American Institute of Mathematical Sciences

January  2016, 36(1): 303-321. doi: 10.3934/dcds.2016.36.303

Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds

 1 Room 216, Lady Shaw Building, The Chinese University of Hong Kong, Shatin, Hong Kong, China 2 Department of Mathematics, Tianjin University, Tianjin, 300072, China, China

Received  April 2014 Revised  March 2015 Published  June 2015

Measure contraction properties are generalizations of the notion of Ricci curvature lower bounds in Riemannian geometry to more general metric measure spaces. In this paper, we give sufficient conditions for a Sasakian manifold equipped with a natural sub-Riemannian distance to satisfy these properties. Moreover, the sufficient conditions are defined by the Tanaka-Webster curvature. This generalizes the earlier work in [2] for the three dimensional case and in [19] for the Heisenberg group. To obtain our results we use the intrinsic Jacobi equations along sub-Riemannian extremals, coming from the theory of canonical moving frames for curves in Lagrangian Grassmannians [24,25]. The crucial new tool here is a certain decoupling of the corresponding matrix Riccati equation. It is also worth pointing out that our method leads to exact formulas for the measure contraction in the case of the corresponding homogeneous models in the considered class of sub-Riemannian structures.
Citation: Paul W. Y. Lee, Chengbo Li, Igor Zelenko. Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 303-321. doi: 10.3934/dcds.2016.36.303
References:
 [1] A. A. Agrachev and R. V. Gamkrelidze, Feedback-invariant optimal control theory - I. Regular extremals, J. Dynamical and Control Systems, 3 (1997), 343-389. doi: 10.1007/BF02463256. [2] A. Agrachev and P. Lee, Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds, Math. Ann., 360 (2014), 209-253. doi: 10.1007/s00208-014-1034-6. [3] A. Agrachev and P. Lee, Bishop and Laplacian comparison theorems on three dimensional contact subriemannian manifolds with symmetry, J. Geom. Anal., 25 (2015), 512-535, arXiv:1105.2206. doi: 10.1007/s12220-013-9437-2. [4] A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, 87. Control Theory and Optimization, II. Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7. [5] A. Agrachev and I. Zelenko, Geometry of Jacobi curves. I, J. Dynamical and Control systems, 8 (2002), 93-140. doi: 10.1023/A:1013904801414. [6] D. Bakry and M. Émery, Diffusions hypercontractives. in Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., 1123, Springer, Berlin, (1985), 177-206. doi: 10.1007/BFb0075847. [7] F. Baudoin, M. Bonnefont and N. Garofalo, A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincare inequality, Math. Ann., 358 (2014), 833-860. doi: 10.1007/s00208-013-0961-y. [8] F. Baudoin and N. Garofalo, Generalized Bochner formulas and Ricci lower bounds for sub-Riemannian manifolds of rank two,, preprint, (). [9] F. Baudoin and N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries,, preprint, (). [10] D. E. Blair, Contact Manifolds in Riemannian Geometry,, Lecture Notes in Mathematics, 509 (). [11] P. Cannarsa and L. Rifford, Semiconcavity results for optimal control problems admitting no singular minimizing controls, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 773-802. doi: 10.1016/j.anihpc.2007.07.005. [12] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkhhäuser, 2004. [13] S. Chanillo and P. Yang, Isoperimetric inequalities & volume comparison theorems on CR manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci., (5) 8 (2009), 279-307. [14] T. Coulhon, I. Holopainen and L. Saloff-Coste, Harnack inequality and hyperbolicity for subelliptic p-Laplacians