Varifold

Abstract: Suppose M is a smooth m dimensional Riemannian manifold and k is a positive integer not exceeding m. Our purpose is to study the first variation of the k dimensional area integrand in M. Our main result is a regularity theorem for weakly defined k dimensional surfaces in M whose first variation of area is summable to a power greater than k. A natural domain for any k dimensional parametric integral in M, among which the simplest is the k dimensional area integral, is the space of k dimensional varifolds in M introduced by Almgren in [AF 1]. Such a varifold is defined to be a Radon measure on the bundle over M whose fiber at each point p of M is the Grassmann manifold of k dimensional linear subspaces of the tangent space to M at p. If V is a varifold in M, let I I V I I be the Radon measure on M obtained from V by ignoring the fiber variables. Naturally injected in the space of k dimensional varifolds in M is the set of k dimensional rectifiable subsets of M, which includes the set of k dimensional submanifolds of M as well as more general k dimensional surfaces in M. A k dimensional varifold in M is said to be rectifiable (integral) if it can be strongly approximated by a positive real (integral) linear combination of varifolds corresponding to continuously differentiable k dimensional submanifolds of M. To any classical k dimensional geometric object in M there corresponds a k dimensional integral varifold in M. If N is a smooth Riemannian manifold and F: M-e N is smooth, then F induces in a natural way a strongly continuous mapping F# of the k dimen-

Abstract: We develop a level set theory for the mean curvature evolution of surfaces with arbitrary co-dimension, thus generalizing the previous work [8, 15] on hypersurfaces. The main idea is to surround the evolving surface of co-dimension k in R by a family of hypersurfaces (the level sets of a function) evolving with normal velocity equal to the sum of the (d — k) smallest principal curvatures. The existence and the uniqueness of a weak (level-set) solution, is easily established using mainly the results of [8] and the theory of viscosity solutions for second order nonlinear parabolic equations. The level set solutions coincide with the classical solutions whenever the latter exist. The proof of this connection uses a careful analysis of the squared distance from the surfaces. It is also shown that varifold solutions constructed by Brakke [7] are included in the level-set solutions. The idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of De Giorgi. An introduction to the theory of barriers and his connection to the level set solutions is also provided.

Abstract: A general theorem characterizing the interaction of concentrations and oscillations effects associated with sequences of gradients bounded in Lp, t p > 1, is proved. The oscillations are recorded in the Young measure while the concentrations are encoded in the varifold.