Gromov-Hausdorff limits of compact Heisenberg manifolds with sub-Riemannian metrics

[1] Agrachev A., Barilari D., Boscain U.: A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge University Press, Cambridge (2019).

[2] Agrachev A., Barilari D., Boscain U.: On the Hausdorff volume in sub- Riemannian geometry, Calc. Var. Partial Differential Equations, 43(3- 4),355388 (2012).

[3] Agrachev A. A., Gamkrelidze R. V.: Feedback-invariant optimal control theory and differential geometry. I. Regular extremals. J. Dynam. Control Systems, 3(3) 343389 (1997). 1997.

[4] Agrachev A. A., Zelenko I.: Geometry of Jacobi curves. I. J. Dynam. Con- trol Systems, 8(1) 93140 (2002).

[5] Agrachev A. A., Sachkov Y.: Control Theory from the Geometric View- point, volume 87 of Encyclopaedia of Mathematical Sciences, Springer- Verlag, New York-Heidelberg-Berlin (2004).

[6] Agrachev A. A., Lee P. W. Y.: Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds. Math. Ann., 360(1- 2), 209253 (2014).

[7] Ambrosio N., Gigli L., Saver´e G., Bakry–E´mery curvature dimension con- dition and Riemannian Ricci curvature bound, Ann. Prob., 43(1), 339-404 (2015).

[8] Ambrosio N., Gigli L., Saver´e G.: Nonlinear diffusion equations and cur- vature conditions in metric measure spaces, Mem. Amer. Math. Soc., in press.

[9] Bakry D.: L’hypercontractivit´e et son utilisation en th´eorie des semi- groupes, Ecole d’E´t´e de Probabilities de St-Flour, Lecture notes in Math (1994).

[10] Bakry D., Gentil I., Ledoux M.: Analysis and geometry of Markov diffusion operator, Grundlehren der Mathematischen Wissenschaften 348. Springer, Cham (2014).

[11] Barilari D., Rizzi L.: Comparison theorems for conjugate points in sub- Riemannian geometry, ESAIM Control Optim. Calc. Var. 22 no. 2, 439472 (2016).

[12] Barilari D., Rizzi L.: Sharp measure contraction property for generalized H-type Carnot groups, Commun. Contemp. Math., 20(6), 1750081, 24pp (2018).

[13] Baudoin F., Bonnefont M., Garofalo N., Munive I.: Volume and distance comparison theorems for sub-Riemannian manifolds, J. Funct. Anal., 267 no.7, 2005-2027 (2014).

[14] Baudoin F., Garofalo N., Curvature-dimension inequalities and Ricci lower bounds for subRiemannian manifolds with transverse symmetries, J. Eur. Math. Soc., 19(1), 151-219 (2017).

[15] Bettiol R., Derdzinski A., Piccione P.: Teichmu¨ller theory and collapse of flat manifolds. Ann. Mat. Pura Appl., 197(4), 12471268 (2018).

[16] Bhatia R.: Matrix analysis, Springer New York (1996).

[17] Boldt S.: Relatively compact sets of Heisenberg manifolds, preprint, arXiv: 2008.05328.

[18] Boldt S.: The height of compact nonsingular Heisenberg-like Nilmanifolds, Dessertation, Doi: 10.18452/18924 (2018).

[19] Cassels, J.W.S.: An introduction to the geometry of numbers. Berlin, Hei- delberg, New York:Springer (1971).

[20] Cheeger J., Gromoll D.: The splitting theorem for manifolds of nonnegative Ricci curvature, Journal of Differential Geometry, 6(1), 119128 (1971).

[21] Cheeger J., Colding T. H.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. of Math. (2)144, 1 189237 (1996).

[22] On the structure of spaces with Ricci curvature bounded below. I. J. Dif- ferential Geom. 46 3, 406480 (1997).

[23] Cheeger J., Colding T. H.: On the structure of spaces with Ricci curvature bounded below. II. J. Differential Geom. 54, 1, 1335 (2000).

