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Surfaces contracting with speed $$| A|^2$$. (English) Zbl 1101.53002
The author investigates families of strictly convex surfaces $$M_t$$ in $$\mathbb{R}^3$$ which satisfy the flow equation $$\frac{d}{dt}X=-| A| ^2\nu$$, where $$X=X(x,t)$$ is the embedding vector of a manifold $$M_t$$ in $$\mathbb{R}^3$$, $$\nu$$ is the outer unit normal vector to $$M_t$$, and $$| A| ^2$$ is the square of the norm of the second fundamental form. The main result is the following (Theorem 1.1):
For any smooth closed strictly convex surface $$M$$ in $$\mathbb{R}^3$$, there exists a smooth family of closed strictly convex surfaces $$M_t$$, $$t\in [0,T)$$, solving the above flow equation with $$M_0=M$$. For $$t \to T$$, $$M_t$$ converges to a point $$Q$$. The rescaled surfaces $$(M_t-Q)\cdot(6(T-t))^{-1/3}$$ converge smoothly to the unit sphere $$\mathbb{S}^2$$.

##### MSC:
 53A05 Surfaces in Euclidean and related spaces 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov) 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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