Angles between Curves in Metric Measure Spaces

Angles between Curves in Metric Measure Spaces References[1] L. Ambrosio, N. Gigli, A. Mondino, and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with infinite measure, Trans. Amer. Math. Soc., 367 (2015), pp. 4661-4701.[2] L. Ambrosio, N. Gigli, and G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., 195 (2014), pp. 289-391.[3] , Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., 163 (2014), pp. 1405-1490.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000336014500004&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[4] L. Ambrosio and S. Honda, New stability results for sequences of metric measure spaces with uniform Ricci bounds from below. Preprint, arXiv:1605.07908, (2016).[5] L. Ambrosio, A. Mondino, and G. Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces, preprint arXiv:1509.07273, to appear in Mem. Amer. Math. Soc.[6] A. Björn and J. Björn, Nonlinear Potential Theory onMetric Spaces,EMS TractsMath., vol.17 of EuropeanMathematical Society (EMS), Zürich, 2011.[7] D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, vol. 33 of Graduate Studies in Mathematics, American Mathematical Society, 2001.[8] M. Biroli and U. Mosco, A Saint-Venant type principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl. (4) 169 (1995), pp 125-181.[9] 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.[10] F. Cavalletti and A. Mondino, Optimal maps in essentially non-branching spaces, preprint arXiv:1609.00782, to appear in Comm. Cont. Math. DOI: 10.1142/S0219199717500079.[11] J. Cheeger, Di_erentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), pp. 428-517.10.1007/s000390050094[12] J. Cheeger and T. Colding, On the structure of spaces with Ricci curvature bounded below I, J. Di_. Geom., 45, (1997), pp. 406 - 480.[13] , On the structure of spaces with Ricci curvature bounded below II, J. Di_. Geom., 54, (2000), pp. 13-35.[14] , On the structure of spaces with Ricci curvature bounded below III, J. Di_. Geom., 54, (2000), pp. 37 - 74.[15] E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications, C. Baiocchi and J. L. Lions, eds., Masson, (1993), pp. 81-98.[16] M. Erbar, K. Kuwada, and K. -T. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner inequality on metric measure spaces, Invent. Math., 201 (2015), pp. 993-1071.[17] N. Gigli, On the di_erential structure of metric measure spaces and applications, Mem. Amer.Math. Soc., 236, (1113), (2015).[18] N. Gigli and B.-X. Han, The continuity equation on metric measure spaces, Calc. Var. Partial Differential Equations, 53 (2015), pp. 149-177.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000352896500006&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[19] N. Gigli and A. Mondino, A PDE approach to nonlinear potential theory in metric measure spaces, J. Math. Pures Appl. 100 (2013), pp. 505-534.[20] N. Gigli, A. Mondino, and G. Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc., 111, (2015), no. 5, pp. 1071-1129.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000368421900004&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[21] N. Gigli, A. Mondino, and T. Rajala, Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below, J. Reine Angew. Math., 705, (2015), pp. 233-244.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000359196100007&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[22] N. Gigli, T. Rajala, and K.T. Sturm, Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below, J. Geom. Anal., 26, (2016), no. 4, pp. 2914-2929.10.1007/s12220-015-9654-yhttp://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000382893800018&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[23] S. Honda, A weakly second-order di_erential structure on rectifiable metric measure spaces, Geom. Topol. 18 (2014), no. 2, pp 633-668.10.2140/gt.2014.18.633[24] S. Lisini, Characterization of absolutely continuous curves in Wasserstein spaces, Calc. Var. Partial Differential Equations, 28 (2007), pp. 85-120.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000242610000005&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[25] J. Lott and C. Villani, Ricci curvature formetric-measure spaces via optimal transport, Ann. ofMath. (2), 169 (2009), pp. 903 -991.[26] A. Mondino, A new notion of angle between three points in a metric space, J. Reine Angew. Math, 706 (2015), pp. 103-121.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000360857300006&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[27] A. Mondino and A. Naber, Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds, preprint arXiv:1405.2222, to appear in Journ. Europ. Math. Soc.[28] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), pp. 101-174.10.1081/PDE-100002243[29] T. Rajala, Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations, Vol. 44, Num. 3, (2012), pp. 477-494.[30] K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math., Vol. 196, (2006), 65-131.[31] , On the geometry of metric measure spaces. II, Acta Math., Vol. 196, (2006), 133-177.[32] C. Villani, Optimal transport. Old and new, Grundlehren derMathematischenWissenschaften, 338, Springer-Verlag, Berlin, (2009). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Geometry in Metric Spaces de Gruyter

