On the First Variation of a Varifold

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-