On the Curvature and Heat Flow on Hamiltonian Systems

Shin-ichi Ohta

Analysis and Geometry in Metric Spaces, Tome 2 (2014), / Harvested from The Polish Digital Mathematics Library

Résumé

We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation.We prove appropriate generalizations of the Bochner-Weitzenböck formula and Laplacian comparison theorem, and study the heat flow.

Détail

Publié le : 2014-01-01

Zbl 1295.53029

EUDML-ID : urn:eudml:doc:267463

Bibliographie

[1] A. A. Agrachev, Curvature and hyperbolicity of Hamiltonian systems (Russian), Tr. Mat. Inst. Steklova 256 (2007), Din. Sist. i Optim., 31-53; translation in Proc. Steklov Inst. Math. 256 (2007), 26-46.

[2] A. A. Agrachev, Geometry of optimal control problems and Hamiltonian systems, Nonlinear and optimal control theory, 1-59, Lecture Notes in Math., 1932, Springer, Berlin, 2008.

[3] A. A. Agrachev and R. V. Gamkrelidze, Feedback-invariant optimal control theory and differential geometry. I. Regular extremals, J. Dynam. Control Systems 3 (1997), 343-389. | Zbl 0952.49019

[4] A. A. Agrachev and P. W. Y. Lee, Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds, Preprint (2009). Available at arXiv:0903.2550

[5] A. A. Agrachev and P.W. Y. Lee, Bishop and Laplacian comparison theorems on three dimensional contact subriemannian manifolds with symmetry, to appear in J. Geom. Anal. Available at arXiv:1105.2206

[6] M. Agueh, Finsler structure in the p-Wasserstein space and gradient flows, C. R.Math. Acad. Sci. Paris 350 (2012), 35-40. | Zbl 1239.53021

[7] J. C. Álvarez Paiva and C. E. Durán, Geometric invariants of fanning curves, Adv. in Appl. Math. 42 (2009), 290-312. | Zbl 1167.53016

[8] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Birkhäuser Verlag, Basel, 2005. | Zbl 1090.35002

[9] L. Ambrosio, N. Gigli and G. Savaré, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, Rev. Mat. Iberoam. 29 (2013), 969-996.[Crossref] | Zbl 1287.46027

[10] 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), 289-391. | Zbl 1312.53056

[11] D. Bakry and M. Émery, Diffusions hypercontractives (French), Séminaire de probabilités, XIX, 1983/84, 177-206, Lecture Notes in Math. 1123, Springer, Berlin, 1985.

[12] D. Bao, S.-S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry, Springer-Verlag, New York, 2000. | Zbl 0954.53001

[13] P. Bernard and B. Buffoni, Optimal mass transportation and Mather theory, J. Eur. Math. Soc. (JEMS) 9 (2007), 85-121.[Crossref] | Zbl 1241.49025

[14] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375-417. | Zbl 0738.46011

[15] H. Brézis, Opérateursmaximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (French), North- Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

[16] D. Cordero-Erausquin, R. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146 (2001), 219-257.

[17] M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265-298. | Zbl 0226.47038

[18] M. Erbar, The heat equation on manifolds as a gradient flow in the Wasserstein space, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), 1-23.

[19] A. Fathi and A. Figalli, Optimal transportation on non-compact manifolds, Israel J. Math. 175 (2010), 1-59. | Zbl 1198.49044

[20] P. Foulon, Géométrie des équations différentielles du second ordre (French), Ann. Inst. H. Poincaré Phys. Théor. 45 (1986), 1-28.

[21] N. Fusco, M. Gori and F. Maggi, A remark on Serrin’s theorem, NoDEA Nonlinear Differential Equations Appl. 13 (2006), 425-433. | Zbl 1215.49024

[22] N. Gigli, On the differential structure of metric measure spaces and applications, Preprint (2012). Available at arXiv:1205.6622

[23] N. Gigli, K. Kuwada and S. Ohta, Heat flow on Alexandrov spaces, Comm. Pure Appl. Math. 66 (2013), 307-331. | Zbl 1267.58014

[24] N. Gigli and S. Ohta, First variation formula in Wasserstein spaces over compact Alexandrov spaces, Canad. Math. Bull. 55 (2012), 723-735.[Crossref] | Zbl 1264.53050

[25] J. Grifone, Structure presque-tangente et connexions. I (French), Ann. Inst. Fourier (Grenoble) 22 (1972), 287-334.[Crossref] | Zbl 0219.53032

[26] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29 (1998), 1-17. | Zbl 0915.35120

[27] N. Juillet, Geometric inequalities and generalized Ricci bounds in the Heisenberg group, Int.Math. Res. Not. IMRN (2009), 2347-2373. | Zbl 1176.53053

