an:05033802
Zbl 1101.53002
Schn??rer, Oliver C.
Surfaces contracting with speed \(| A|^2\)
EN
J. Differ. Geom. 71, No. 3, 347-363 (2005).
00124800
2005
j
53A05 53C45 53C44
flow equations; convex surfaces; the second fundamental form
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 \(\mathbb{S}^2\).
Yurii G. Nikonorov (Rubtsovsk)