In this paper, we derive a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold M for which there exists a bounded vector field X such that \(\langle \nabla f,X\rangle \ge 0\) on M and \(\rm {div} X\ge af\) outside a suitable compact subset of M, for some constant \(a>0\), under the assumption that M has either polynomial or exponential volume growth. We then use it to obtain some Bernstein-type results for hypersurfaces immersed into a Riemannian manifold endowed with a Killing vector field, as well as to some results on the existence and size of minimal submanifolds immersed into a Riemannian manifold endowed with a conformal vector field.

在本文中，我们推导出一种新形式的光滑函数在完全非紧黎曼流形M上的极大值原理，其中存在一个有界向量场X使得\(\langle \nabla f,X\rangle \ge 0\)在M和\(\rm {div} X\ge af\)在M的合适紧致子集之外，对于一些常数\(a>0\)，假设M有多项式或指数式的体积增长。然后，我们使用它来获得一些关于浸入具有 Killing 向量场的黎曼流形中的超曲面的 Bernstein 类型的结果，以及关于浸入具有共形向量场的黎曼流形中的最小子流形的存在和大小的一些结果.