[28] M. Kell, On interpolation and curvature via Wasserstein geodesics, Preprint (2013). Available at arXiv:1311.5407

[29] M. Kell, q-heat flow and the gradient flow of the Renyi entropy in the p-Wasserstein space, Preprint (2014). Available at arXiv:1401.0840

[30] P. W. Y. Lee, A remark on the potentials of optimal transport maps, Acta Appl. Math. 115 (2011), 123-138. | Zbl 1223.49050

[31] P. W. Y. Lee, Displacement interpolations from a Hamiltonian point of view, J. Funct. Anal. 265 (2013), 3163-3203. | Zbl 1285.53011

[32] P. W. Y. Lee, C. Li and I. Zelenko, Measure contraction properties of contact sub-Riemannian manifolds with symmetry, Preprint (2013). Available at arXiv:1304.2658

[33] J. Lott and C. Villani, Weak curvature conditions and functional inequalities, J. Funct. Anal. 245 (2007), 311-333. | Zbl 1119.53028

[34] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169 (2009), 903-991. | Zbl 1178.53038

[35] J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal. 261 (2011), 2250-2292. | Zbl 1237.60058

[36] U. F. Mayer, Gradient flows on nonpositively curved metric spaces and harmonic maps, Comm. Anal. Geom. 6 (1998), 199-253. | Zbl 0914.58008

[37] R. J. McCann, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal. 11 (2001), 589-608.[Crossref] | Zbl 1011.58009

[38] S. Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82 (2007), 805-828. | Zbl 1176.28016

[39] S. Ohta, Gradient flows on Wasserstein spaces over compact Alexandrov spaces, Amer. J. Math. 131 (2009), 475-516. | Zbl 1169.53053

[40] S. Ohta, Uniform convexity and smoothness, and their applications in Finsler geometry,Math. Ann. 343 (2009), 669-699. | Zbl 1160.53033

[41] S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations 36 (2009), 211-249. | Zbl 1175.49044

[42] S. Ohta, Vanishing S-curvature of Randers spaces, Differential Geom. Appl. 29 (2011), 174-178.[Crossref] | Zbl 1215.53067

[43] S. Ohta, Splitting theorems for Finslermanifolds of nonnegative Ricci curvature, to appear in J. Reine Angew.Math. Available at arXiv:1203.0079

[44] S. Ohta and K.-T. Sturm, Heat flow on Finsler manifolds, Comm. Pure Appl. Math. 62 (2009), 1386-1433. | Zbl 1176.58012

[45] S. Ohta and K.-T. Sturm, Non-contraction of heat flow on Minkowski spaces, Arch. Ration. Mech. Anal. 204 (2012), 917-944. | Zbl 1257.53098

[46] S. Ohta and K.-T. Sturm, Bochner-Weitzenböck formula and Li-Yau estimates on Finslermanifolds, Adv.Math. 252 (2014), 429-448. | Zbl 1321.53089

[47] S. Ohta and A. Takatsu, Displacement convexity of generalized relative entropies, Adv. Math. 228 (2011), 1742-1787. | Zbl 1223.53032

[48] S. Ohta and A. Takatsu, Displacement convexity of generalized relative entropies. II, Comm. Anal. Geom. 21 (2013), 687-785.[Crossref] | Zbl 1315.53015

[49] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), 361-400. | Zbl 0985.58019

[50] Z. Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser. (2) 48 (1997), 235-242. | Zbl 0902.53032

[51] M.-K. von Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy and Ricci curvature, Comm. Pure Appl. Math. 58 (2005), 923-940. | Zbl 1078.53028

[52] G. Savaré, Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, C. R. Math. Acad. Sci. Paris 345 (2007), 151-154. | Zbl 1125.53064

[53] J. Serrin, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc. 101 (1961), 139-167. | Zbl 0102.04601

[54] Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry, Adv. Math. 128 (1997), 306-328. | Zbl 0919.53021

[55] Z. Shen, Lectures on Finsler geometry, World Scientific Publishing Co., Singapore, 2001. | Zbl 0974.53002

[56] K.-T. Sturm, On the geometry of metric measure spaces. I, Acta Math. 196 (2006), 65-131.

[57] K.-T. Sturm, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), 133-177.

[58] C. Villani, Optimal transport, old and new, Springer-Verlag, Berlin, 2009.

[59] G. Wang and C. Xia, A sharp lower bound for the first eigenvalue on Finsler manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), 983-996. | Zbl 1286.35179

[60] G. Wei and W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), 377-405. | Zbl 1189.53036

[61] C. Xia, Local gradient estimate for harmonic functions on Finsler manifolds, to appear in Calc. Var. Partial Differential Equations. Available at arXiv:1308.3609