We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a ${𝒞}^1$ dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a ${𝒞}^1$ neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem [] which applies only to calibrated submanifolds of special holonomy ambient manifolds.

中文翻译：

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最小子流形上的强稳定性条件及其含义

我们确定了最小子流形上的强稳定性条件，这暗示了唯一性和动态稳定性。特别是，我们证明了唯一性定理和${𝒞}^{\mathrm{1个}}$满足此条件的最小子流形的平均曲率流的动力稳定性定理。后一个定理指出，一个子流形中任何其他子流形的平均曲率流。${𝒞}^{\mathrm{1个}}$这种最小子流形的邻域一直存在，并且以指数形式收敛到最小子流形。这扩展了我们先前的唯一性和稳定性定理[]，该定理仅适用于特殊完整环境歧管的校准子流形。