Abstract
By studying the group of rigid motions, PSH(1), in the 3D-Heisenberg group H1,we define a density and a measure in the set of horizontal lines. We show that the volume of a convex domain D ⊂ H1 is equal to the integral of the length of chords of all horizontal lines intersecting D. As in classical integral geometry, we also define the kinematic density for PSH(1) and show that the measure of all segments with length l intersecting a convex domain D ⊂ H1 can be represented by the p-area of the boundary ∂D, the volume of D, and 2l. Both results show the relationship between geometric probability and the natural geometric quantity in [10] derived by using variational methods. The probability that a line segment be contained in a convex domain is obtained as an application of our results.
References
[1] A. Baddeley, A Fourth Note on Recent Research in Geometrical Probability, Adv. in Appl. Probability, Vol. 9, No. 4 (1977), pp. 824-860. Search in Google Scholar
[2] M. Beals, C. Fefferman, R. Grossman, Strictly Pseudoconvex Domains in Cn, Bull. Amer.Math. Soc. (N.S.) V(8), No.2, (1983), 125-322. 10.1090/S0273-0979-1983-15087-5Search in Google Scholar
[3] W. Blaschke, Integralgeometrie 2:Zu Ergebnissen von M. W. Crofton, Bull. Math. Soc. Roumaine Sci., 37, 1935, 3-7. Search in Google Scholar
[4] O. Calin, D.C. Chang, P. Greiner, Geometric Analysis on the Heisenberg Group and Its Generalizations, AMS Studies in Adv. Math., 40, 2007. 10.1090/amsip/040Search in Google Scholar
[5] L. Capogna, D. Danielli, N. Garofalo, An isoperimetric inequality and the geometric Sobolev embedding for vector fields, Math. Res. Lett. 1 (2), (1994), 203-215. 10.4310/CAG.1994.v2.n2.a2Search in Google Scholar
[6] L. Capogna, D. Danielli, S.D. Pauls, J. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progress in Mathematics, 2007. Search in Google Scholar
[7] S. Chanillo, P. Yang, Isoperimetric inequality & volume comparison theorems on CR manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009), no. 2, 279-307. Search in Google Scholar
[8] S.S. Chern, Lectures on integral geometry, notes by Hsiao, H.C., Academia Sinica, National Taiwan University and National Tsinghua University, (1965). Search in Google Scholar
[9] S.S. Chern, J.K. Moser, Real Hyperserufaces in Complex Manifolds, Acta Math. 133, (1974), 219-271. 10.1007/BF02392146Search in Google Scholar
[10] J.H.Cheng, J.F.Hwang, A.Malchiodi, P.Yang,Minimal surfaces in pseudohermitian geometry, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Vol. IV(2005), 129-177. 10.2422/2036-2145.2005.1.05Search in Google Scholar
[11] J.H. Cheng, J.F. Hwang, A. Malchiodi, P. Yang, A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group, J. Reine Angew. Math., 671, (2012), 131–198. 10.1515/CRELLE.2011.159Search in Google Scholar
[12] J.H. Cheng, H.L. Chiu, J.F. Hwang, P. Yang, Umbilic hypersurfaces of constant sigma-k curvature in the Heisenberg group, Calc. Var. Partial Differential Equations 55 (2016), no. 3, Art.66, 25 pages. 10.1007/s00526-016-1006-7Search in Google Scholar
[13] J.H. Cheng, J.F. Hwang, P. Yang, Regularity of C1 smooth surfaces with prescribed p-mean curvature in the Heisenberg group, Math. Ann., 344, (2009), no. 1, 1–35. Search in Google Scholar
[14] J.H. Cheng, J.F. Hwang, P. Yang, Existence and uniqueness for p-area minimizers in the Heisenberg group, Math. Ann., 337, (2007), no. 2, 253–293. Search in Google Scholar
