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On the mean curvature flow for \(\sigma_k\)-convex hypersurfaces. (English) Zbl 1076.53080

The authors show that the singular limit set of a closed \(\sigma_k\)-convex hypersurface flowing by mean curvature flow has Hausdorff dimension at most \(n-k\). They give an example that shows that the theorem is sharp. More precisely, the \(\sigma_k\) curvature of a closed hypersurface \(M^n\) is defined to be \[ \sigma_k^M (x)= \sum_{1\leq i_k <\cdots <i_k \leq n} \lambda_{i_1}(x)\cdots \lambda_{i_k}(x), \] where \(\lambda_{i_k}(x)\) are the principal curvatures of \(M\) at \(x\). \(M\) is \(\sigma_k\)-convex if \(\min_{x\in M}\sigma_k^M(x)\geq0. \) The authors prove the following:
Theorem: Let \(F_0 :M^n \rightarrow \mathbb R^{n+1}\) be a smooth immersion, \(M_t = F(\;,t)(M^n)\) satisfy the mean curvature flow, and \(T\) be the first time at which singularities occur. Assume that \(M_0\) is \(\sigma_k\)-convex for some \(1\leq k \leq n\). Then the singular set of \(M_T\) has Hausdorff dimension at most \(n-k\).
Note that the \(k=n\) case is well-known [G. Huisken, J. Differ. Geom. 20, 237–266 (1984; Zbl 0556.53001)] .

MSC:

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

Citations:

Zbl 0556.53001
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