Nonlinear equations on Carnot groups and curvature problems for CR manifolds

Lanconelli, Ermanno

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 14 (2003), p. 227-238 / Harvested from Biblioteca Digitale Italiana di Matematica

Access to full text
Full (PDF)
Full (DjVu)

Résumé

We give a short overview of sub-Laplacians on Carnot groups starting from a result by Caccioppoli dated 1934. Then we show that sub-Laplacians on Carnot groups of step one arise in studying curvature problems for $C R$ manifolds. We restrict our presentation to the cases of the Webster-Tanaka curvature problem for the $C R$ sphere and of the Levi-curvature equation for strictly pseudoconvex functions.

Publié le : 2003-09-01

MR 2064269 | Zbl 1225.35059

@article{RLIN_2003_9_14_3_227_0,
     author = {Ermanno Lanconelli},
     title = {Nonlinear equations on Carnot groups and curvature problems for CR manifolds},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {14},
     year = {2003},
     pages = {227-238},
     zbl = {1225.35059},
     mrnumber = {2064269},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_2003_9_14_3_227_0}
}

Lanconelli, Ermanno. Nonlinear equations on Carnot groups and curvature problems for CR manifolds. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 14 (2003) pp. 227-238. http://gdmltest.u-ga.fr/item/RLIN_2003_9_14_3_227_0/

Bibliographie

[1] Ambrosetti, A. - Badiale, M., Homoclinics: Poincaré-Melnikov type results via a variational approach. Ann. Inst. Henry Poincaré, Analyse non Linéaire, 15, 1998, 233-252. | MR 1614571 | Zbl 1004.37043

[2] Bedford, E. - Gaveau, B., Hypersurfaces with bounded Levi form. Ind. Univ. Math. J., 27, 1978, 867-877. | MR 499287 | Zbl 0365.32011

[3] Caccioppoli, R., Sui teoremi di esistenza di Riemann. Rend. Acc. Sc. Napoli, 4, 1934, 49-54. | JFM 60.0310.06 | Zbl 0010.16804

[4] Citti, G., $C^{\infty}$ -regularity of solutions of the Levi equation. Ann. Inst. Henry Poincaré, Analyse non Linéaire, 15, 1998, 517-534. | MR 1632929 | Zbl 0921.35033

[5] Citti, G. - Lanconelli, E. - Montanari, A., Smoothness of Lipschitz-continuous graphs with non-vanishing Levi curvature. Acta Math., 118, 2002, 87-128. | MR 1947459 | Zbl 1030.35084

[6] Folland, G.B., Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Math., 13, 1975, 161-207. | MR 494315 | Zbl 0312.35026

[7] Gallardo, L., Capacités, mouvement Brownien et problème de l’épine de Lebesgue sur le groups de Lie nilpotents. Proc. VII Oberwolfach Conference on Probability measures on groups. Lectures Notes in Math., 928, Springer-Verlag, Berlin-New York1982, 96-120. | MR 669065 | Zbl 0483.60072

[8] Gamara, N., The $C R$ Yamabe conjecture: the case $n = 1$ . J. Eur. Math. Soc., 3, 2001, 105-137. | MR 1831872 | Zbl 0988.53013

[9] Gamara, N. - Yacoub, R., $C R$ Yamabe conjecture: the conformally flat case. Pacific J. of Math., 201, 2001, 121-175. | MR 1867895 | Zbl 1054.32020

[10] Hebey, E., Changement de métriques conformes sur la sphére. Le problème de Nirenberg. Bull. Sc. Math., 114, 1990, 215-242. | MR 1056162 | Zbl 0713.53023

[11] Hormander, L., Hypoelliptic second order differential equations. Acta Math., 119, 1967, 147-171. | MR 222474 | Zbl 0156.10701

[12] Jerison, D. - Lee, J.M., Intrinsic $C R$ normal coordinates and the $C R$ Yamabe problem. J. Amer. Math. Soc., 1, 1988, 1-13. | MR 982177 | Zbl 0634.32016

[13] Jerison, D. - Lee, J.M., Extremals of the Sobolev inequality on the Heisenberg group and the $C R$ Yamabe problem. J. of Diff. Geom., 29, 1989, 303-343. | MR 924699 | Zbl 0671.32016

[14] Kohn, J.J. - Nirenberg, L., A pseudoconvex domain not admitting a holomorphic support function. Math. Ann., 201, 1973, 265-268. | MR 330513 | Zbl 0248.32013

[15] Kolmogorov, A., Zufällige Bewegungen. Ann. of Math., 35, 1934, 116-117. | JFM 60.1159.01 | MR 1503147 | Zbl 0008.39906

[16] Lascialfari, F. - Montanari, A., Smooth regularity for solutions of the Levi Monge-Ampère equations. Rend. Mat. Acc. Lincei, s. 9, v. 12, 2001, 1633-1664. | MR 1898454 | Zbl 1019.35056

[17] Levi, E.E., Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse. Ann. di Mat. Pura ed Appl., 17, 1910, 61-87. | JFM 41.0487.01

[18] Levi, E.E., Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiere del campo di esistenza di una funzione analitica di due variabili complesse. Ann. di Mat. Pura ed Appl., 18, 1911, 69-79. | JFM 42.0449.02

[19] Montanari, A. - Lanconelli, E., Strong comparison principle for symmetric functions in the eigenvalues of the Levi form. Preprint.

[20] Malchiodi, A. - Uguzzoni, F., A perturbation result for the Webster scalar curvature problem on the $C R$ sphere. J. Math. Pures Appl., 81, 2002, 983-997. | MR 1946912 | Zbl 1042.53025

[21] Rothschild, L.P. - Stein, E.M., Hypoelliptic differential operators and nilpotent groups. Acta Math., 137, 1976, 247-320. | MR 436223 | Zbl 0346.35030

[22] Slodkowski, Z. - Tomassini, G., Weak solutions for the Levi equation and envelope of holomorphy. J. Funct. Anal., 101, 1991, 392-407. | MR 1136942 | Zbl 0744.35015

[23] Slodkowski, Z. - Tomassini, G., The Levi equation in higher dimension and relationships to the envelope of holomorphy. Amer. J. of Math., 116, 1994, 392-407. | MR 1269612 | Zbl 0802.35050

[24] Weyl, H., The method of the orthogonal projection in Potential Theory. Duke Math. J., 7, 1940, 411-444. | JFM 66.0444.01 | MR 3331