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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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