The volume-preserving mean curvature flow in Euclidean space
The author proves a convergence of the volume-preserving mean curvature flow of hypersurfaces in Euclidean space under some initial integral pinching conditions on the second fundamental form. Let \(M\) be a closed oriented immersed surface in \({\mathbb R}^{n+1}\) given locally by an immersion \(F_0\). Let \(F_t(x)\) be a flow satisfying \[ \partial _t F_t(x) = (h-H)\nu(x,t),\quad F_t(x)|_{t=0} = f_0(x), \] where \(\nu(x,t)\) is the outer unit normal vector on \(M_t = F_t(M)\), \(H(x,t)\) is the mean curvature, and \(h(t)\) is its average defined by \(h(t) = \frac{1}{V_t}\int H(x,t)\, dv_t(x)\). Here, \(V_t\) denotes the volume of \(M_t\) with respect to the induced metric. The main result is the following. Given positive real numbers \(v, \Lambda, \gamma\), there exists a positive real number \(\varepsilon = \varepsilon(v, \Lambda, \gamma, n)>0\) such that if \(M^n\) is a closed oriented immersed surface in \({\mathbb R}^{n+1}\) satisfying \[ {\text{vol}}(M) \leq v,\quad |A| \leq \Lambda,\quad h \geq \gamma,\quad \int_M |{\AA}|^2 \leq \varepsilon,\tag{1} \] then the volume-preserving mean curvature flow with the initial surface \(M\) converges exponentially fast to a round sphere. Here, \({\AA} = A - \frac{H}{n} g\) is the traceless part of the second fundamental form \(=A\), and \(g\) is the induced metric. The condition (1) implies that both \(|\AA|\) and \(|H-h|\) are small after a short time and so decay exponentially in a short time provide that the average of the mean curvature is strictly positive. On the other hand, that they decay exponentially implies that they are both small in the next time interval. Finally, using iteration arguments, long time existence and exponential decay for all time follow. The condition on \(A\) can be weakened in the case \(n=2\). Namely, for given positive constants \(v_0, v_1, \Lambda, \gamma\) such that \(M^2 \subset {\mathbb R}^3\) is a compact smoothly immersed surface satisfying \[ v_0 < {\text{vol}}(M) < v,\quad \gamma < h < \Lambda, \] and \[ \int_M |H-h|^2 \leq \varepsilon, \quad \int_M |{\AA}|^2 \leq \varepsilon, \] the volume-preserving mean curvature flow with initial surface \(M\) converges exponentially fast to a round sphere. For higher-dimensional cases, assuming the \(L^n\) norms \(|\AA|\) and \(|H-h|\) are small, the same conclusion holds.
scientific article; zbMATH DE number 5632309