[24] Cheeger J., Colding T. H.: On the structure of spaces with Ricci curvature bounded below. III, J. Differential Geom. 54, 1, 3774 (2000).

[25] Eberlein P.: Geometry of 2-step nilpotent Lie groups with a left invariant metric, Ann. sci. Ec. Norm. Sup., 27, 611-660 (1994).

[26] Erbar M., Kuwada K., Sturm K. T.: On the Equivalence of the Entropic Curvature Dimension Condition and Bochner ’s Inequality on Metric Mea- sure Space, Invent. Math., 201, no. 3, 9931071 (2015).

[27] Fukaya K., Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math. 87,517-547 (1987).

[28] Gordon C. S., Wilson E. N.: The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J.,33, 253271 (1986).

[29] Gordon C. S., Wilson E. N.: Isometry groups of Riemannian solvmanifolds, Trans. Amer. Math. Soc. 307, no. 1, 245269, (1988).

[30] Gornet R., Mast M. B.: Length minimizing geodesics and the length spec- trum of Riemannian two-step nilmanifolds, J. Geom. Anal., 13(1), 107143, (2003).

[31] Gromov M.: Structures m´etriques pour les vari´et´es riemanniannes, Textes Math´ematiques, 1, Paris, 1-120, (1981).

[32] Juillet N.: Geometric inequalities and generalized Ricci bounds in the Heisenberg group, Int. Math. Res. Not. IMRN, (13), 23472373 (2009).

[33] Kivioja V., Le Donne E.: Isometries of nilpotent metric groups. J. c. poly- tech. Math., 4, 473482 (2017).

[34] Lott L., Villani C.: Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169, 903991, (2009).

[35] Mahler K.: On lattice points in n-dimensional star bodies. I. Existence the- orems, Proceedings of the Royal Society of London. Series A: Mathematical, Phisical and Engineering Sciences, 187, 151-187, (1946).

[36] Montgomery R.: Abnormal minimizers, SIAM J. Control Optim., 32(6), 1605-1620 (1994).

[37] Montgomery R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society (Mathematical Surveys and Monographs Vol. 91) (2000).

[38] Myers S. B.: Riemannian manifolds with positive mean curvature, Duke Mathematical Journal, 8(2), 401404 (1941).

[39] Ohta S.: On the measure contraction property of metric measure spaces, Comment. Math. Helv. 82, no. 4, 805828 (2007).

[40] Rifford L.: Ricci curvatures in Carnot groups. Math. Control Relat. Fields, 3(4), 467487, (2013).

[41] Rizzi L.: Measure contraction properties of Carnot groups, Calc. Var. Par- tial Differential Equations, 55(3):Art. 60, 20pp, (2016).

[42] Sormani C., Wei G.: Hausdorff Convergence and Universal Covers, Trans. Amer. Math. Soc., 353, 35853602 (2001).

[43] Sturm T.: On the geometry of metric measure spaces. I, Acta Math., 196, 65131 (2006).

[44] Sturm T.: On the geometry of metric measure spaces. II, Acta Math., 196, 133177 (2006).

[45] Tashiro K.: Precompactness theorem for compact Heisenberg mani- folds with sub-Riemannian metrics and the Gromov–Hausdorff topology, preprint, arXiv:2004.09407.

[46] Terras A.: Harmonic analysis on symmetric spaces and applications. II, Springer New York (1988).

[47] Wilson E. N.: Isometry groups on homogeneous nilmanifolds, Geom. Ded- icata 12, no. 3, 337346 (1982).

[48] Wolf J.A.: Local and global equivalence for flat affine manifolds with par- allel geometric structures, Geom. Dedicata, 2, 127132 (1973).

[49] Zelenko I., Li C.: Differential geometry of curves in Lagrange Grassman- nians with given Young diagram. Differential Geom. Appl., 27(6), 723742 (2009).