Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

Abstract

Abstract. We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time T : (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter γ > 1/2, there is a solution with the highest curvature blowing up at the rate (T − t) −(γ+1/2) . (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the "Grim Reaper" solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ.

Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up. II

“…The new scaling of y implies changes in the rescaled PDEs for MCF; e.g., equations (1.3) and (1.4), cf. the same-numbered equations in [15]. In particular, we note that the change occurs in the first-order term in equation (1.3), or equivalently in the zerothorder term in equation (1.3).…”

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We continue the study, initiated by the first two authors in [15], of Type-II curvature blow-up in mean curvature flow of complete noncompact embedded hypersurfaces. In particular, we construct mean curvature flow solutions, in the rotationally symmetric class, with the following precise asymptotics near the "vanishing" time T : (1) The highest curvature concentrates at the tip of the hypersurface (an umbilical point) and blows up at the rate (T − t) −1 .(2) In a neighbourhood of the tip, the solution converges to a translating soliton known as the bowl soliton.

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“…Having established the existence of compact MCF solutions with Type-II curvature blow-up, it is natural to seek noncompact counterparts, as first realized in [15] by the first two authors of this paper. More precisely, for each real number γ > 1/2, we have constructed MCF of noncompact rotationally symmetric embedded hypersurfaces that are complete convex graphs over a shrinking ball and asymptotically approach a shrinking cylinder near spatial infinity.…”

“…Following the set up in [15], in this paper we consider mean curvature flow of rotationally symmetric hypersurfaces embedded in Euclidean space. For any point (x 0 , x 1 , .…”

“…Indeed, to capture the borderline case γ = 1/2, taking the limit γ → 1 2 in [15] is insufficient. The new scaling of y, on the other hand, is natural because in [15] the asymptotic cylindricality is measured precisely by 2(n−1)−φ 2 ∼ y (1/2−γ) −1 and if we let γ → 1/2, then we expect 2(n − 1) − φ 2 to decay faster than any arbitrarily large power of y; i.e., we have exponential decay in y, as is captured by the asymptotic property (3) of Theorem 1.1. The new scaling of y implies changes in the rescaled PDEs for MCF; e.g., equations (1.3) and (1.4), cf.…”