AI 中文总结
该研究构造表面扩散流，证明其可驱动光滑闭嵌入环面有限时间内捏裂，刻画了该流的曲率奇点剖面，严格实现了经典锥形相似剖面。
AI 中文摘要
我们构造了表面扩散流 $f:\mathbb{T}^2\times[0,T)\to\mathbb{R}^3$，该流驱动光滑闭嵌入环面在有限时间内发生捏裂。当 $t\nearrow T$ 时，流在 $t\in[0,T)$ 内保持嵌入，仅在特定点 $p$ 处发展出曲率奇点，而在 $(p,T)$ 之外，流随 $t\nearrow T$ 光滑收敛。我们刻画了奇点剖面：若 $A(t)$ 表示腰部半径，且表面在腰部附近局部由径向图 $(z,t)\mapsto r(z,t)$ 的旋转给出，则存在常数 $\mu>0$ 和光滑函数 $U:\mathbb{R}\to\mathbb{R}$，使得 $A(t)=\{4\mu(T-t)\}^{1/4}(1+o(1))$，且 $A(t)^{-1}r(A(t)\zeta,t)\stackrel{C^\infty_{loc}}{\longrightarrow} U(\zeta)$。此处 $U$ 是 Wong、Miksis、Voorhees 和 Davis 首次数值计算，后经 Bernoff、Bertozzi 和 Witelski 分析的经典基本正偶锥形相似剖面的严格实现。
英文摘要
We construct surface diffusion flows $f:\mathbb{T}^2\times[0,T)\to\mathbb{R}^3$ that drive smooth closed embedded tori to pinchoff in finite time. The flow remains embedded for $t\in[0,T)$ and develops a curvature singularity only at a distinguished point $p$ as $t\nearrow T$. Away from $(p,T)$ the flow converges smoothly as $t\nearrow T$. We characterise the singularity profile: If $A(t)$ denotes the radius of the waist, and the surface is given locally near the waist by the rotation of a radial graph $(z,t)\mapsto r(z,t)$, then there exists a constant $μ>0$ and smooth function $U:\mathbb R\to\mathbb R$ such that \[ A(t)=\{4μ(T-t)\}^{1/4}(1+o(1)), \qquad A(t)^{-1}r(A(t)ζ,t)\stackrel{C^\infty_{loc}}{\longrightarrow} U(ζ). \] Here $U$ is a rigorous realisation of the classical fundamental positive even conical similarity profile first computed numerically by Wong, Miksis, Voorhees and Davis and subsequently analysed by Bernoff, Bertozzi and Witelski.
Comments48 pages, 1 figure