[15] H.L.Chiu, Y.C. Huang, S.H.Lai, The application of the moving frame method to Integral Geometry in Heisenberg group, Preprint, http://arxiv.org/abs/1509.00950. Search in Google Scholar
[16] B. Franchi, R. Serapioni, F. S. Cassano, Rectifiability and perimeter in the Heisenberg group, Math. Ann. 321 (3), (2001), 479-531. 10.1007/s002080100228Search in Google Scholar
[17] R. Howard, The Kinematic Formula in Riemannian Homogeneous Spaces, Mem. Amer. Math. Soc. No. 509, V106, 1993. 10.1090/memo/0509Search in Google Scholar
[18] G.P. Leonardi, S.Masnou, On the isoperimetric problemin the Heisenberg group Hn, Annali diMatematica Pura ed Applicata, V184,4, (2005), pp 533-553. 10.1007/s10231-004-0127-3Search in Google Scholar
[19] D.V. Little, A Third Note on Recent Research in Geometrical Probability, Adv. in Appl. Probability, Vol. 6, No. 1 (1974), pp. 103-130. Search in Google Scholar
[20] A. Malchiodi, Minimal surfaces in three dimensional pseudo-hermitian manifolds, Lect. Notes Semin. Interdiscip. Mat., 6, Semin. Interdiscip. Mat. (S.I.M.), Potenza, 2007. Search in Google Scholar
[21] P.A.P. Moran, A Note on Recent Research in Geometrical Probability, J. of Appl. Probability, Vol. 3, No. 2 (1966), pp. 453-463. Search in Google Scholar
[22] P.A.P. Moran, A Second Note on Recent Research in Geometrical Probability, Adv. in Appl. Probability, Vol. 1, No. 1 (1969), pp. 73-89. Search in Google Scholar
[23] P. Pansu, Une ingalit isoprimtrique sur le groupe de Heisenberg, C. R. Acad. Sci. Paris Sr. IMath. 295 (1982), no.. 2, 127-130. Search in Google Scholar
[24] M.Ritoré, C. Rosales, Area-stationary surfaces in the Heisenberg group H1, Adv. in Math., 219 (2008), pp.633-671. Search in Google Scholar
[25] M. Ritoré, C. Rosales, Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group Hn, J. Geom. Anal., 16(4), (2006), 703-720. 10.1007/BF02922137Search in Google Scholar
[26] L. A. Santaló, Integral Geometry and Geometric Probability, Cambridge Mathematical Library, 2nd edition, 2004. 10.1017/CBO9780511617331Search in Google Scholar
[27] R.Schneider, W. Weil, Stochastic and Integral Geometry, Springer, 2008. 10.1007/978-3-540-78859-1Search in Google Scholar
[28] N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds, Khokuniya, Yokyo, 1975. 10.3792/pja/1195518896Search in Google Scholar
[29] S. Webster, Pseudo-hermitian structures of a real hyper-surfaces, J. Diff. Geo., 13(1978), 25-41. Search in Google Scholar
[30] P. Yang, Minimal Surfaces in CR geometry, Geometric analysis and PDEs, 253–273, Lecture Notes in Math., 1977, Springer, 2009, 10.1007/978-3-642-01674-5_6Search in Google Scholar
[31] P. Yang, Pseudo-Hermitian Geometry in 3-D, Milan J. Math. V(79), (2011), 181-191. 10.1007/s00032-011-0156-5Search in Google Scholar
[32] G.Y. Zhang, A lecture on Integral Geometry, Proceedings of the fourteenth Internatioanl Workshop on Diff. Geom. 14(2010), 13-30. Search in Google Scholar
[33] J.Z. Zhou, Kinematic formulas in Riemannian spaces, Proceedings of the tenth Internatioanl Workshop on Diff. Geom. 10(2006), 39-55. Search in Google Scholar
[34] J.Z. Zhou, Kinematic Formulas for Mean Curvature Powers of Hypersurfaces and Hadwiger’s Theorem in R2n, Trans. Amer. Math. Soci., v345, No1, 1994, pp. 243-262. 10.1090/S0002-9947-1994-1250829-7Search in Google Scholar
[35] J.Z. Zhou, M. Li, Kinematic Formulas of Total Mean Curvatures for Hypersurfaces, Chinese Ann.Math. 37B(1), 2016, 137–148. 10.1007/s11401-015-0952-2Search in Google Scholar
© 2016 Yen-Chang Huang
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
