带边流形上k-Yamabe方程的变分结构

Variational structure of k-Yamabe problem on manifolds with boundary

本文考虑了n维带边流形上的k-Yamabe问题.当2 k≤n/2时,假设方程是具有变分结构的.在之前的文献中,人们都是假设流形是局部共形平坦的.当k=2时,在较弱的假设下,本文可以推广已有的k-Yamabe问题的结论.特别地,本文证明了,当n>4、边界的平均曲率非负、且流形的Yamabe常数比标准半球面的Yamabe常数严格小时,2-Yamabe问题是可解的.

This paper considers the k-Yamabe problem on manifolds with boundary. We investigate the problem under the assumption that the k-Yamabe equation has a variational structure instead of that the manifold is locally conformally flat, where 2 ≤ k 〈 n/2, n is the dimension of the manifold. In particular, we show that the 2-Yamabe problem is solvable on the manifold of dimension n 〉 4 with umbilic boundary if its Yamabe constant is smaller than that of the standard half unit sphere and the mean curvature of the boundary is nonnegative.