加权保体积曲率流近圆柱定常解的稳定性
Stability of near cylindrical stationary solutions to weighted-volume preserving curvature flows
摘要：
本文考虑一类作用在两个平行超平面内且满足正交边界条件的超曲面上的加权保体积曲率流。在作者的论文中，考虑了圆柱在流动下的稳定性，发现只要半径满足一定的条件，它们是稳定的。这里我们考虑的流动，有一个临界半径，使得圆柱体只在临界值的一侧稳定，权函数是初等对称函数的线性组合，这样选择的原因在附录中说明得很清楚。我们发现，在这种情况下，从圆柱形固定的解决方案发生分歧的临界半径，我们确定的速度和权函数，使附近的非圆柱形固定的解决方案是稳定的流动下的条件。我们还将突出齐次速度函数和混合体积保持中曲率流的具体情况。
英文摘要：
In this paper we consider a class of weighted-volume preserving curvature flows acting on hypersurfaces that are trapped within two parallel hyperplanes and satisfy an orthogonal boundary condition. In the author's thesis the stability of cylinders under the flows was considered; it was found that they are stable provided the radius satisfies a certain condition. Here we consider flows where there is a critical radius such that the cylinders are only stable on one side of the critical value, and the weight function is a linear combination of the elementary symmetric functions, the reason for such a choice is made clear in the appendix. We find that in such instances bifurcation from the cylindrical stationary solutions occurs at the critical radius and we determine a condition on the speed and weight functions such that the nearby non-cylindrical stationary solutions are stable under the flow. We will also highlight the specific cases of homogeneous speed functions and the mixed-volume preserving mean curvature flows.
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