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 & Continuous Dynamical Systems - A, 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.  doi: 10.1007/BF02463256.  Google Scholar

[2]

A. Agrachev and P. Lee, Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds,, Math. Ann., 360 (2014), 209.  doi: 10.1007/s00208-014-1034-6.  Google Scholar

[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.  doi: 10.1007/s12220-013-9437-2.  Google Scholar

[4]

A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint,, Encyclopaedia of Mathematical Sciences, (2004).  doi: 10.1007/978-3-662-06404-7.  Google Scholar

[5]

A. Agrachev and I. Zelenko, Geometry of Jacobi curves. I,, J. Dynamical and Control systems, 8 (2002), 93.  doi: 10.1023/A:1013904801414.  Google Scholar

[6]

D. Bakry and M. Émery, Diffusions hypercontractives., in Séminaire de probabilités, 1123 (1985), 177.  doi: 10.1007/BFb0075847.  Google Scholar

[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.  doi: 10.1007/s00208-013-0961-y.  Google Scholar

[8]

F. Baudoin and N. Garofalo, Generalized Bochner formulas and Ricci lower bounds for sub-Riemannian manifolds of rank two,, preprint, ().   Google Scholar

[9]

F. Baudoin and N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries,, preprint, ().   Google Scholar

[10]

D. E. Blair, Contact Manifolds in Riemannian Geometry,, Lecture Notes in Mathematics, 509 ().   Google Scholar

[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.  doi: 10.1016/j.anihpc.2007.07.005.  Google Scholar

[12]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,, Birkhhäuser, (2004).   Google Scholar

[13]

S. Chanillo and P. Yang, Isoperimetric inequalities & volume comparison theorems on CR manifolds,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 279.   Google Scholar

[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.  doi: 10.1007/s00039-001-8227-3.  Google Scholar

[15]

D. B. A. Epstein, Complex hyperbolic geometry,, in Analytical and Geometric Aspects of Hyperbolic Space (ed. D.B.A. Epstein), 111 (1987), 93.   Google Scholar

[16]

A. Figalli and L. Rifford, Mass Transportation on sub-Riemannian Manifolds,, Geom. Funct. Anal., 20 (2010), 124.  doi: 10.1007/s00039-010-0053-z.  Google Scholar

[17]

K. Hughen, The Geometry of Sub-Riemannian Three-Manifolds,, Ph.D. Dissertation, (1995).   Google Scholar

[18]

D. Jerison, The Poincaŕe inequality for vector fields satisfying the Hörmander condition,, Duke Math. J., 53 (1986), 503.  doi: 10.1215/S0012-7094-86-05329-9.  Google Scholar

[19]

N. Juillet, Geometric inequalities and generalized ricci bounds in the heisenberg group,, Int. Math. Res. Not. IMRN, (2009), 2347.  doi: 10.1093/imrn/rnp019.  Google Scholar

[20]

P. W. Y. Lee, Displacement interpolations from a Hamiltonian point of view,, J. Func. Anal., 265 (2013), 3163.  doi: 10.1016/j.jfa.2013.08.022.  Google Scholar

[21]

P. W. Y. Lee and C. Li, Bishop and Laplacian comparison theorems on Sasakian manifolds, preprint,, , (2013).   Google Scholar

[22]

P. W. Y. Lee, C. Li and I. Zelenko, Measure contraction properties of contact sub-Riemannian manifolds with symmetry, preprint,, , ().   Google Scholar

[23]

J. J. Levin, On the matrix Riccati equation,, Proc. Amer. Math. Soc., 10 (1959), 519.  doi: 10.1090/S0002-9939-1959-0108628-X.  Google Scholar

[24]

C. Li and I. Zelenko, Parametrized curves in Lagrange Grassmannians,, C.R. Acad. Sci. Paris, 345 (2007), 647.  doi: 10.1016/j.crma.2007.10.034.  Google Scholar

[25]

C. Li and I. Zelenko, Differential geometry of curves in Lagrange Grassmannians with given Young diagram,, Differ. Geom. Appl., 27 (2009), 723.  doi: 10.1016/j.difgeo.2009.07.002.  Google Scholar

[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.  doi: 10.1016/j.geomphys.2010.12.009.  Google Scholar

[27]

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport,, Ann. of Math. (2), 169 (2009), 903.  doi: 10.4007/annals.2009.169.903.  Google Scholar

[28]

