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The surface area preserving mean curvature flow. (English) Zbl 1078.53067
Summary: Under the surface area preserving mean curvature flow, formulated by D. M. Pihan in [A length preserving geometric heat flow for curves, PhD thesis, University of Melbourne (1998)], the family of maps \(F_t=F (\cdot,t)\) evolves according to \[ \frac{\partial}{\partial t}F(x,t)= \bigl \{1-h(t)H(x,t)\bigr\}\nu (x,t),\quad x\in U,\;0\leq t\leq T\leq\infty,\tag{1} \] \(F(\cdot,0)=F_0\), where \(H\) is the mean curvature of \(M_t= F_t (U)\), \(\nu\) is the outer unit normal to \(M_t\) and \[ h(t)=\frac {\int_{M_t}Hd\mu_t}{\int_{M_t}H^2d\mu_t},\tag{2} \] where \(d\mu_t\) is the surface area element on \(M_t\). Pihan studied basic properties of this flow for general \(n\) and showed that (1) has a unique solution for a short time. He also proved for \(n=1\) that an initially closed, convex curve in the plane converges exponentially to a circle with the same length as the initial curve. Analogous to this result and those of Huisken in [G. Huisken, J. Differ. Geom. 20, 237–266 (1984; Zbl 0556.53001) and J. Reine Angew. Math. 382, 35–48 (1987; Zbl 0621.53007)] for the mean curvature flow and the volume preserving mean curvature flow, we show here a similar result for the surface area preserving flow, when \(n\geq 2\).

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