Reese Harvey

Removable singularities of solutions of linear partial differential equations

Suppose P(x, D) is a linear partial differential operator on an open set ~ contained in R ~ and that A is a closed subset of ~. Given a class ~(~) of distributions on ~, the set A is said to be removable for ~(~) if e ach /E ~(~), which satisfies P(x, D ) / = 0 in ~ A , also satisfies P(x, D)/= 0 in ~. The problem considered in this paper is the following. Given a class ~(~) of distributions on ~, what restriction on the size of A will ensure that A is removable for ~(~). We obtain results for Lroc (~) (p ~< ~ ) , C(~), and Lipa (~). The first result of this kind was the Riemann removable singularity theorem: if a function / is holomorphic in the punctured unit disk a n d / ( z ) = o ( H -1) as z approaches zero, then / is holomorphic in the whole disk. Bochner [1] generalized Riemanns result by considering the class ~(~) of functions f on ~ such that ](x)=o(d(x, A) -q) uniformly for x in compact subsets of ~, and giving a condition on the size of A which insures tha t A is removable for ~(~) (Theorem 2.5 below). Bochners theorem is remarkable in that the condition on the size of A only depends on the order of the operator P(x, D). The theorem applies, therefore, to systems of differential operators, such as exterior differentiation in R n and ~ (the Cauchy-Riemann operator) in C n. The same can be said for the other results in this paper. The proof of Bochners theorem provided the motivation for our results. I t is interesting to note tha t a very general result (Corollary 2.4) f o r / ~ (~) (due to Li t tman [7]) is an easy corollary of Bochners work. Here the condition on the singular set A is expressed in terms of Minkowski content. In section 4 the case of Ll~oc(~) is studied again, and results in section 2 are improved by replacing Minkowski content with Hausdorff measure. In addition, the cases C(~)

An intrinsic characterization of Kähler manifolds

On etudie la condition de Kahler pour une variete complexe compacte. On caracterise les varietes complexes compactes qui admettent des metriques de Kahler

Extending Minimal Varieties

Introduction The purpose of this paper is to study the general problem of extending minimal varieties across closed subsets in riemannian manifolds. The analogous theory in the complex analytic case falls into two categories: extension theorems for functions (Hartogs phenomenon) and extension theorems for varieties (the RemmertStein-Shiffman Theorem). The discussion here splits along similar lines. In w 1 we analyze the Bers, deGiorgi-Stampacchia results on removing singularities of solutions to the non-parametric minimal surface equation in codimension one. This theorem does not carry over directly to higher codimension. However, certain parts of their argument can be generalized and lead to a strong reflection principle for minimal submanifolds and a uniqueness result for minimal cones. Also a theorem for surfaces in general codimension is proved. The remainder of the paper is devoted to proving extension theorems of the following type for varieties. Let M be a riemannian manifold and A ~ M a compact subset of sufficient smoothness and appropriate Hausdorff dimension. Then any k-dimensional minimal variety which is stationary (or area minimizing) in M A extends to a minimal variety which is stationary (or, resp., area minimizing) in M. The proof falls into two parts. The first is to show that the mass of the variety is finite across A so that an extension exists (w 3). The second is to prove the properties of minimality for the extension. The main conclusions are stated precisely in w 5.

Stiefel—Whitney currents

A canonically defined mod 2 linear dependency current is associated to each collection v of sections, v1,…,vm, of a real rank n vector bundle. This current is supported on the linear dependency set of v. It is defined whenever the collection v satisfies a weak measure theoretic condition called “atomicity.” Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2 d-closed locally integrally flat current of degree q = n −m + 1 and hence determines a ℤ2-cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover, it is the qth Stiefel-Whitney class of the vector bundle.More is true if q is odd or q = n. In this case a linear dependency current which is twisted by the orientation of the bundle can be associated to the collection v. The mod 2 reduction of this current is the mod 2 linear dependency current. The cohomology class of the linear dependency current is 2-torsion and is the qth twisted integral Stiefel-Whitney class of the bundle.In addition, higher dependency and general degeneracy currents of bundle maps are studied, together with applications to singularities of projections and maps.These results rely on a theorem of Federer which states that the complex of integrally flat currents mod p computes cohomology mod p. An alternate approach to Federer’s theorem is offered in an appendix. This approach is simpler and is via sheaf theory.

The Pontryagin

The unit 4-planes on which the first Pontryagin form of the Grassmann manifolds achieves its maximum are determined. This is a shorter and unified proof of results first obtained in 1995 by H. Gluck et al. and in 1998 by W. Gu.