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Construction of complete embedded self-similar surfaces under mean curvature flow. II. (English) Zbl 1200.53061
The author studies surfaces in \(\mathbb R^3\) which satisfy the equation
\[ H + X\cdot \nu = 0, \]
where \(X\), \(\nu\) and \(H\) are the position-vector, the unit normal and the mean curvature respectively. Such surfaces are called shrinking self-similar surfaces or self-shrinkers because the mean curvature flow does not change their shape and merely contracts them.
The graph of a function \(u\) over a domain \(\Omega\subset\mathbb R^2\) satisfies the self-shrinker equation if and only if \(u\) satisfies the equation
\[ {\mathcal E} (u)=\left(\delta^{ij}-\frac{D_iuD_ju}{1+| Du|^2}\right)\left( Du(\xi) D_{ij}u(\xi) - \xi\cdot Du(\xi) +u(\xi)\right) = 0,\,\, \xi\in\Omega. \]
The author discusses a Dirichlet problem for this equation on an unbounded domain. The main result states that, for small enough boundary conditions with some symmetries on the circle \(C_R=\partial D_R\) of radius \(R\) in the plane, there exists a function \(u\) matching the boundary conditions such that the graph of \(u\) outside the disc \(D_R\) is a self-shrinker.
Theorem. Let \(\sqrt{3}/2<R<2\) and \(N\geq 5\). There is an \(\varepsilon_0>0\) depending on \(R\) and \(N\) such that, for any \(f\in C^4([0,2\pi])\) with \(|| f||_{C^4([0,2\pi])}=\varepsilon\leq\varepsilon_0\) and satisfying the symmetries \(f(\theta)=-f(-\theta)=f(\pi/N-\theta)\), there exists a function \(u\) on \(\Omega=\mathbb R^2\setminus D_R\) such that
\[ {\mathcal E}(u) = 0 \quad \text{in }\Omega, \]
\[ u=f\quad \text{on }\partial D_R, \]
\[ u(r,\theta) = -u(r,-\theta) = u(r,\pi/N - \theta) \quad \text{for } r>R,\;\theta [0,2\pi). \]
Moreover, we can choose the constant \(\varepsilon_0\) uniformly for all \(R\in(\sqrt{3}/2,2)\).
Here \(r,\theta\) are the polar coordinates in \(\mathbb R^2\) whose pole coincides with the center of \(D_R\), so \(u=f\) on \(\partial D_R\) means that \(u(R,\theta) = f(\theta)\).
In the proof the solution to \({\mathcal E} (u) = 0\) is found considering the limit for time \(t\) going to infinity of a solution to the parabolic equation \(\partial_t u = {\mathcal E} (u)\).
The author suggests that the result obtained may be applied to construct new examples of complete embedded self-similar surfaces under mean curvature flow [cf. N. Kapouleas, J. Differ. Geom. 47, No. 1, 95–169 (1997; Zbl 0936.53006) and the first part, X. H. Nguyen, Trans. Am. Math. Soc. 361, No. 4, 1683–1701 (2009; Zbl 1166.53046)].

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
Full Text: arXiv