Angles between Curves in Metric Measure Spaces

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References[1] L. Ambrosio, N. Gigli, A. Mondino, and T. Rajala, Riemannian Ricci curvature lower bounds in metric measure spaces with infinite measure, Trans. Amer. Math. Soc., 367 (2015), pp. 4661-4701.[2] L. Ambrosio, N. Gigli, and G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., 195 (2014), pp. 289-391.[3] , Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., 163 (2014), pp. 1405-1490.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000336014500004&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[4] L. Ambrosio and S. Honda, New stability results for sequences of metric measure spaces with uniform Ricci bounds from below. Preprint, arXiv:1605.07908, (2016).[5] L. Ambrosio, A. Mondino, and G. Savaré, Nonlinear diffusion equations and curvature conditions in metric measure spaces, preprint arXiv:1509.07273, to appear in Mem. Amer. Math. Soc.[6] A. Björn and J. Björn, Nonlinear Potential Theory onMetric Spaces,EMS TractsMath., vol.17 of EuropeanMathematical Society (EMS), Zürich, 2011.[7] D. Burago, Y. Burago, and S. Ivanov, A course in metric geometry, vol. 33 of Graduate Studies in Mathematics, American Mathematical Society, 2001.[8] M. Biroli and U. Mosco, A Saint-Venant type principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl. (4) 169 (1995), pp 125-181.[9] 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.[10] F. Cavalletti and A. Mondino, Optimal maps in essentially non-branching spaces, preprint arXiv:1609.00782, to appear in Comm. Cont. Math. DOI: 10.1142/S0219199717500079.[11] J. Cheeger, Di_erentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), pp. 428-517.10.1007/s000390050094[12] J. Cheeger and T. Colding, On the structure of spaces with Ricci curvature bounded below I, J. Di_. Geom., 45, (1997), pp. 406 - 480.[13] , On the structure of spaces with Ricci curvature bounded below II, J. Di_. Geom., 54, (2000), pp. 13-35.[14] , On the structure of spaces with Ricci curvature bounded below III, J. Di_. Geom., 54, (2000), pp. 37 - 74.[15] E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications, C. Baiocchi and J. L. Lions, eds., Masson, (1993), pp. 81-98.[16] M. Erbar, K. Kuwada, and K. -T. Sturm, On the equivalence of the entropic curvature-dimension condition and Bochner inequality on metric measure spaces, Invent. Math., 201 (2015), pp. 993-1071.[17] N. Gigli, On the di_erential structure of metric measure spaces and applications, Mem. Amer.Math. Soc., 236, (1113), (2015).[18] N. Gigli and B.-X. Han, The continuity equation on metric measure spaces, Calc. Var. Partial Differential Equations, 53 (2015), pp. 149-177.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000352896500006&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[19] N. Gigli and A. Mondino, A PDE approach to nonlinear potential theory in metric measure spaces, J. Math. Pures Appl. 100 (2013), pp. 505-534.[20] N. Gigli, A. Mondino, and G. Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc., 111, (2015), no. 5, pp. 1071-1129.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000368421900004&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[21] N. Gigli, A. Mondino, and T. Rajala, Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below, J. Reine Angew. Math., 705, (2015), pp. 233-244.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000359196100007&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[22] N. Gigli, T. Rajala, and K.T. Sturm, Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below, J. Geom. Anal., 26, (2016), no. 4, pp. 2914-2929.10.1007/s12220-015-9654-yhttp://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000382893800018&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[23] S. Honda, A weakly second-order di_erential structure on rectifiable metric measure spaces, Geom. Topol. 18 (2014), no. 2, pp 633-668.10.2140/gt.2014.18.633[24] S. Lisini, Characterization of absolutely continuous curves in Wasserstein spaces, Calc. Var. Partial Differential Equations, 28 (2007), pp. 85-120.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000242610000005&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[25] J. Lott and C. Villani, Ricci curvature formetric-measure spaces via optimal transport, Ann. ofMath. (2), 169 (2009), pp. 903 -991.[26] A. Mondino, A new notion of angle between three points in a metric space, J. Reine Angew. Math, 706 (2015), pp. 103-121.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000360857300006&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[27] A. Mondino and A. Naber, Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds, preprint arXiv:1405.2222, to appear in Journ. Europ. Math. Soc.[28] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 26 (2001), pp. 101-174.10.1081/PDE-100002243[29] T. Rajala, Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations, Vol. 44, Num. 3, (2012), pp. 477-494.[30] K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math., Vol. 196, (2006), 65-131.[31] , On the geometry of metric measure spaces. II, Acta Math., Vol. 196, (2006), 133-177.[32] C. Villani, Optimal transport. Old and new, Grundlehren derMathematischenWissenschaften, 338, Springer-Verlag, Berlin, (2009).

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