一类带外力场的逆平均曲率流的梯度估计

Gradient Estimate for Inverse Mean Curvature Flow with Forced Term

本文研究由逆平均曲率加上一个常量支配的星形超曲面的发展运动.首先,我们利用曲面的星形性质给出了流的等价同解方程,由此给出了流的短时间存在结果.接着,我们给出了流在特殊外力项下的梯度估计,并同时证明在这个外力场下,曲面的星形性质是保持的.

In this paper,we mainly study the evolution of star-shaped hypersurface by inverse mean curvature minus a constant.We assume the initial hypersurface is star-shaped with respect to the origin.By expressing the position vector by a positive function,we get an equivalent single equation of the evolution equation,from which we obtain the shorttime existence of the flow.And we also obtain that the hypersurface is still star-shaped during the flow when the constant is non-positive.Under the condition that the constant is non-positive,we study the equivalent equation.We obtain the maximum estimate and gradient estimate,and prove that the maximum and the gradient are determined by the initial data.The method we use here is mainly Hamilton's maximum principle and the idea is from Urbas who gave the same results for inverse mean curvature flow.