论文标题
关于由高斯曲率的力量翻译的超曲面的存在
On existence of hypersurfaces translating by powers of Gauss curvature
论文作者
论文摘要
在本文中，我们在$ \ mathbb r^{n+1} $中构造完整的凸出曲面，该$在powers $α\ in（0，\ frac1 {n+2}）$的流下翻译为高斯曲线的$。每种解决方案的水平集渐近地缩小了由$ \ Mathbb r^n $的高斯曲率的功率$ \fracα{1-α} $的流量缩小的孤子。例如，我们的构造揭示了翻译人员的存在，其水平集合到球体，单纯形，超立方体等。翻译孤子作为一个家庭存在，其参数对应于雅各比字段，这是渐近谱周围线性化方程的解决方案。
In this paper we construct complete convex hypersurfaces in $\mathbb R^{n+1}$ which translate under the flow by powers $α\in (0, \frac1{n+2})$ of the Gauss curvature. The level set of each solution is asymptotic to a shrinking soliton for the flow by power $\frac α{1-α}$ of the Gauss curvature in $\mathbb R^n$. For example, our construction reveals the existence of translators whose level set converges to the sphere, simplex, hypercube and so on. The translating solitons exist as a family whose parameters correspond to Jacobi fields, solutions to linearized equation around the asymptotic profile.