J. Lott and C. Villani, Weak curvature conditions and functional inequalities,, J. Funct. Anal., 245 (2007), 311.  doi: 10.1016/j.jfa.2006.10.018.  Google Scholar

[29]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications,, Mathematical Surveys and Monographs, (2002).   Google Scholar

[30]

S. Ohta, On the measure contraction property of metric measure spaces,, Comment. Math. Helv., 82 (2007), 805.  doi: 10.4171/CMH/110.  Google Scholar

[31]

S. Ohta, Finsler interpolation inequalities,, Calc. Var. Partial Differential Equations, 36 (2009), 211.  doi: 10.1007/s00526-009-0227-4.  Google Scholar

[32]

H. L. Royden, Comparison theorems for the matrix Riccati equation,, Comm. Pure Appl. Math., 41 (1988), 739.  doi: 10.1002/cpa.3160410512.  Google Scholar

[33]

T. Sakai, Riemannian Geometry,, Translations of Mathematical Monographs, (1996).   Google Scholar

[34]

K. T. Sturm, On the geometry of metric measure spaces,, Acta Math., 196 (2006), 65.  doi: 10.1007/s11511-006-0002-8.  Google Scholar

[35]

K. T. Sturm, On the geometry of metric measure spaces II,, Acta Math., 196 (2006), 133.  doi: 10.1007/s11511-006-0003-7.  Google Scholar

[36]

N. Tanaka, A Differential Geometric Study on Strongly Pseudo-Convex Manifold,, Kinokunya Book Store Co., (1975).   Google Scholar

[37]

S. Tanno, Variational problems on contact Riemannian manifolds,, Trans. Amer. Math. Soc., 314 (1989), 349.  doi: 10.1090/S0002-9947-1989-1000553-9.  Google Scholar

[38]

C. Villani, Optimal Transport. Old and new,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

[39]

J. Wang, Sub-Riemannian Heat Kernels on Model Spaces and Curvature-Dimension Inequalities on Contact Manifolds,, Ph.D. Dissertation, (2014).   Google Scholar

[40]

S. M. Webster, Pseudo-Hermitian structures on a real hypersurface,, J. Differential Geometry, 13 (1978), 25.   Google Scholar

show all references

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.  doi: 10.1007/BF02463256.  Google Scholar

[2]

A. Agrachev and P. Lee, Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds,, Math. Ann., 360 (2014), 209.  doi: 10.1007/s00208-014-1034-6.  Google Scholar

[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.  doi: 10.1007/s12220-013-9437-2.  Google Scholar

[4]

A. Agrachev and Y. Sachkov, Control Theory from the Geometric Viewpoint,, Encyclopaedia of Mathematical Sciences, (2004).  doi: 10.1007/978-3-662-06404-7.  Google Scholar

[5]

A. Agrachev and I. Zelenko, Geometry of Jacobi curves. I,, J. Dynamical and Control systems, 8 (2002), 93.  doi: 10.1023/A:1013904801414.  Google Scholar

[6]

D. Bakry and M. Émery, Diffusions hypercontractives., in Séminaire de probabilités, 1123 (1985), 177.  doi: 10.1007/BFb0075847.  Google Scholar

[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.  doi: 10.1007/s00208-013-0961-y.  Google Scholar

[8]

F. Baudoin and N. Garofalo, Generalized Bochner formulas and Ricci lower bounds for sub-Riemannian manifolds of rank two,, preprint, ().   Google Scholar

[9]

F. Baudoin and N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries,, preprint, ().   Google Scholar

[10]

D. E. Blair, Contact Manifolds in Riemannian Geometry,, Lecture Notes in Mathematics, 509 ().   Google Scholar

[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.  doi: 10.1016/j.anihpc.2007.07.005.  Google Scholar

[12]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,, Birkhhäuser, (2004).   Google Scholar

[13]

S. Chanillo and P. Yang, Isoperimetric inequalities & volume comparison theorems on CR manifolds,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 279.   Google Scholar

[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.  doi: 10.1007/s00039-001-8227-3.  Google Scholar

[15]

D. B. A. Epstein, Complex hyperbolic geometry,, in Analytical and Geometric Aspects of Hyperbolic Space (ed. D.B.A. Epstein), 111 (1987), 93.   Google Scholar

[16]

A. Figalli and L. Rifford, Mass Transportation on sub-Riemannian Manifolds,, Geom. Funct. Anal., 20 (2010), 124.  doi: 10.1007/s00039-010-0053-z.  Google Scholar

