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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)].

MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
Keywords:
self-shrinkers; Dirichlet problem
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