The Existence of Some Metrics on Riemannian Warped Product Manifolds with Fiber Manifold of Class (B)

Abstract

One of the basic problems in the dierential geometry is studying the set of curvature functions which a given manifold possesses. The well-known problem in dierential geometry is that of whether there exists a warping function of warped metric with some prescribed scalar curvature function. One of the main methods of studying dierential geometry is the existence and the nonexistence of Riemannian warped metric with prescribed scalar curvature functions on some Riemannian warped product manifolds. In order to study these kinds of problems, we need some analytic methods in dierential geometry. For Riemannian manifolds, warped products have been useful in producing examples of spectral behavior, examples of manifolds of negative curvature (cf. [B.K.], [B.O.], [D.D.], [G.L.], [K.K.P.], [L.M.], [M.M.]), and also in studying L2-cohomology (cf. [Z.]). In a study [L. 1, 2], M.C. Leung have studied the problem of scalar curvature functions on Riemannian warped product manifolds and obtained partial results about the existence and the nonexistence of Riemannian warped metric with some prescribed scalar curvature function. In this paper, we also study the existence and the nonexistence of Riemannian warped product metric with prescribed scalar curvature functions on some Riemannian warped product manifolds. So, using upper solution and lower solution methods, we consider the solution of some partial dierential equations on a warped product manifold. That is, we express the scalar curvature of a warped product manifold M = B f N in terms of its warping function f and the scalar curvatures of B and N. By the results of Kazdan and Warner (cf. [K.W. 1, 2, 3]), if N is a compact Riemannian n-manifold without boundary, n 3; then N belongs to one of the following three categories: (A) A smooth function on N is the scalar curvature of some Riemannian metric on N if and only if the function is negative somewhere. (B) A smooth function on N is the scalar curvature of some Riemannian metric on N if and only if the function is either identically zero or strictly negative everywhere. (C) Any smooth function on N is the scalar curvature of some Riemannian metric on N. This completely answers the question of which smooth functions are scalar curvatures of Riemannian metrics on a compact manifold N. In [K.W. 1, 2, 3], Kazdan and Warner also showed that there exists some obstruction of a Riemannian metric with positive scalar curvature (or zero scalar curvature) on a compact manifold. For noncompact Riemannian manifolds, many important works have been done on the question how to determine which smooth functions are scalar curvatures of complete Riemannian metrics on open manifold. Results of Gromov and Lawson (cf. [G.L.]) show that some open manifolds cannot carry complete Riemannian metrics of positive scalar curvature, for example, weakly enlargeable manifolds. Furthermore, they show that some open manifolds cannot even admit complete Riemannian metrics with scalar curvatures uniformly positive outside a compact set and with Ricci curvatures bounded (cf. [G.L.], [L.M., p.322]). On the other hand, it is well known that each open manifold of dimension bigger than 2 admits a complete Riemannian metric of constant negative scalar curvature (cf. [B.K.]). It follows from the results of Aviles and McOwen (cf.[A.M.]) that any bounded negative function on an open manifold of dimension bigger than 2 is the scalar curvature of a complete Riemannian metric. In this paper, when N is a compact Riemannian manifold, we discuss the method of using warped products to construct Riemannian metrics on M = [a;1) f N with specic scalar curvatures, where a is a positive constant. It is shown that if the ber manifold N belongs to class (B), then M admits a Riemannian metric with some prescribed scalar curvature outside a compact set. That is, suppose that R(g) = 0. and that R(t) 2 C1([a;1)) is a function such that 4n n + 1 c 4 1 t2 > R(t) - 4n n + 1 et for t > t0; where t0 > a, > 0 and 0 < c < 1 are constants. Then equation (3.4) has a positive solution on [a;1): These results are extensions of the results in [J.L.K.L.]. Although we will assume throughout this paper that all data (M, metric g, and curvature, etc.) are smooth, this is merely for convenience. Our arguments go through with little or no change if one makes minimal smoothness hypotheses, such as assuming that the given data is H older continuous.| 미분기하학에서 기본적인 문제 중의 하나는 미분다양체가 가지고 있는 곡률 함수에 관한 연구이다. 연구방법으로는 종종 해석적인 방법을 적용하여 다양체 위에서의 편미분방정식을 유도하여 해의 존재성을 보인다. Kazdan and Warner ([K.W.1,2,3])의 결과에 의하면 위의 함수 가 위의 Riemannian metric의 scalar curvature가 되는 세 가지 경우의 타입이 있는 데 먼저 (A) 위의 함수 가 Riemannian metric의 scalar curvature이면 그 함수 가 적당한 점에서 일 때이다. 즉, 위에 nagative constant scalar curvature를 갖는 Riemannian metric이 존재하는 경우이다. (B) 위의 함수 가 Riemannian metric의 scalar curvature이면 그 함수 가 항등적으로 이거나 모든 점에서 인 경우이다. 이 경우에는, 위에서 zero scalar curvature를 갖는 Riemannian metric이 존재하는 경우이다. (C) 위의 임의의 함수 를 scalar curvature로 갖는 Riemannian metric 이 존재하는 경우이다. 본 논문에서는 엽다양체 이 (B)에 속하는 compact Riemannian manifold 일 때, Riemannian warped product manifold인 위에 함수 가 적당한 조건을 만족하면 가 Riemannian warped product metric의 scalar curvature가 될 수 있는 warping function 가 존재할 수 있음을 상해·하해 방법을 이용하여 증명하였다.