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Tytuł artykułu

An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature

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Tom
2
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1
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Daty
wydano
2014-01-01
otrzymano
2013-05-21
zaakceptowano
2014-04-18
online
2014-05-17
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autor
Bibliografia
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  • [3] L. Ambrosio, N. Gigli, A. Mondino, and T. Rajala, Riemannian ricci curvature lower bounds in metric measure spaces with ff-ffnite measure. Preprint, arXiv:1207.4924, 2011.
  • [4] L. Ambrosio, N. Gigli, and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, second ed., 2008.
  • [5] , Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Accepted by Revista Matemática Iberoamericana, arXiv:1111.3730, 2011.
  • [6] , Metric measure spaces with riemannian Ricci curvature bounded from below. Preprint, arXiv:1109.0222, 2011.
  • [7] , Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Preprint, arXiv:1209.5786, 2012.
  • [8] L. Ambrosio, N. Gigli, and G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below, Inventiones mathematicae, (2013), pp. 1-103.
  • [9] L. Ambrosio, A. Mondino, and G. Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces. Preprint, 2013.
  • [10] K. Bacher and K.-T. Sturm, Localization and tensorization properties of the curvature-dimension condition for metric measure spaces, J. Funct. Anal., 259 (2010), pp. 28-56.
  • [11] A. Björn and J. Björn, Nonlinear potential theory on metric spaces, vol. 17 of EMS Tracts in Mathematics, European Mathematical Society (EMS), Zürich, 2011.
  • [12] F. Cavalletti and K.-T. Sturm, Local curvature-dimension condition implies measure-contraction property, J. Funct. Anal., 262 (2012), pp. 5110-5127.
  • [13] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), pp. 428-517.[Crossref]
  • [14] J. Cheeger and T. H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2), 144 (1996), pp. 189-237.
  • [15] , On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., 46 (1997), pp. 406-480.
  • [16] , On the structure of spaces with Ricci curvature bounded below. II, J. Differential Geom., 54 (2000), pp. 13-35.
  • [17] , On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom., 54 (2000), pp. 37-74.
  • [18] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry, 6 (1971/72), pp. 119-128.
  • [19] D. L. Cohn, Measure theory, Birkhäuser Boston Inc., Boston, MA, 1993. Reprint of the 1980 original.
  • [20] M. Erbar, K. Kuwada, and K.-T. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Preprint, arXiv:1303.4382, 2013.
  • [21] N. Gigli, On the heat flow on metric measure spaces: existence, uniqueness and stability, Calc. Var. PDE, 39 (2010), pp. 101-120.
  • [22] , On the differential structure of metric measure spaces and applications. Preprint, arXiv:1205.6622, 2012.
  • [23] , Optimal maps in non branching spaces with Ricci curvature bounded from below, Geom. Funct. Anal., 22 (2012), pp. 990-999.
  • [24] , The splitting theorem in non-smooth context. Preprint, arXiv:1302.5555, 2013.
  • [25] N. Gigli, K. Kuwada, and S.-i. Ohta, Heat flow on Alexandrov spaces, Communications on Pure and Applied Mathematics, 66 (2013), pp. 307-331.
  • [26] N. Gigli and A. Mondino, A PDE approach to nonlinear potential theory in metric measure spaces. Accepted at JMPA, arXiv:1209.3796, 2012.
  • [27] N. Gigli, A. Mondino, and G. Savaré, A notion of convergence of non-compact metric measure spaces and applications. Preprint, 2013.
  • [28] N. Gigli and S. Mosconi, The Abresch-Gromoll inequality in a non-smooth setting. Accepted at DCDS-A, arXiv:1209.3813, 2012.
  • [29] N. Gigli, T. Rajala, and K.-T. Sturm, Optimal maps and exponentiation on ffnite dimensional spaces with Ricci curvature bounded from below. Preprint, 2013.
  • [30] A. Grigor0yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.), 36 (1999), pp. 135-249.[Crossref]
  • [31] J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001.
  • [32] , Nonsmooth calculus, Bull. Amer. Math. Soc. (N.S.), 44 (2007), pp. 163-232.
  • [33] R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), pp. 1-17.
  • [34] B. Kleiner and J. Mackay, Differentiable structure on metric measure spaces: a primer. Preprint, arXiv:1108.1324, 2011.
  • [35] K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal., 258 (2010), pp. 3758-3774.
  • [36] J. Lott and C. Villani, Weak curvature conditions and functional inequalities, J. Funct. Anal., 245 (2007), pp. 311-333.
  • [37] , Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), pp. 903-991.
  • [38] S.-i. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations, 36 (2009), pp. 211-249.
  • [39] S.-i. Ohta and K.-T. Sturm, Heat flow on Finsler manifolds, Comm. Pure Appl. Math., 62 (2009), pp. 1386-1433.
  • [40] , Non-contraction of heat flow on Minkowski spaces, Arch. Ration. Mech. Anal., 204 (2012), pp. 917-944.
  • [41] Y. Otsu and T. Shioya, The Riemannian structure of Alexandrov spaces, J. Differential Geom., 39 (1994), pp. 629-658.
  • [42] G. Perelman, Dc structure on Alexandrov Space. Unpublished preprint, available online at http://www.math.psu.edu/petrunin/papers/alexandrov/Cstructure.pdf.
  • [43] A. Petrunin, Alexandrov meets Lott-Villani-Sturm, Münster J. Math., 4 (2011), pp. 53-64.
  • [44] T. Rajala, Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm, J. Funct. Anal., 263 (2012), pp. 896-924.
  • [45] , Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations, 44 (2012), pp. 477-494. [46] T. Rajala and K.-T. Sturm, Non-branching geodesics and optimalmaps in strong CD(K,1)-spaces. Preprint, arXiv:1207.6754, 2012.
  • [47] N. Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana, 16 (2000), pp. 243-279.
  • [48] K.-T. Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties, J. Reine Angew. Math., 456 (1994), pp. 173-196.
  • [49] , Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality, J. Math. Pures Appl. (9), 75 (1996), pp. 273-297.
  • [50] , On the geometry of metric measure spaces. I, Acta Math., 196 (2006), pp. 65-131.
  • [51] , On the geometry of metric measure spaces. II, Acta Math., 196 (2006), pp. 133-177.
  • [52] C. Villani, Optimal transport. Old and new, vol. 338 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009.
  • [53] N. Weaver, Lipschitz algebras and derivations. II. Exterior differentiation, J. Funct. Anal., 178 (2000), pp. 64-112.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_agms-2014-0006
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