with applications to Picard type theorems, Geom. Funct. Anal., 11 (2001), 1139-1191. doi: 10.1007/s00039-001-8227-3. [15] D. B. A. Epstein, Complex hyperbolic geometry, in Analytical and Geometric Aspects of Hyperbolic Space (ed. D.B.A. Epstein), London Mathematical Society Lecture Notes Series, 111 (1987), 93-111. [16] A. Figalli and L. Rifford, Mass Transportation on sub-Riemannian Manifolds, Geom. Funct. Anal., 20 (2010), 124-159. doi: 10.1007/s00039-010-0053-z. [17] K. Hughen, The Geometry of Sub-Riemannian Three-Manifolds, Ph.D. Dissertation, Duke University, 1995. [18] D. Jerison, The Poincaŕe inequality for vector fields satisfying the Hörmander condition, Duke Math. J., 53 (1986), 503-523. doi: 10.1215/S0012-7094-86-05329-9. [19] N. Juillet, Geometric inequalities and generalized ricci bounds in the heisenberg group, Int. Math. Res. Not. IMRN, (2009), 2347-2373. doi: 10.1093/imrn/rnp019. [20] P. W. Y. Lee, Displacement interpolations from a Hamiltonian point of view, J. Func. Anal., 265 (2013), 3163-3203. doi: 10.1016/j.jfa.2013.08.022. [21] P. W. Y. Lee and C. Li, Bishop and Laplacian comparison theorems on Sasakian manifolds, preprint, arXiv:1310.5322 (2013), 38pp. [22] P. W. Y. Lee, C. Li and I. Zelenko, Measure contraction properties of contact sub-Riemannian manifolds with symmetry, preprint,, , (). [23] J. J. Levin, On the matrix Riccati equation, Proc. Amer. Math. Soc., 10 (1959), 519-524. doi: 10.1090/S0002-9939-1959-0108628-X. [24] C. Li and I. Zelenko, Parametrized curves in Lagrange Grassmannians, C.R. Acad. Sci. Paris, Ser. I, 345 (2007), 647-652. doi: 10.1016/j.crma.2007.10.034. [25] C. Li and I. Zelenko, Differential geometry of curves in Lagrange Grassmannians with given Young diagram, Differ. Geom. Appl., 27 (2009), 723-742. doi: 10.1016/j.difgeo.2009.07.002. [26] C.Li and I. Zelenko, Jacobi equations and comparison theorems for corank 1 sub-Riemannian structures with symmetries, J. Geom. Phys., 61 (2011), 781-807. doi: 10.1016/j.geomphys.2010.12.009. [27] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903. [28] J. Lott and C. Villani, Weak curvature conditions and functional inequalities, J. Funct. Anal., 245 (2007), 311-333. doi: 10.1016/j.jfa.2006.10.018. [29] R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, 91, American Mathematical Society, Providence, RI, 2002. [30] S. Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv., 82 (2007), 805-828. doi: 10.4171/CMH/110. [31] S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations, 36 (2009), 211-249. doi: 10.1007/s00526-009-0227-4. [32] H. L. Royden, Comparison theorems for the matrix Riccati equation, Comm. Pure Appl. Math., 41 (1988), 739-746. doi: 10.1002/cpa.3160410512. [33] T. Sakai, Riemannian Geometry, Translations of Mathematical Monographs, 149. American Mathematical Society, Providence, RI, 1996. [34] K. T. Sturm, On the geometry of metric measure spaces, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8. [35] K. T. Sturm, On the geometry of metric measure spaces II, Acta Math., 196 (2006), 133-177. doi: 10.1007/s11511-006-0003-7. [36] N. Tanaka, A Differential Geometric Study on Strongly Pseudo-Convex Manifold, Kinokunya Book Store Co., Ltd., Kyoto, 1975. [37] S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc., 314 (1989), 349-379. doi: 10.1090/S0002-9947-1989-1000553-9. [38] C. Villani, Optimal Transport. Old and new, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. [39] J. Wang, Sub-Riemannian Heat Kernels on Model Spaces and Curvature-Dimension Inequalities on Contact Manifolds, Ph.D. Dissertation, Purdue University, 2014. [40] S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry, 13 (1978), 25-41.

