Beyond traditional Curvature-Dimension I: New model spaces for isoperimetric and concentration inequalities in negative dimension
HTML articles powered by AMS MathViewer
- by Emanuel Milman PDF
- Trans. Amer. Math. Soc. 369 (2017), 3605-3637
Abstract:
We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative and, more generally, is in the range $N \in (-\infty ,1)$, extending the scope from the traditional range $N \in [n,\infty ]$. In particular, we identify the correct one-dimensional model-spaces under an additional diameter upper bound and discover a new case yielding a single model space (besides the previously known $N$-sphere and Gaussian measure when $N \in [n,\infty ]$): a (positively curved) sphere of (possibly negative) dimension $N \in (-\infty ,1)$. When curvature is non-negative, we show that arbitrarily weak concentration implies an $N$-dimensional Cheeger isoperimetric inequality and derive various weak Sobolev and Nash-type inequalities on such spaces. When curvature is strictly positive, we observe that such spaces satisfy a Poincaré inequality uniformly for all $N \in (-\infty ,1-\varepsilon ]$ and enjoy a two-level concentration of the type $\exp (-\min (t,t^2))$. Our main technical tool is a generalized version of the Heintze–Karcher theorem, which we extend to the range $N \in (-\infty ,1)$.References
- Dominique Bakry, L’hypercontractivité et son utilisation en théorie des semigroupes, Lectures on probability theory (Saint-Flour, 1992) Lecture Notes in Math., vol. 1581, Springer, Berlin, 1994, pp. 1–114 (French). MR 1307413, DOI 10.1007/BFb0073872
- D. Bakry and Michel Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206 (French). MR 889476, DOI 10.1007/BFb0075847
- D. Bakry and M. Ledoux, Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (1996), no. 2, 259–281. MR 1374200, DOI 10.1007/s002220050026
- Dominique Bakry and Zhongmin Qian, Volume comparison theorems without Jacobi fields, Current trends in potential theory, Theta Ser. Adv. Math., vol. 4, Theta, Bucharest, 2005, pp. 115–122. MR 2243959
- Franck Barthe and Emanuel Milman, Transference principles for log-Sobolev and spectral-gap with applications to conservative spin systems, Comm. Math. Phys. 323 (2013), no. 2, 575–625. MR 3096532, DOI 10.1007/s00220-013-1782-2
- Christophe Bavard and Pierre Pansu, Sur le volume minimal de $\textbf {R}^2$, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 479–490 (French). MR 875084
- V. Bayle, Propriétés de concavité du profil isopérimétrique et applications, PhD thesis, Institut Joseph Fourier, Grenoble, 2004.
- S. Bobkov, Extremal properties of half-spaces for log-concave distributions, Ann. Probab. 24 (1996), no. 1, 35–48. MR 1387625, DOI 10.1214/aop/1042644706
- Sergey G. Bobkov, Large deviations and isoperimetry over convex probability measures with heavy tails, Electron. J. Probab. 12 (2007), 1072–1100. MR 2336600, DOI 10.1214/EJP.v12-440
- Serguei G. Bobkov and Christian Houdré, Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc. 129 (1997), no. 616, viii+111. MR 1396954, DOI 10.1090/memo/0616
- Sergey G. Bobkov and Michel Ledoux, Weighted Poincaré-type inequalities for Cauchy and other convex measures, Ann. Probab. 37 (2009), no. 2, 403–427. MR 2510011, DOI 10.1214/08-AOP407
- C. Borell, Convex set functions in $d$-space, Period. Math. Hungar. 6 (1975), no. 2, 111–136. MR 404559, DOI 10.1007/BF02018814
- Stéphane Boucheron, Gábor Lugosi, and Pascal Massart, Concentration inequalities, Oxford University Press, Oxford, 2013. A nonasymptotic theory of independence; With a foreword by Michel Ledoux. MR 3185193, DOI 10.1093/acprof:oso/9780199535255.001.0001
- Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366–389. MR 0450480, DOI 10.1016/0022-1236(76)90004-5
- Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
- Isaac Chavel, Riemannian geometry—a modern introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. MR 1271141
- Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
- Dario Cordero-Erausquin, Robert J. McCann, and Michael Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146 (2001), no. 2, 219–257. MR 1865396, DOI 10.1007/s002220100160
- Dario Cordero-Erausquin, Robert J. McCann, and Michael Schmuckenschläger, Prékopa-Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 4, 613–635 (English, with English and French summaries). MR 2295207
- Matthias Erbar, Kazumasa Kuwada, and Karl-Theodor Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces, Invent. Math. 201 (2015), no. 3, 993–1071. MR 3385639, DOI 10.1007/s00222-014-0563-7
- Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
- Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian geometry, 3rd ed., Universitext, Springer-Verlag, Berlin, 2004. MR 2088027, DOI 10.1007/978-3-642-18855-8
- D. J. H. Garling, Inequalities: a journey into linear analysis, Cambridge University Press, Cambridge, 2007. MR 2343341, DOI 10.1017/CBO9780511755217
- Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437
- M. Gromov, Paul Lévy’s isoperimetric inequality, preprint, I.H.E.S., 1980.
