Variational problem and evolution equation of curves and surfaces

点击翻译按钮获取中文摘要

The most natural variational problem of closed submanifolds in the 3-euclidean space is the elastic curves in 1-dimensional case, and the constant mean curvature surfaces in 2-dimensional case. These problems are extended naturally to n-euclidean spaces as curves or hyper surfaces. However, we don't have good variational problem of closed mid-dimensional submanifold in n-dimensional euclidean spaces In this research, we defined the following new good variational problem. Consider pairs (S, dS) of minimal submanifold S and its boundary dS. Given the volume of dS, we seek a minimal submanifold S whose volume attains maximum. A solution of this variational problem is called a max-min submanifold. We got the following results.1. The pair of a totally geodesic submanifold and its constant mean curvature surfaces is max-min submanifold.2. In particular, a round sphere of any dimensional Euclidean space with any codimension is max-min submanifold.3. The solution (2) is stable.4. A pair of a minimal cone C and the intersection of C and the unit sphere is max-min submanifold.5. We can construct non-homogeneous examples in the torus.6. Since the variational problem is conditional, two stabilities are defined. Let k be the trace of second fundamental form for outer unit vector. If k is positive, then the solution is A-unstable. If k is negative, then A-stability and B-stability are equivalent.