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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.

There have been many attempts to define quasilocal energy/mass for a spacelike 2-surface in a spacetime by the Hamilton–Jacobi method. The main difficulty in this approach is the subtle choice of the background configuration to be subtracted from the physical Hamiltonian. Quasilocal mass should be positive for general surfaces, but on the other hand should be zero for surfaces in the flat spacetime. In this paper, we survey the work in a series of papers [6, 25–27] in which a new notion of quasilocal mass/energy–momentum is proposed and investigated. In particular, the notion of energy observed by a 'quasilocal observer' will be discussed.

Smooth valuations on manifolds are studied by establishing a link with the Rumin-de Rham complex of the co-sphere bundle. Several operations on differential forms induce operations on smooth valuations: signature operator, Rumin-Laplace operator, Euler-Verdier involution and derivation operator. As an application, Alesker’s Hard Lefschetz Theorem for even translation invariant valuations on a finite-dimensional Euclidean space is generalized to all translation invariant valuations. The proof uses K¨ahler identities, the Rumin-de Rham complex and spectral geometry.