If ψ : M → R is a smooth immersed closed hypersurface, we consider the functional Fm(ψ) = ∫ M 1 + |∇ν| dμ, where ν is a local unit normal vector along ψ, ∇ is the Levi–Civita connection of the Riemannian manifold (M, g), with g the pull–back metric induced by the immersion and μ the associated volume measure. We prove that if m > ⌊n/2⌋ then the unique globally defined smooth solution to the L–gradient flow of Fm, for every initial hypersurface, smoothly converges asymptotically to a critical point of Fm, up to diffeomorphisms. The proof is based on the application of a Łojasiewicz–Simon gradient inequality for the functional Fm.

中文翻译：

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演化超曲面的渐近收敛

如果 ψ : M → R 是一个光滑的浸没闭合超曲面，我们考虑泛函 Fm(ψ) = ∫ M 1 + |∇ν| dμ，其中 ν 是沿 ψ 的局部单位法线向量，∇ 是黎曼流形 (M, g) 的 Levi-Civita 连接，其中 g 是由浸没引起的回拉度量， μ 是相关的体积度量。我们证明，如果 m > ⌊n/2⌋，那么对于每个初始超曲面，Fm 的 L 梯度流的唯一全局定义平滑解会平滑地渐近收敛到 Fm 的临界点，直至微分同胚。该证明基于对泛函 Fm 应用 Łojasiewicz–Simon 梯度不等式。