- M. Gromov and V. D. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), no. 4, 843–854. MR 708367, DOI 10.2307/2374298
- Ernst Heintze and Hermann Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 451–470. MR 533065
- R. Kannan, L. Lovász, and M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13 (1995), no. 3-4, 541–559. MR 1318794, DOI 10.1007/BF02574061
- L. Kennard and W. Wylie, Positive weighted sectional curvature, arXiv:1410.1558, 2014.
- Bo’az Klartag, Needle decompositions in Riemannian geometry, Mem. Amer. Math. Soc., to appear.
- Alexander V. Kolesnikov and Emanuel Milman, Brascamp–Lieb-type inequalities on weighted Riemannian manifolds with boundary, J. Geom. Anal., to appear, arXiv:1310.2526.
- Ernst Kuwert, Note on the isoperimetric profile of a convex body, Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 2003, pp. 195–200. MR 2008339
- Michel Ledoux, The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 2, 305–366 (English, with English and French summaries). Probability theory. MR 1813804
- Michel Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001. MR 1849347, DOI 10.1090/surv/089
- Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23, Springer-Verlag, Berlin, 1991. Isoperimetry and processes. MR 1102015, DOI 10.1007/978-3-642-20212-4
- André Lichnerowicz, Variétés riemanniennes à tenseur C non négatif, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A650–A653 (French). MR 268812
- André Lichnerowicz, Variétés kählériennes à première classe de Chern non negative et variétés riemanniennes à courbure de Ricci généralisée non negative, J. Differential Geometry 6 (1971/72), 47–94 (French). MR 300228
- John Lott, Some geometric properties of the Bakry-Émery-Ricci tensor, Comment. Math. Helv. 78 (2003), no. 4, 865–883. MR 2016700, DOI 10.1007/s00014-003-0775-8
- John Lott and Cédric Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), no. 3, 903–991. MR 2480619, DOI 10.4007/annals.2009.169.903
- V. G. Maz′ja, Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl. 1 (1960), 882–885. MR 0126152
- V. G. Maz′ja, The negative spectrum of the higher-dimensional Schrödinger operator, Dokl. Akad. Nauk SSSR 144 (1962), 721–722 (Russian). MR 0138880
- Emanuel Milman, On the role of convexity in isoperimetry, spectral gap and concentration, Invent. Math. 177 (2009), no. 1, 1–43. MR 2507637, DOI 10.1007/s00222-009-0175-9
- Emanuel Milman, Isoperimetric and concentration inequalities: equivalence under curvature lower bound, Duke Math. J. 154 (2010), no. 2, 207–239. MR 2682183, DOI 10.1215/00127094-2010-038
- Emanuel Milman, Isoperimetric bounds on convex manifolds, Concentration, functional inequalities and isoperimetry, Contemp. Math., vol. 545, Amer. Math. Soc., Providence, RI, 2011, pp. 195–208. MR 2858533, DOI 10.1090/conm/545/10772
- Emanuel Milman, Properties of isoperimetric, functional and transport-entropy inequalities via concentration, Probab. Theory Related Fields 152 (2012), no. 3-4, 475–507. MR 2892954, DOI 10.1007/s00440-010-0328-1
- Emanuel Milman, Sharp isoperimetric inequalities and model spaces for the curvature-dimension-diameter condition, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 5, 1041–1078. MR 3346688, DOI 10.4171/JEMS/526
- Emanuel Milman, Beyond traditional Curvature-Dimension II: Graded Curvature-Dimension condition and applications, manuscript, 2015.
