Immersed hypersurfaces with constant Weingarten curvature

Abstract

In a recent paper Korevaar 1-5] used the Alexandrov reflection principle to show that closed embedded hypersurfaces in N 1, lHn+ 1 or the upper hemisphere of S~ + 1 are umbilic spheres provided a certain function f of the principal curvatures ,t=().t,22 .. . . ,)~,) is constant. He only had to assume that f is positive on the positive cone ~={212i>0 Vi} and that f is elliptic on the component F of {21 f(,~)>0} which contains ~. Here f is said to be elliptic if 0f/d21>0 for all i, 1 <iNn . This generalization of earlier sphere theorems (see [9] and [10] for references) cannot be extended to hypersurface immersions in view of recent counterexamples, 1-11]. However, assuming additional curvature conditions Walter derived in [10] global results for hypersurface immersions in a space N~+l(c) of constant curvature c, which have a constant higher mean curvature function H,. Here H, is the r-th symmetric function of the principal curvatures. It was shown that such hypersurfaces are of constant mean curvature H1, provided they have nonnegative sectional curvature and non-negative principal curvatures. As a consequence they have to be isoparametric with at most two distinct principal curvatures and can therefore be completely classified. Here we show that it is not necessary to assume all principal curvatures to be non-negative. Moreover we extend Waiter's result to general symmetric functions f = f(2). Let M ~ be a smooth, connected and compact manifold without boundary. Then we have the following result.