Unique tangent flows at the first surface singularity

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Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in ℝ³. At every singular point, all fixed-center backward tangent flows agree as area measures at every negative time in the original ambient coordinates, without mean-convexity or a prescribed tangent model.

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Theorem 1.1

In Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow, 24 September 2026

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Let M_0\subset\R^3 be a smooth compact connected embedded surface without boundary, and let (M_t)_{0\le t<T} be its maximal smooth mean curvature flow. At every singular point (x_0,T) at its first singular time there is a smooth properly embedded self-shrinker S such that, as \lambda\to\infty, the full family (1.1) converges locally smoothly in space and with multiplicity one to \sqrt{-s}\,S, uniformly for s in compact subintervals of (-\infty,0). The convergence is in the original ambient coordinates, including the axes and position of the limit. Every backward integral Brakke tangent in the stated convention therefore equals the area-measure flow of \sqrt{-s}\,S at every negative time.

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Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow

24 September 2026, current edition.

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We prove that, at each singular point of the first singular time of the mean-curvature flow of a smooth compact connected embedded surface without boundary in ℝ³, all fixed-center rescalings converge locally smoothly on compact negative-time intervals to one multiplicity-one homothetic self-shrinker flow. The limit is unique in the original ambient coordinates, including its position and axes. No mean-convexity assumption or prescribed tangent model is required.

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