Surfaces contracting with speed \(| A|^2\). (English) Zbl 1101.53002

Schnürer, Oliver C.

J. Differ. Geom. 71, No. 3, 347-363 (2005).

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 \(\mathbb{S}^2\).

Reviewer: Yurii G. Nikonorov (Rubtsovsk)

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)

Keywords:

flow equations; convex surfaces; the second fundamental form