平均曲率流的平移解的伯恩斯坦问题
A Bernstein problem for translating solutions to the mean curvature flow
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- University of Bath（巴斯大学）
- Chinese University of Hong Kong（香港中文大学）
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中文总结 AI 辅助
该研究针对平均曲率流的平移解的伯恩斯坦问题，在N≥8时构造出非凸的平均凸整体图平移子，完善了相关刚性理论。
中文摘要 AI 辅助
我们研究平均曲率流的整体图平移解，其在ℝⁿ中满足方程：div(∇G/√(1+|∇G|²)) = 1/√(1+|∇G|²)。每个此类图都是平均凸的，因为其平均曲率是单位法向量的垂直分量。在二维时，平均凸平移孤子是凸的，因此整体图平移子是旋转对称的碗状孤子；在更高维度，Wang构造了非旋转的整体凸平移图。我们证明在伯恩斯坦维度会出现进一步的刚性损失：对于每个N≥8，存在单参数族的整体图平移子，它们是平均凸但非凸的。该构造从ℝ⁸中的Bombieri-De Giorgi-Giusti（BDG）整体极小图出发，围绕其发展平移子方程的奇异扰动理论。主要新特征是Simons锥附近的过渡层：平移项打破极小图的奇对称性，经适当重新中心化后，匹配问题由抛物内方程控制。对该层的详细分析，结合BDG图上的加权雅可比理论和全局障碍，得到所需的整体解。
英文摘要
We study entire graphical translating solutions of the mean curvature flow, \[ \operatorname{div}\!\left(\frac{\nabla G}{\sqrt{1+|\nabla G|^2}}\right) =\frac{1}{\sqrt{1+|\nabla G|^2}} \qquad\text{in }\mathbb R^N. \] Every such graph is mean-convex, since its mean curvature is the vertical component of its unit normal. In dimension two, mean-convex translating solitons are convex, and an entire graphical translator is therefore the rotationally symmetric bowl soliton. In higher dimensions Wang constructed non-rotational entire convex translating graphs. We prove that a further loss of rigidity occurs at the Bernstein dimension: for every $N\ge 8$ there exists a one-parameter family of entire graphical translators that are mean-convex but not convex. The construction starts from the Bombieri--De Giorgi--Giusti (BDG) entire minimal graph in $\mathbb R^8$ and develops a singular perturbation theory for the translator equation around it. The main new feature is a transition layer near Simons' cone: the translating term breaks the odd symmetry of the minimal graph, and after a suitable recentering the matching problem is governed by a parabolic inner equation. A detailed analysis of this layer, together with weighted Jacobi theory on the BDG graph and global barriers, yields the desired entire solutions.
