H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 (1997), no. 3, 477–501. MR 1454700, DOI 10.2307/2951841
Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
Robert E. Greene and Peter Petersen V, Little topology, big volume, Duke Math. J. 67 (1992), no. 2, 273–290. MR 1177307, DOI 10.1215/S0012-7094-92-06710-X
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
Mikhael Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR 682063
Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
Misha Gromov, Plateau-Stein manifolds, Cent. Eur. J. Math. 12 (2014), no. 7, 923–951. MR 3188456, DOI 10.2478/s11533-013-0387-5
Lan-Hsuan Huang, Dan A. Lee, and Christina Sormani, Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space, J. Reine Angew. Math. 727 (2017), 269–299. MR 3652253, DOI 10.1515/crelle-2015-0051
J. Jauregui, On the lower semicontinuity of the adm mass, arXiv:1411.3699, 2016.
Sajjad Lakzian, Intrinsic flat continuity of Ricci flow through neckpinch singularities, Geom. Dedicata 179 (2015), 69–89. MR 3424658, DOI 10.1007/s10711-015-0068-6
Sajjad Lakzian, On diameter controls and smooth convergence away from singularities, Differential Geom. Appl. 47 (2016), 99–129. MR 3504922, DOI 10.1016/j.difgeo.2016.01.003
Sajjad Lakzian and Christina Sormani, Smooth convergence away from singular sets, Comm. Anal. Geom. 21 (2013), no. 1, 39–104. MR 3046939, DOI 10.4310/CAG.2013.v21.n1.a2
Dan A. Lee and Christina Sormani, Near-equality of the Penrose inequality for rotationally symmetric Riemannian manifolds, Ann. Henri Poincaré 13 (2012), no. 7, 1537–1556. MR 2982632, DOI 10.1007/s00023-012-0172-1
Dan A. Lee and Christina Sormani, Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds, J. Reine Angew. Math. 686 (2014), 187–220. MR 3176604, DOI 10.1515/crelle-2012-0094
Philippe G. LeFloch and Christina Sormani, The nonlinear stability of rotationally symmetric spaces with low regularity, J. Funct. Anal. 268 (2015), no. 7, 2005–2065. MR 3315585, DOI 10.1016/j.jfa.2014.12.012
N. Li and R. Perales, On the Sormani–Wenger intrinsic flat convergence of Alexandrov spaces, arXiv:1411.6854, 2015.
Fanghua Lin and Xiaoping Yang, Geometric measure theory—an introduction, Advanced Mathematics (Beijing/Boston), vol. 1, Science Press Beijing, Beijing; International Press, Boston, MA, 2002. MR 2030862
Rostislav Matveev and Jacobus W. Portegies, Intrinsic flat and Gromov-Hausdorff convergence of manifolds with Ricci curvature bounded below, J. Geom. Anal. 27 (2017), no. 3, 1855–1873. MR 3667412, DOI 10.1007/s12220-016-9742-7
Frank Morgan, Geometric measure theory, 4th ed., Elsevier/Academic Press, Amsterdam, 2009. A beginner’s guide. MR 2455580
M. Munn, Intrinsic flat convergence with bounded Ricci curvature, arXiv:1405.3313, 2014.
Raquel del Carmen Perales Perales Aguilar, Convergence of Manifolds and Metric Spaces with Boundary, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–State University of New York at Stony Brook. MR 3407469
Raquel Perales, Volumes and limits of manifolds with Ricci curvature and mean curvature bounds, Differential Geom. Appl. 48 (2016), 23–37. MR 3534433, DOI 10.1016/j.difgeo.2016.05.004
Jacobus W. Portegies, Semicontinuity of eigenvalues under intrinsic flat convergence, Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1725–1766. MR 3396431, DOI 10.1007/s00526-015-0842-1
H. L. Royden, Real analysis, 3rd ed., Macmillan Publishing Company, New York, 1988. MR 1013117
Christina Sormani, The tetrahedral property and a new Gromov-Hausdorff compactness theorem, C. R. Math. Acad. Sci. Paris 351 (2013), no. 3-4, 119–122 (English, with English and French summaries). MR 3037999, DOI 10.1016/j.crma.2013.02.011
—, Intrinsic flat Arzela–Ascoli theorems, Comm. Anal. Geom. (in preparation); arXiv: 1402.6066, 2014.
Christina Sormani, Scalar curvature and intrinsic flat convergence, Measure theory in non-smooth spaces, Partial Differ. Equ. Meas. Theory, De Gruyter Open, Warsaw, 2017, pp. 288–338. MR 3701743
Christina Sormani and Stefan Wenger, Weak convergence of currents and cancellation, Calc. Var. Partial Differential Equations 38 (2010), no. 1-2, 183–206. With an appendix by Raanan Schul and Wenger. MR 2610529, DOI 10.1007/s00526-009-0282-x
Christina Sormani and Stefan Wenger, The intrinsic flat distance between Riemannian manifolds and other integral current spaces, J. Differential Geom. 87 (2011), no. 1, 117–199. MR 2786592
Stefan Wenger, Flat convergence for integral currents in metric spaces, Calc. Var. Partial Differential Equations 28 (2007), no. 2, 139–160. MR 2284563, DOI 10.1007/s00526-006-0034-0
Stefan Wenger, Compactness for manifolds and integral currents with bounded diameter and volume, Calc. Var. Partial Differential Equations 40 (2011), no. 3-4, 423–448. MR 2764913, DOI 10.1007/s00526-010-0346-y
Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148