A flow approach to Nirenberg's problem

We describe an alternative approach to the existence results of S.-Y. A. Chang and P. C. Yang for metrics of prescribed scalar curvature on S 2 via the prescribed curvature flow. Moreover, we give an example showing that the results of these authors in general cannot be improved upon.

[1]  Min Ji On positive scalar curvature on S2 , 2004 .

[2]  P. Baird,et al.  The evolution of the scalar curvature of a surface to a prescribed function , 2004 .

[3]  S. Brendle Global existence and convergence for a higher order flow in conformal geometry , 2003, math/0404415.

[4]  Hartmut Schwetlick,et al.  Convergence of the Yamabe flow for ``large'' energies , 2003 .

[5]  S. Brendle Prescribing a higher order conformal invariant on S^n , 2003 .

[6]  M. Struwe Curvature flows on surfaces , 2002 .

[7]  Xiuxiong Chen Weak limits of Riemannian metrics in surfaces with integral curvature bound , 1998, math/0009243.

[8]  S. Chang,et al.  The scalar curvature equation on 2- and 3-spheres , 1993 .

[9]  Kung-Ching Chang,et al.  ON NIRENBERG'S PROBLEM , 1993 .

[10]  J. Smoller,et al.  Conformal metrics with prescribed Gaussian curvature on S2 , 1993 .

[11]  Paul C. Yang,et al.  A perturbation result in prescribing scalar curvature on $S^n$ , 1991 .

[12]  Wenxiong Chen,et al.  Classification of solutions of some nonlinear elliptic equations , 1991 .

[13]  Zheng-chao Han Prescribing Gaussian curvature on $S^2$ , 1990 .

[14]  Paul Yang,et al.  Conformal deformation of metrics on $S^2$ , 1988 .

[15]  S. Chang,et al.  Prescribing Gaussian curvature on S2 , 1987 .

[16]  E. Onofri On the positivity of the effective action in a theory of random surfaces , 1982 .

[17]  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 , 1979 .

[18]  F. W. Warner,et al.  Curvature Functions for Compact 2-Manifolds , 1974 .

[19]  J. Moser On a Nonlinear Problem in Differential Geometry , 1973 .