C. Enéralisés, A VALEURS COMPLEXES serait alors intéressant d'avoir une expression pour la courbure de Ricci associéè a une submersion semi-conforme (M 4 , g) ? (N 2 , h). classe d'applications semi-conformes de (M 4 , g) ? (N 2 , h) : les applications semi-conformes ? : (M 4 , g) ? (N 2 , h) qui se factorisent comme ? = ? ?? o` u ? : (M 4 , g) ? (P 3 , k) est un morphisme harmonique et ? : (P 3 , k) ? (N 2 , h) est semi'idées, P. Baird, A. Fardoun et S. Ouakkas ont montré que cela est vrai, mais la conditionétantconditionétant tellement compliquée, ils n'ont pas pu en déduire des conséquences harmonique généralisé, il nous semble toutàtoutà fait abordable, de chercherà chercherà caractériser ces morphismes d

C. Complexes,

C. Complexes,

R. Ababou, P. Baird, and J. Brossard, Polynômes semi-conformes et morphismes harmoniques, Math.Z, vol.231, pp.589-604, 1999.

P. Baird, Harmonic morphisms onto Riemann surfaces and generalized analytic functions, Ann. Inst. Fourier (Grenoble), vol.37, issue.1, pp.135-173, 1987.

P. Baird, Explicit constructions of Ricci solitons, Variational Problems in Differential Geometry, vol.394, pp.37-55, 2012.

P. Baird and J. Burel, Conformal subfoliations of prescribed geodesic curvature, Mathematica Scandinavica, vol.93, pp.221-239, 2003.

P. Baird and L. Danielo, Three-dimensional Ricci solitons which project to surfaces, vol.608, pp.65-91, 2007.

P. Baird and M. G. Eastwood, CR geometry and conformal foliations, vol.44, pp.73-90, 2013.

P. Baird and M. G. Eastwood, On functions with a conjugate, Ann. Inst. Fourier (Gernoble), vol.65, issue.1, pp.277-314, 2015.

P. Baird and J. Eells, A conservation law for harmonic maps, Geometry Symp, vol.894, p.125, 1980.

P. Baird, A. Fardoun, and S. Ouakkas, Conformal and semi-conformal biharmonic maps, Ann. Glob. Anal. Geom, vol.34, pp.403-414, 2008.

P. Baird and R. Pantilie, Harmonic morphisms on heaven spaces, Bull. London Math. Soc, vol.41, issue.2, pp.198-204, 2009.

P. Baird and J. C. Wood, Bernstein Theorems for harmonic morphisms from R 3 and S 3, Math. Ann, vol.280, pp.579-603, 1988.

P. Baird and J. C. Wood, Harmonic morphisms between Riemannian manifolds, New Series, vol.29, 2003.

A. Bernard, E. Campbell, and A. M. Davie, Brownian motion and generalized analytic and inner functions, Ann. Inst. Fourier (Grenoble), vol.29, issue.1, pp.207-228, 1979.

C. Böhm and R. Lafuente, Immortal homogeneous Ricci flows, 2017.

T. P. Branson, L. Fontana, and C. Morpurgo, Moser-Trudinger and Beckner-Onofri's inequalities on the CR sphere, Annals of Mathematics, vol.177, issue.1, pp.1-52, 2013.

M. Brelot, Lectures on potential theory, 1967.

R. L. Bryant, Harmonic morphisms with fibres of dimension one, Comm. Anal. Geom, vol.8, pp.219-265, 2000.

B. Chow and D. Knopf, The Ricci flow : an introduction, Mathematical Surveys and Monographs, vol.110, 2004.
DOI : 10.1090/surv/110

B. Chow, S. Chu, D. Glickenstein, C. Guenther, J. Isenberg et al., The Ricci Flow : Techniques and Applications, vol.135, 2007.

B. Chow, S. Chu, D. Glickenstein, C. Guenther, J. Isenberg et al., The Ricci Flow : Techniques and Applications, vol.144, 2008.
DOI : 10.1090/surv/163

C. Constantinescu and A. Cornea, Compactifications of harmonic spaces, Nagoya Math. J, vol.25, pp.1-57, 1965.
DOI : 10.1017/s0027763000011454

URL : https://www.cambridge.org/core/services/aop-cambridge-core/content/view/367B49D8130787844F83EA49BF5221F1/S0027763000011454a.pdf/div-class-title-compactifications-of-harmonic-spaces-div.pdf

L. Danileo, Structures Conformes, 2004.

L. Danielo, Construction de métriques d'EinsteinàEinsteinà partir de transformations biconformes, Ann. Fac. des Sciences de Toulouse, Sér, vol.6, issue.3, pp.553-588, 2006.
DOI : 10.5802/afst.1129

URL : http://afst.cedram.org/cedram-bin/article/AFST_2006_6_15_3_553_0.pdf

J. Dieudonné, Foundation of Modern Analysis, 1969.

J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, American Journal of Mathematics, vol.86, pp.109-160, 1964.

