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Prescribing Gaussian curvature on S 2. (English) Zbl 0636.53053
On the standard two sphere S 2 the Gaussian curvature after a conformal change ds $$2=e^{2u}ds$$ $$2_ 0$$ is determined by the following equation: $(1.1)\quad \Delta u+Ke^{2u}=1\quad on\quad S^ 2$ where $$\Delta$$ denotes the Laplacian relative to the standard metric. The question raised by L. Nirenberg is: which function K can be prescribed so that (1.1) has a solution? An obvious necessary condition is given by the Gauss-Bonnet theorem implying that K must be positive somewhere. Some further necessary condition has been noted by Kazdan-Warner [J. Kazdan and F. Warner, Ann. Math., II. Ser. 99, 14-47 (1974; Zbl 0273.53034)]. For each eigenfunction $$x_ j$$ with $$\Delta x_ j+2x_ j=0$$ $$(j=1,2,3)$$, the Kazdan-Warner condition states that $(1.3)\quad \int_{S^ 2}<\nabla K,\quad \nabla x_ j>\quad e^{2u} d\mu =0\quad j=1,2,3.$ Thus functions of the form $$K=\psi \circ x_ j$$ where $$\psi$$ is any monotonic function defined on [-1,1] do not admit solutions. When K is an even function on S 2, (1.1) was interpreted as some Euler equation by J. Moser [On a nonlinear problem in differential geometry, Dyn. Syst., Proc. Sympos. Univ. Bahia, Salvador 1971, 273-280 (1973; Zbl 0275.53027).
In a previous paper on the subject, we gave the corresponding version of Moser’s result for those K satisfying a reflection symmetry about some plane (e.g. $$K(x_ 1,x_ 2,x_ 3)=K(x_ 1,x_ 2,-x_ 3))$$ [J. Moser, Indiana Univ. Math. J. 20, 1077-1092 (1971; Zbl 0203.437)]. In this paper, we give two sufficient conditions for the existence of solutions to the equation (1.1). The first is an attempt to generalize Moser’s result:
Theorem 1. Let K be a smooth positive function with two nondegenerate local maxima located at the north and south poles N, S. Let $$\phi_ t$$ be the one-parameter group of conformal transformations given in terms of stereographic complex coordinates (with $$z=\infty$$ corresponding to N and $$z=0$$ corresponding to S) by $$\phi_ t(z)=tz$$, $$0<t<\infty$$. Assume $(1.6)\quad \inf_{0<t<\infty}\frac{1}{4\pi}\int_{S^ 2}K\circ \phi_ t d\mu \quad >\quad \max \{K(Q)| \quad \nabla K(Q)=0,\quad Q\neq N,S\})$ then (1.1) admits a solution.
Theorem 2. Let K be a positive smooth function with only nondegenerate critical points, and in addition $$\nabla K(Q)\neq 0$$ where Q is any critical point. Suppose there are at least two local maximum points of K, and at all saddle points of K, $$\nabla K(Q)>0$$, then K admits a solution to the equation (1.1).
Based on the analysis of this present paper, we know the precise behavior of concentration near the saddle points of K where $$\nabla K(Q)<0$$. A stronger version of Theorem 2 has appeared in a separate article [“Conformal deformation of metrics on S 2”, Differ. Geom. 27, 259-296 (1988)].
Reviewer: S.A.Chang

##### MSC:
 53C20 Global Riemannian geometry, including pinching 58E35 Variational inequalities (global problems) in infinite-dimensional spaces 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 58J90 Applications of PDEs on manifolds 45C05 Eigenvalue problems for integral equations
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##### References:
 [1] Aubin, T., Meilleures constantes dans le théorème d’inclusion de Sobolev et un théorème de Fredholm non linéaire par la transformation conforme de la courbure scalaire.J. Funct. Anal., 32 (1979), 148–174. · Zbl 0411.46019 · doi:10.1016/0022-1236(79)90052-1 [2] Aubin, T.,The scalar curvature in differential geometry and relativity. Holland 1976, pp. 5–18. [3] Bahri, A. &Coron, J. M., Une théorie des points critiques à l’infini pour l’equation de Yamabe et le problème de Kazdan-Warner.C. R. Acad. Sci. Paris Sér. I, 15 (1985), 513–516. · Zbl 0585.58005 [4] Chang, S. Y. A. & Yang, P. C., Conformal deformation of metric onS 2. To appear inJournal of Diff. Geometry. [5] Escobar, J. F. & Schoen, R., Conformal metrics with prescribed scalar curvature. Preprint. · Zbl 0628.53041 [6] Hartman, P.,Ordinary differential equations. Basel Birkhäuser (1982). · Zbl 0476.34002 [7] Hersch, J., Quatre propriétés isopérimétriques de membranes sphériques homogènes.C. R. Acad. Sci. Paris Ser. I, 270 (1970), 1645–1648. · Zbl 0224.73083 [8] Hong, C. W., A best constant and the Gaussian curvature. Preprint. · Zbl 0603.58056 [9] Kazdan, J. &Warner, F., Curvature functions for compact 2-manifold.Ann. of Math. (2), 99 (1974), 14–47. · Zbl 0273.53034 · doi:10.2307/1971012 [10] – Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature.Ann. of Math. (2), 101 (1975), 317–331. · Zbl 0297.53020 · doi:10.2307/1970993 [11] Moser, J., A sharp form of an inequality by N. Trudinger.Indiana Univ. Math. J., 20 (1971), 1077–1091. · Zbl 0213.13001 · doi:10.1512/iumj.1971.20.20101 [12] – On a non-linear problem in differential geometry.Dynamical Systems (M. Peixoto, editor), Academic Press, N.Y. (1973). [13] Mostow, G. D., Some new decomposition theorems for semi-simple groups.Mem. Amer. Math. Soc., 14 (1955). · Zbl 0064.25901 [14] Onofri, E., On the positivity of the effective action in a theory of random surface.Comm. Math. Phys., 86 (1982), 321–326. · Zbl 0506.47031 · doi:10.1007/BF01212171 [15] Onofri, E. &Virasoro, M., On a formulation of Polyakov’s string theory with regular classical solutions.Nuclear Phys. B, 201 (1982), 159–175. · doi:10.1016/0550-3213(82)90378-9 [16] Schoen, R., Conformal deformation of a Riemannian metric to constant scalar curvature.J. Differential Geom., 20 (1985), 479–495. · Zbl 0576.53028
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