Approximation in Sobolev Spaces between two manifolds and homotopy groups

Abstract

We consider two compact manifolds M n and N k and the Sobolev spaces W 1,p(M n, N k),for 1 < p < n = dim M n. We give a necessary and sufficient condition for smooth maps between M n and N k to be dense in W 1,p(M n, N k). This condition can be simply stated in terms of homotopy groups, and is π[p](N k)= 0. In cases where such a condition does not hold, we show that we can approximate maps in W 1,p(M n, N k) by maps smooth except on a singular set which has a simple shape. We consider also the problem of the weak density of smooth maps.KeywordsFinite NumberSobolev SpacePoint SingularityWeak TopologyHomotopy GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.