AI 中文总结
该研究针对任意高阶几何演化方程建立集中紧性理论，以三维欧氏空间闭浸入曲面的几何多重调和热流为模型，结合多种估计与性质，证明满足无迹曲率阈值条件的连通初始浸入对应的解全局存在且指数收敛到圆球面。
AI 中文摘要
我们针对任意高阶几何演化方程建立了集中紧性理论，以三维欧氏空间中闭浸入曲面的几何多重调和热流作为模型案例。该流是(2p+2)阶法向演化：∂_t f=(-1)^(p+1)Δ^p H ν，其中p≥1，当p=1时包含表面扩散流。我们证明了带精确截断记账的局部能量估计、内部估计、寿命/集中二择一、无迹曲率ε正则性估计，以及定常解的间隙定理。随后将这些工具与爆破论证、带符号包围体积的守恒性、面积的单调性相结合，以排除无迹曲率阈值以下的奇点。因此，对于满足‖A^o‖_2²<ε（ε>0仅依赖于流的阶数）的连通初始浸入，解存在所有时间，并在C^∞中指数收敛到具有守恒包围体积的圆球面。
英文摘要
We develop a concentration-compactness theory for geometric evolution equations of arbitrarily high order, using the geometric polyharmonic heat flow of closed immersed surfaces in \(\R^3\) as the model case. The flow is the \((2p+2)\)-order normal evolution \[ \partial_t f=(-1)^{p+1}Δ^p H\,ν,\qquad p\geq1, \] which includes the surface diffusion flow when \(p=1\). We prove localised energy estimates with sharp cut-off bookkeeping, interior estimates, a lifespan/concentration alternative, tracefree-curvature \(\varepsilon\)-regularity estimates, and a gap theorem for stationary solutions. These tools are then combined with a blowup argument, the preservation of signed enclosed volume, and the monotonicity of area to rule out singularities below a small tracefree-curvature threshold. Consequently, for connected initial immersions satisfying \(\|A^o\|_2^2<\varepsilon\), where \(\varepsilon>0\) depends only on the order of the flow, the solution exists for all time and converges exponentially in \(C^\infty\) to a round sphere with the preserved enclosed volume.
Comments58 pages
