Uniqueness of tangent flows at the first singular time of mean curvature flow of closed embedded surfaces

Claims uniqueness of tangent flows, without mean-convexity or a prescribed tangent model, at every singular point of the first singular time for smooth compact connected embedded surfaces in $\mathbb R^3$: all fixed-center rescalings converge to one multiplicity-one self-shrinker flow in the original coordinates, and every backward Brakke tangent agrees with it. The key step is a finite-jet Lojasiewicz analysis handling shrinkers with both conical and cylindrical ends. It does NOT cover later singular times, higher dimensions, or moving-center blowups.

Claims uniqueness of tangent flows, without mean-convexity or a prescribed tangent model, at every singular point of the first singular time for smooth compact connected embedded surfaces in $\mathbb R^3$: all fixed-center rescalings converge to one multiplicity-one self-shrinker flow in the original coordinates, and every backward Brakke tangent agrees with it. The key step is a finite-jet Lojasiewicz analysis handling shrinkers with both conical and cylindrical ends. It does NOT cover later singular times, higher dimensions, or moving-center blowups.