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The mean curvature flow smoothes Lipschitz submanifolds. (English) Zbl 1059.53053
Let \(F:\Sigma^n\to \mathbb{R}^N\) be an isometric immersion of a compact manifold \(\Sigma\) in the Euclidean space. The mean curvature flow of \(F\) is a family of immersion \(F_t:\Sigma^n\to \mathbb{R}^N\) that satisfies \(\frac{d}{dt}F_t(x) = H(x,t)\) and \(F_0=F\), where \(H(x,t)\) is the mean curvature vector of \(F_t(\Sigma)\subset E^N\). The mean curvature flow is the gradient flow of the volume functional, minimal submanifolds are its stationary solutions.
The author generalizes the known result of Ecker and Huisken on the smoothness of a complete hypersurface in \(\mathbb{R}^N\) along the mean curvature flow. The main result is the following theorem:
Let \(\Sigma\) be a compact \(n\)-dimensional Lipschitz submanifold of \(\mathbb{R}^{n+m}\). There exists a positive constant \(K\) depending on \(n\) and \(m\) such that if \(\Sigma\) satisfies the \(K\) local Lipschitz condition, then the mean curvature flow of \(\Sigma\) has a smooth solution on some interval \([0,T]\).
By definition, a submanifold \(\Sigma\subset \mathbb{R}^{n+m}\) satisfies the \(K\) local Lipschitz condition if there exists an \(r_0>0\) such that for each \(q\in\Sigma\) the piece of \(\Sigma\) inside the ball \(B_q(r_0)\), i.e. \(\Sigma\cup B_q(r_0)\), can be written as the graph of a vector valued Lipschitz function \(f_q\) over an \(n\)-dimensional affine space through \(q\) with \(\frac{1}{\sqrt{\text{det}(I+(df_q)^Tdf_q)}}>K\). Roughly speaking, this means that \(\Sigma\) is allowed to have some corners which are not too sharp.
Naturally, any \(C^1\) submanifold satisfies the \(K\) local Lipschitz condition. So the demonstrated result may be viewed as a generalization of a classical theorem of Morrey on the smoothness of \(C^1\) minimal submanifolds.

MSC:
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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