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References:
 [1] A. A. Agrachev and R. V. Gamkrelidze, Feedback-invariant optimal control theory - I. Regular extremals, J. Dynamical and Control Systems, 3 (1997), 343-389. doi: 10.1007/BF02463256. [2] A. Agrachev and P. Lee, Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds, Math. Ann., 360 (2014), 209-253. doi: 10.1007/s00208-014-1034-6. [3] A. Agrachev and P. Lee, Bishop and Laplacian comparison theorems on three dimensional contact subriemannian manifolds with symmetry, J. Geom. Anal., 25 (2015), 512-535, arXiv:1105.2206. doi: 10.1007/s12220-013-9437-2. [4] A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, 87. Control Theory and Optimization, II. Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-06404-7. [5] A. Agrachev and I. Zelenko, Geometry of Jacobi curves. I, J. Dynamical and Control systems, 8 (2002), 93-140. doi: 10.1023/A:1013904801414. [6] D. Bakry and M. Émery, Diffusions hypercontractives. in Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., 1123, Springer, Berlin, (1985), 177-206. doi: 10.1007/BFb0075847. [7] F. Baudoin, M. Bonnefont and N. Garofalo, A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincare inequality, Math. Ann., 358 (2014), 833-860. doi: 10.1007/s00208-013-0961-y. [8] F. Baudoin and N. Garofalo, Generalized Bochner formulas and Ricci lower bounds for sub-Riemannian manifolds of rank two,, preprint, (). [9] F. Baudoin and N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries,, preprint, (). [10] D. E. Blair, Contact Manifolds in Riemannian Geometry,, Lecture Notes in Mathematics, 509 (). [11] P. Cannarsa and L. Rifford, Semiconcavity results for optimal control problems admitting no singular minimizing controls, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 773-802. doi: 10.1016/j.anihpc.2007.07.005. [12] P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkhhäuser, 2004. [13] S. Chanillo and P. Yang, Isoperimetric inequalities & volume comparison theorems on CR manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci., (5) 8 (2009), 279-307. [14] T. Coulhon, I. Holopainen and L. Saloff-Coste, Harnack inequality and hyperbolicity for subelliptic p-Laplacians with applications to Picard type theorems, Geom. Funct. Anal., 11 (2001), 1139-1191. doi: 10.1007/s00039-001-8227-3. [15] D. B. A. Epstein, Complex hyperbolic geometry, in Analytical and Geometric Aspects of Hyperbolic Space (ed. D.B.A. Epstein), London Mathematical Society Lecture Notes Series, 111 (1987), 93-111. [16] A. Figalli and L. Rifford, Mass Transportation on sub-Riemannian Manifolds, Geom. Funct. Anal., 20 (2010), 124-159. doi: 10.1007/s00039-010-0053-z. [17] K. Hughen, The Geometry of Sub-Riemannian Three-Manifolds, Ph.D. Dissertation, Duke University, 1995. [18] D. Jerison, The Poincaŕe inequality for vector fields satisfying the Hörmander condition, Duke Math. J., 53 (1986), 503-523. doi: 10.1215/S0012-7094-86-05329-9. [19] N. Juillet, Geometric inequalities and generalized ricci bounds in the heisenberg group, Int. Math. Res. Not. IMRN, (2009), 2347-2373. doi: 10.1093/imrn/rnp019. [20] P. W. Y. Lee, Displacement interpolations from a Hamiltonian point of view, J. Func. Anal., 265 (2013), 3163-3203. doi: 10.1016/j.jfa.2013.08.022. [21] P. W. Y. Lee and C. Li, Bishop and Laplacian comparison theorems on Sasakian manifolds, preprint, arXiv:1310.5322 (2013), 38pp. [22] P. W. Y. Lee, C. Li and I. Zelenko, Measure contraction properties of contact sub-Riemannian manifolds with symmetry, preprint,, , (). [23] J. J. Levin, On the matrix Riccati equation, Proc. Amer. Math. Soc., 10 (1959), 519-524. doi: 10.1090/S0002-9939-1959-0108628-X. [24] C. Li and I. Zelenko, Parametrized curves in Lagrange Grassmannians, C.R. Acad. Sci. Paris, Ser. I, 345 (2007), 647-652. doi: 10.1016/j.crma.2007.10.034. [25] C. Li and I. Zelenko, Differential geometry of curves in Lagrange Grassmannians with given Young diagram, Differ. Geom. Appl., 27 (2009), 723-742. doi: 10.1016/j.difgeo.2009.07.002. [26] C.Li and I. Zelenko, Jacobi equations and comparison theorems for corank 1 sub-Riemannian structures with symmetries, J. Geom. Phys., 61 (2011), 781-807. doi: 10.1016/j.geomphys.2010.12.009. [27] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2), 169 (2009), 903-991. doi: 10.4007/annals.2009.169.903. [28] J. Lott and C. Villani, Weak curvature conditions and functional inequalities, J. Funct. Anal., 245 (2007), 311-333. doi: 10.1016/j.jfa.2006.10.018. [29] R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, 91, American Mathematical Society, Providence, RI, 2002. [30] S. Ohta, On the measure contraction property of metric measure spaces, Comment. Math. Helv., 82 (2007), 805-828. doi: 10.4171/CMH/110. [31] S. Ohta, Finsler interpolation inequalities, Calc. Var. Partial Differential Equations, 36 (2009), 211-249. doi: 10.1007/s00526-009-0227-4. [32] H. L. Royden, Comparison theorems for the matrix Riccati equation, Comm. Pure Appl. Math., 41 (1988), 739-746. doi: 10.1002/cpa.3160410512. [33] T. Sakai, Riemannian Geometry, Translations of Mathematical Monographs, 149. American Mathematical Society, Providence, RI, 1996. [34] K. T. Sturm, On the geometry of metric measure spaces, Acta Math., 196 (2006), 65-131. doi: 10.1007/s11511-006-0002-8. [35] K. T. Sturm, On the geometry of metric measure spaces II, Acta Math., 196 (2006), 133-177. doi: 10.1007/s11511-006-0003-7. [36] N. Tanaka, A Differential Geometric Study on Strongly Pseudo-Convex Manifold, Kinokunya Book Store Co., Ltd., Kyoto, 1975. [37] S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc., 314 (1989), 349-379. doi: 10.1090/S0002-9947-1989-1000553-9. [38] C. Villani, Optimal Transport. Old and new, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. [39] J. Wang, Sub-Riemannian Heat Kernels on Model Spaces and Curvature-Dimension Inequalities on Contact Manifolds, Ph.D. Dissertation, Purdue University, 2014. [40] S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geometry, 13 (1978), 25-41.
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