We consider a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space,through higher order equations. We prove that the regularized problems converge to the mean curvature flow for all times before the first singularity.

The curvature estimates for k curvature equations with general right-hand sides is a longstanding problem. In this paper, we completely solve the problem when k . We also discuss some applications of our estimates.

Let M⊂CN be a generic real-analytic submanifold of finite type, M′⊂CN′ be a real-analytic set, and p∈M, where we assume that N,N′⩾2. Let H:(CN,p)→CN′ be a formal holomorphic mapping sending M into M′, and let EM′ denote the set of points in M′ through which there passes a complex-analytic subvariety of positive dimension contained in M′. We show that, if H does not send M into EM′, then H must be convergent. As a consequence, we derive the convergence of all formal holomorphic mappings when M′ does not contain any complex-analytic subvariety of positive dimension, answering by this a long-standing open question in the field. More generally, we establish necessary conditions for the existence of divergent formal maps, even when the target real-analytic set is foliated by complex-analytic subvarieties, allowing us to settle additional convergence problems such as e.g. for transversal formal maps between Levi-non-degenerate hypersurfaces and for formal maps with range in the tube over the light cone.

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Eigenvalues and capacitors. Let X be a Riemannian manifold and be the Laplace operator on X. It is well-known that if X is compact then the spectrum of is discrete and consists of an increasing sequence {k} k= 1 of the eigenvalues (counted with the multiplicities) where 1= 0 and k as k. Moreover, if n= dim X then Weyls asymptotic formula says that (1.1) k cn