[17]

K. Hughen, The Geometry of Sub-Riemannian Three-Manifolds,, Ph.D. Dissertation, (1995).   Google Scholar

[18]

D. Jerison, The Poincaŕe inequality for vector fields satisfying the Hörmander condition,, Duke Math. J., 53 (1986), 503.  doi: 10.1215/S0012-7094-86-05329-9.  Google Scholar

[19]

N. Juillet, Geometric inequalities and generalized ricci bounds in the heisenberg group,, Int. Math. Res. Not. IMRN, (2009), 2347.  doi: 10.1093/imrn/rnp019.  Google Scholar

[20]

P. W. Y. Lee, Displacement interpolations from a Hamiltonian point of view,, J. Func. Anal., 265 (2013), 3163.  doi: 10.1016/j.jfa.2013.08.022.  Google Scholar

[21]

P. W. Y. Lee and C. Li, Bishop and Laplacian comparison theorems on Sasakian manifolds, preprint,, , (2013).   Google Scholar

[22]

P. W. Y. Lee, C. Li and I. Zelenko, Measure contraction properties of contact sub-Riemannian manifolds with symmetry, preprint,, , ().   Google Scholar

[23]

J. J. Levin, On the matrix Riccati equation,, Proc. Amer. Math. Soc., 10 (1959), 519.  doi: 10.1090/S0002-9939-1959-0108628-X.  Google Scholar

[24]

C. Li and I. Zelenko, Parametrized curves in Lagrange Grassmannians,, C.R. Acad. Sci. Paris, 345 (2007), 647.  doi: 10.1016/j.crma.2007.10.034.  Google Scholar

[25]

C. Li and I. Zelenko, Differential geometry of curves in Lagrange Grassmannians with given Young diagram,, Differ. Geom. Appl., 27 (2009), 723.  doi: 10.1016/j.difgeo.2009.07.002.  Google Scholar

[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.  doi: 10.1016/j.geomphys.2010.12.009.  Google Scholar

[27]

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport,, Ann. of Math. (2), 169 (2009), 903.  doi: 10.4007/annals.2009.169.903.  Google Scholar

[28]

J. Lott and C. Villani, Weak curvature conditions and functional inequalities,, J. Funct. Anal., 245 (2007), 311.  doi: 10.1016/j.jfa.2006.10.018.  Google Scholar

[29]

R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications,, Mathematical Surveys and Monographs, (2002).   Google Scholar

[30]

S. Ohta, On the measure contraction property of metric measure spaces,, Comment. Math. Helv., 82 (2007), 805.  doi: 10.4171/CMH/110.  Google Scholar

[31]

S. Ohta, Finsler interpolation inequalities,, Calc. Var. Partial Differential Equations, 36 (2009), 211.  doi: 10.1007/s00526-009-0227-4.  Google Scholar

[32]

H. L. Royden, Comparison theorems for the matrix Riccati equation,, Comm. Pure Appl. Math., 41 (1988), 739.  doi: 10.1002/cpa.3160410512.  Google Scholar

[33]

T. Sakai, Riemannian Geometry,, Translations of Mathematical Monographs, (1996).   Google Scholar

[34]

K. T. Sturm, On the geometry of metric measure spaces,, Acta Math., 196 (2006), 65.  doi: 10.1007/s11511-006-0002-8.  Google Scholar

[35]

K. T. Sturm, On the geometry of metric measure spaces II,, Acta Math., 196 (2006), 133.  doi: 10.1007/s11511-006-0003-7.  Google Scholar

[36]

N. Tanaka, A Differential Geometric Study on Strongly Pseudo-Convex Manifold,, Kinokunya Book Store Co., (1975).   Google Scholar

[37]

S. Tanno, Variational problems on contact Riemannian manifolds,, Trans. Amer. Math. Soc., 314 (1989), 349.  doi: 10.1090/S0002-9947-1989-1000553-9.  Google Scholar

[38]

C. Villani, Optimal Transport. Old and new,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (2009).  doi: 10.1007/978-3-540-71050-9.  Google Scholar

[39]

J. Wang, Sub-Riemannian Heat Kernels on Model Spaces and Curvature-Dimension Inequalities on Contact Manifolds,, Ph.D. Dissertation, (2014).   Google Scholar

[40]

S. M. Webster, Pseudo-Hermitian structures on a real hypersurface,, J. Differential Geometry, 13 (1978), 25.   Google Scholar

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