no. 2
Isoperimetric inequalities & volume comparison theorems on CR manifolds
Chanillo, Sagun1 ; Yang, Paul2
1 Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854
2 Department of Mathematics, Princeton University, Princeton, NJ 08544
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 2, pp. 279-307.

In this article we study the Jacobi equation associated with the geodesics in a pseudo-hermitian manifold wish vanishing Webster torsion. We develop integral geometric formula generalizing the well known Santalo formula in Riemannian geometry. As applications we obtain volume comparison results under suitable curvature assumptions as well as isoperimetric inequalities for domains in such manifolds.

Classification : 32V20, 32V05, 53C17, 53C21
Chanillo, Sagun 1 ; Yang, Paul 2

1 Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854
2 Department of Mathematics, Princeton University, Princeton, NJ 08544 
@article{ASNSP_2009_5_8_2_279_0,
     author = {Chanillo, Sagun and Yang, Paul},
     title = {Isoperimetric inequalities & volume comparison theorems on {CR} manifolds},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {279--307},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {2},
     year = {2009},
     mrnumber = {2548248},
     zbl = {1176.32014},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2009_5_8_2_279_0/}
}
TY  - JOUR
AU  - Chanillo, Sagun
AU  - Yang, Paul
TI  - Isoperimetric inequalities & volume comparison theorems on CR manifolds
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2009
SP  - 279
EP  - 307
VL  - 8
IS  - 2
PB  - Scuola Normale Superiore, Pisa
UR  - http://www.numdam.org/item/ASNSP_2009_5_8_2_279_0/
LA  - en
ID  - ASNSP_2009_5_8_2_279_0
ER  - 
%0 Journal Article
%A Chanillo, Sagun
%A Yang, Paul
%T Isoperimetric inequalities & volume comparison theorems on CR manifolds
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2009
%P 279-307
%V 8
%N 2
%I Scuola Normale Superiore, Pisa
%U http://www.numdam.org/item/ASNSP_2009_5_8_2_279_0/
%G en
%F ASNSP_2009_5_8_2_279_0
Chanillo, Sagun; Yang, Paul. Isoperimetric inequalities & volume comparison theorems on CR manifolds. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 2, pp. 279-307. http://www.numdam.org/item/ASNSP_2009_5_8_2_279_0/

[1] A. Bellaiche, The tangent space in Sub-Riemannian Geometry, in Sub-Riemannian Geometry, Progr. Math., Birkhäuser, Basel 144 (1996), 1–78. | MR | Zbl

[2] J.-H. Cheng, J.-F. Hwang, A. Malchiodi and P. Yang, Minimal surfaces in pseudohermitian geometry, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (2005), 129–177. | EuDML | Numdam | MR | Zbl

[3] C. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Ecole Norm. Sup. 13 (1980), 419–435. | EuDML | Numdam | MR | Zbl

[4] S. S. Chern and R. S. Hamilton, On Riemannian metrics adapted to three dimensional contact manifolds, with an appendix by A. Weinstein, Lecture Notes in Math., Vol. 1111, Springer Berlin, 1985, 279–308. | MR

[5] P. Pansu Une inegalite isoperimetrique sur le group de Heisenberg, C.R. Acad. Sc. Paris 295 (1982), 127–130. | MR | Zbl

[6] M. Rumin, Forms differentielles sur les varíetés de contact, J. Differential Geom. 39 (1994), 281–330. | MR | Zbl

[7] N. Tanaka, “A Differential Geometric Study on Strongly Pseudo-convex Manifolds”, Kinokuniya, Tokyo, 1975. | MR | Zbl

[8] F. Treves “Introduction to Pseudo-differential and Fourier Integral Operators”, Vol. 1, Plenum Press. | MR

[9] N. Varopoulos; Sobolev inequalities on Lie groups and symmetric spaces, J. Funct. Anal. 86 (1989), 19–40. | MR | Zbl

[10] S. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25–41. | MR | Zbl