L. Euler, Methods inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti, chapter Additamentum 1. eulerarchive. org E065, p.1744

A. Fedotov, P. Giuseppe, D. Manifolds, and . Gruyter, , 2018.

B. Fuglede, Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble), vol.28, issue.2, pp.107-144, 1978.
DOI : 10.5802/aif.691

URL : https://aif.centre-mersenne.org/article/AIF_1978__28_2_107_0.pdf

S. Gudmundsson and J. C. Wood, Multivalued harmonic morphisms, Math. Scand, vol.73, pp.127-155, 1993.

C. Guenther, J. Isenberg, and D. Knopf, Linear stability of homogeneous Ricci solitons, Int. Math. Res. Not, vol.30, p.pp, 2006.

R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom, vol.17, issue.2, pp.255-306, 1982.
DOI : 10.4310/jdg/1214436922

URL : https://doi.org/10.4310/jdg/1214436922

L. Hömander, The analysis of linear partial differential operators, vol.1, 1983.

T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ, vol.19, pp.215-229, 1979.

C. Jacobi, Ueber eine particulre Lsung der partiellen Differentialgleichung ? 2 V ?x 2 + ? 2 V ?y 2 + ? 2 V ?z 2 = 0. Cr, vol.36, pp.113-134, 1848.

T. Ivey, Local existence of Ricci solitons, Manuscripta Math, vol.91, pp.151-162, 1996.
DOI : 10.1007/bf02567946

J. Kazdan and F. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geom, vol.10, pp.113-134, 1975.

S. Kowalevsky, Zur Theorie der partiellen Differentialgleichung, Journal für die reine und angewandte Mathematik, vol.80, p.132, 1875.

G. Lamé, Leçons sur la théorie analytique de la chaleur, 1861.

E. Loubeau and Y. Ou, The characterization of biharmonic morphisms, Differential geometry and its applications, vol.3, pp.31-41, 2001.

E. Loubeau and Y. Ou, Biharmonic maps and morphisms from conformal mappings, Tohoku Math J, vol.62, issue.1, pp.55-73, 2010.

J. Lott, On the long-time behavior of type-III Ricci flow solutions, Math. Annalen, vol.339, pp.627-666, 2007.

M. Nakai and L. Sario, Biharmonic classification of Riemannian manifolds, Bull. Amer. Math. Soc, vol.77, pp.432-436, 1971.

P. Nurowski, Construction of conjugate functions, Ann. Global Anal. Geom, vol.37, pp.321-326, 2010.

Y. Ou, On constructions of harmonic morphisms into Euclidean spaces, J. Guangxi University for Nationalities, vol.2, pp.1-6, 1996.

Y. Ou, Biharmonic morphisms between Riemannian manifolds, Geometry and topology of submanifolds, X (Beijing, pp.231-239, 1999.

Y. Ou, Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces, Jour. Geom. Phys, vol.62, pp.751-762, 2012.

Y. Ou and Z. Wang, Biharmonic Riemannian submersions from 3-manifolds, Math Zeitschrift, vol.269, issue.3, pp.917-925, 2011.

S. M. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds (summary), SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, vol.4, 2008.

G. Perelman, The entropy formula for the Ricci flow and its geometric applications

G. Perelman, Ricci flow with surgery on three-manifolds

G. Perelman, Finite extinction time for solutions to the Ricci flow on certain threemanifolds

P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc, vol.15, pp.401-487, 1983.

. Spivak, , vol.IV, pp.455-500, 1979.

W. P. Thurston, Three-dimensional Geometry and Topology, vol.1, 1997.

M. Ville, Harmonic morphisms from Einstein 4-manifolds to Riemann surfaces, Internat. J. Math, vol.14, issue.3, pp.327-337, 2003.

M. Wehbe, Aspects twistoriels des applications semi-conformes, Thesis. Mathématiques, 2009.

J. B. Wilker, Inversive Geometry, 1981.

J. C. Wood, Harmonic morphisms and Hermitian structures on Einstein 4manifolds, Internat. J. Math, vol.3, pp.415-439, 1992.