The evolution of the scalar curvature of a surface to a prescribed function

Baird, Paul; Fardoun, Ali; Regbaoui, Rachid

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 1, pp. 17-38.

We investigate the gradient flow associated to the prescribed scalar curvature problem on compact riemannian surfaces. We prove the global existence and the convergence at infinity of this flow under sufficient conditions on the prescribed function, which we suppose just continuous. In particular, this gives a uniform approach to solve the prescribed scalar curvature problem for general compact surfaces.

Classification : 58E20

[1] T. Aubin, Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J.Math.Pures Appl. 55 (1976), 269-296. | MR | Zbl

[2] T. Aubin, Meilleures constantes dans le théorème d'inclusion de Sobolev et un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire prescrite, J. Funct. Anal. 32 (1979), 148-174. | MR | Zbl

[3] T. Aubin, Sur le problème de la courbure scalaire prescrite, Bull. Sci. Math. (5) 118 (1994), 465-474. | MR | Zbl

[4] J. Bartz - M. Struwe - R. Ye, A new approach to the Ricci flow on $S^2$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), 475-482. | Numdam | MR | Zbl

[5] S. Bismuth, Prescribed scalar curvature on a $C^{\infty }$ compact Riemannian manifold of dimension two, Bull. Sci. Math. 124 (2000), 239-248. | MR | Zbl

[6] S. Y. A. Chang - P. C. Yang, Prescribing Gaussian curvature on $S^2$, Acta Math. 159 (1987), 215-259. | MR | Zbl

[7] S. Y. A. Chang - P. C. Yang, Conformal deformations of metric on $S^2$, J. Differential Geom. 27 (1988), 259-296. | MR | Zbl

[8] S. Y. A. Chang - M. J. Gursky - P. C. Yang, The scalar curvature equation on 2- and 3-spheres, Calc. Var. Partial Differerential Equations 1 (1993), 205-229. | MR | Zbl

[9] W. Chen - C. Li, A note on Kazdan-Warner type conditions, J. Differential Geom. 41 (1995), 259-268. | MR | Zbl

[10] W. Chen - C. Li, A necessary and sufficient condition for the Nirenberg Problem, Comm. Pure Appl. Math. 47 (1995), 657-667. | MR | Zbl

[11] W. Chen - W. X. Ding, Scalar curvature on $S^2$, Trans. Amer. Math. Soc. 303 (1987), 365-382. | MR | Zbl

[12] X. X. Chen, Calabi flows in Riemannian surfaces revisited; a new point of view, Int. Math. Res. Not. 6 (2001), 275-297. | MR | Zbl

[13] B. Chow, The Ricci-Hamilton flow on the 2-sphere, J. Differential Geom. 24 (1986), 153-179.

[14] R. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71 (1988), 237-262. | MR | Zbl

[15] M. A. Jendoubi, A simple unified approach to some convergence Theorems of L.Simon, J. Funct. Anal. 153 (1998), 187-202. | MR | Zbl

[16] J. Kazdan - F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math. 99 (1974), 14-47. | MR | Zbl

[17] J. Kazdan - F. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math. 101 (1975), 317-331. | MR | Zbl

[18] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077-1092. | MR | Zbl

[19] J. Moser, On a nonlinear problem in differential geometry, In: “Dynamical systems” (M. Peixoto eds.), Academic Press, 1973. | MR | Zbl

[20] E. Onofri, On the positivity of the effective action in a theory of random surfaces, Comm. Math. Physics 86 (1982), 321-326. | MR | Zbl

[21] L. Simon, Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math. 118 (1983), 525-571. | MR | Zbl

[22] M. Struwe, Curvature flows on surfaces, Ann. Scuola Norm. Sup. Pisa (5) 1 (2002), 247-274. | EuDML | Numdam | MR | Zbl

[23] X. Xu - P. Yang, Remarks on prescribing Gauss Curvature, Trans. Amer. Math. Soc. 336 (1993), 831-840. | MR | Zbl