- Emanuel Milman, Harmonic measures on the sphere via curvature-dimension, Annales de la Faculté des Sciences de Toulouse, to appear, arXiv:1505.04335.
- Emanuel Milman and Liran Rotem, Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures, Adv. Math. 262 (2014), 867–908. MR 3228444, DOI 10.1016/j.aim.2014.05.023
- Frank Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), no. 8, 853–858. MR 2161354
- Frank Morgan, Geometric measure theory, 4th ed., Elsevier/Academic Press, Amsterdam, 2009. A beginner’s guide. MR 2455580
- Frank Morgan and David L. Johnson, Some sharp isoperimetric theorems for Riemannian manifolds, Indiana Univ. Math. J. 49 (2000), no. 3, 1017–1041. MR 1803220, DOI 10.1512/iumj.2000.49.1929
- Van Hoang Nguyen, Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa’s theorem, J. Funct. Anal. 266 (2014), no. 2, 931–955. MR 3132733, DOI 10.1016/j.jfa.2013.11.003
- Shin-ichi Ohta, $(K,N)$-convexity and the curvature-dimension condition for negative $N$, J. Geom. Anal. 26 (2016), no. 3, 2067–2096. MR 3511469, DOI 10.1007/s12220-015-9619-1
- Shin-ichi Ohta and Asuka Takatsu, Displacement convexity of generalized relative entropies, Adv. Math. 228 (2011), no. 3, 1742–1787. MR 2824568, DOI 10.1016/j.aim.2011.06.029
- Shin-Ichi Ohta and Asuka Takatsu, Displacement convexity of generalized relative entropies. II, Comm. Anal. Geom. 21 (2013), no. 4, 687–785. MR 3078941, DOI 10.4310/CAG.2013.v21.n4.a1
- G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arxiv.org/abs/math/0211159, 2002.
- Zhongmin Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser. (2) 48 (1997), no. 190, 235–242. MR 1458581, DOI 10.1093/qmath/48.2.235
- Peter Sternberg and Kevin Zumbrun, On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint, Comm. Anal. Geom. 7 (1999), no. 1, 199–220. MR 1674097, DOI 10.4310/CAG.1999.v7.n1.a7
- Karl-Theodor Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), no. 1, 65–131. MR 2237206, DOI 10.1007/s11511-006-0002-8
- Max-K. von Renesse and Karl-Theodor Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math. 58 (2005), no. 7, 923–940. MR 2142879, DOI 10.1002/cpa.20060
- Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377–405. MR 2577473, DOI 10.4310/jdg/1261495336
- William Wylie, Sectional curvature for Riemannian manifolds with density, Geom. Dedicata 178 (2015), 151–169. MR 3397488, DOI 10.1007/s10711-015-0050-3
- W. Wylie, A warped product version of the Cheeger-Gromoll splitting theorem, arXiv: 1506.03800, 2015.
Additional Information
- Emanuel Milman
- Affiliation: Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel
- MR Author ID: 696280
- Email: emilman@tx.technion.ac.il
- Received by editor(s): March 24, 2015
- Received by editor(s) in revised form: July 7, 2015
- Published electronically: December 27, 2016
- Additional Notes: The author was supported by ISF (grant no. 900/10), BSF (grant no. 2010288) and Marie-Curie Actions (grant no. PCIG10-GA-2011-304066)
- © Copyright 2016 by the author
- Journal: Trans. Amer. Math. Soc. 369 (2017), 3605-3637
- MSC (2010): Primary 32F32, 53C21, 39B62, 58J50
- DOI: https://doi.org/10.1090/tran/6796
- MathSciNet review: 3605981