Stability of the surface area preserving mean curvature flow in Euclidean space

Abstract

Abstract. The surface area preserving mean curvature flow is a mean curvature type flow with a global forcing term to keep the hypersurface area fixed. By iteration techniques, we show that the surface area preserving mean curvature flow in Euclidean space exists for all time and converges exponentially to a round sphere, if initially the L 2 -norm of the traceless second fundamental form is small (but the initial hypersurface is not necessarily convex).

Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries

“…For n ≥ 2, Mccoy in [16] showed that if the initial n-dimensional hypersurface M 0 is strictly convex then the evolution equation (1.3) has a smooth solution M t for all time 0 ≤ t < ∞, and M t converge, as t → ∞, in the C ∞ -topology, to a sphere with the same surface area as M 0 . In [11], Huang and Lin use the idea of iteration of Li in [14] and Ye in [23], in cases of volume preserving mean curvature flow and Ricci flow, respectively. And obtained the same result, by assuming that the initial hypersurface M 0 satisfies h(0) > 0 and…”

Blow-up of the mean curvature at the first singular time of the mean curvature flow

“…Kuwert and Schätzle [KS01] showed that the Willmore flow (i.e., the gradient flow of the Willmore energy |Å| 2 dµ) starting from a surface with small |Å| 2 dµ in R 3 exists for all time and converges to a round sphere. The surface area preserving mean curvature flow starting from hypersurfaces with small |Å| 2 dµ in R n was recently investigated by Huang and the first author in [HL12].…”