Singular set and curvature blow-up rate of the level set flow

Under certain conditions such as the $2$-convexity, a singularity of the level set flow is of type I (in the sense that the rate of curvature blow-up is constrained before and after the singular time) if and only if the flow shrinks to either a round point or a $C^{1}$ curve near that singular point. Analytically speaking, the arrival time is $C^{2}$ near a critical point if and only if it satisfies a Lojasiewicz inequality near the point

On the Existence of Soap Bubbles in the Cylinder

水管中泡沫的純在性從數學上的角度來看，是一個常均曲率曲面滿足邊界垂直管壁的問題，本文是用固定體積下最小曲面方法來證明這類曲面的純在性。Mathematically, the existence of soap bubbles in the cylinder is the existence of a constant mean curvature surface which is perpendicular to the wall on boundary. In this article, we use the method of minimizing area with fixed volume to solve the problem