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

Analysis and Geometry in Metric Spaces (2014)

- Volume: 2, Issue: 1, page 169-213, electronic only
- ISSN: 2299-3274

## Access Full Article

top## How to cite

topNicola Gigli. "An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature." Analysis and Geometry in Metric Spaces 2.1 (2014): 169-213, electronic only. <http://eudml.org/doc/266707>.

@article{NicolaGigli2014,

author = {Nicola Gigli},

journal = {Analysis and Geometry in Metric Spaces},

keywords = {Sobolev spaces; measure valued Laplacian; optimal transport; condition; Busemann function; Bakry-Émery contraction; Dirichlet energy},

language = {eng},

number = {1},

pages = {169-213, electronic only},

title = {An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature},

url = {http://eudml.org/doc/266707},

volume = {2},

year = {2014},

}

TY - JOUR

AU - Nicola Gigli

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

JO - Analysis and Geometry in Metric Spaces

PY - 2014

VL - 2

IS - 1

SP - 169

EP - 213, electronic only

LA - eng

KW - Sobolev spaces; measure valued Laplacian; optimal transport; condition; Busemann function; Bakry-Émery contraction; Dirichlet energy

UR - http://eudml.org/doc/266707

ER -

## References

top- [1] U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc., 3 (1990), pp. 355-374.[Crossref] Zbl0704.53032
- [2] L. Ambrosio and N. Gigli, A user’s guide to optimal transport. Modelling and Optimisation of Flows on Networks, Lecture Notes in Mathematics, Vol. 2062, Springer, 2011.
- [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. Zbl1317.53060
- [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. Zbl1145.35001
- [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. Zbl1335.35088
- [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. Zbl1196.53027
- [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. Zbl1231.31001
- [12] F. Cavalletti and K.-T. Sturm, Local curvature-dimension condition implies measure-contraction property, J. Funct. Anal., 262 (2012), pp. 5110-5127. Zbl1244.53050
- [13] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), pp. 428-517.[Crossref] Zbl0942.58018
- [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. Zbl0865.53037
- [15] , On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., 46 (1997), pp. 406-480. Zbl0902.53034
- [16] , On the structure of spaces with Ricci curvature bounded below. II, J. Differential Geom., 54 (2000), pp. 13-35. Zbl1027.53042
- [17] , On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom., 54 (2000), pp. 37-74. Zbl1027.53043
- [18] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry, 6 (1971/72), pp. 119-128. Zbl0223.53033
- [19] D. L. Cohn, Measure theory, Birkhäuser Boston Inc., Boston, MA, 1993. Reprint of the 1980 original. Zbl0860.28001
- [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. Zbl1329.53059
- [21] N. Gigli, On the heat flow on metric measure spaces: existence, uniqueness and stability, Calc. Var. PDE, 39 (2010), pp. 101-120. Zbl1200.35178
- [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. Zbl1257.53055
- [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. Zbl1267.58014
- [26] N. Gigli and A. Mondino, A PDE approach to nonlinear potential theory in metric measure spaces. Accepted at JMPA, arXiv:1209.3796, 2012. Zbl1283.31002
- [27] N. Gigli, A. Mondino, and G. Savaré, A notion of convergence of non-compact metric measure spaces and applications. Preprint, 2013. Zbl06522136
- [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. Zbl0985.46008
- [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. Zbl0915.35120
- [34] B. Kleiner and J. Mackay, Differentiable structure on metric measure spaces: a primer. Preprint, arXiv:1108.1324, 2011. Zbl06586482
- [35] K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal., 258 (2010), pp. 3758-3774. Zbl1194.53032
- [36] J. Lott and C. Villani, Weak curvature conditions and functional inequalities, J. Funct. Anal., 245 (2007), pp. 311-333. Zbl1119.53028
- [37] , Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), pp. 903-991. Zbl1178.53038
- [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. Zbl1176.58012
- [40] , Non-contraction of heat flow on Minkowski spaces, Arch. Ration. Mech. Anal., 204 (2012), pp. 917-944. Zbl1257.53098
- [41] Y. Otsu and T. Shioya, The Riemannian structure of Alexandrov spaces, J. Differential Geom., 39 (1994), pp. 629-658. Zbl0808.53061
- [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. Zbl1247.53038
- [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. Zbl1260.53076
- [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. Zbl1250.53040
- [47] N. Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana, 16 (2000), pp. 243-279. Zbl0974.46038
- [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. Zbl0806.53041
- [49] , Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality, J. Math. Pures Appl. (9), 75 (1996), pp. 273-297. Zbl0854.35016
- [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. Zbl0979.46035

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.