Introduction to the theory of currents

Introduction to the theory of currentsTien-Cuong Dinh and Nessim SibonySeptember 21, 200523This course is an introduction to the theory of currents. We give here the mainnotions with examples, exercises and we state the basic results. The proofs oftheorems are not written. We hope that this will be done in the next version.All observations and remarks are welcome.dinh@math.jussieu.fr and nessim.sibony@math.u psud.fr4ContentsNotations 71 Measures 91.1 Borel -algebra and measurable maps . . . . . . . . . . . . . . . . 91.2 Positive measures and integrals . . . . . . . . . . . . . . . . . . . 121.3 Locally nite measures . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Outer measures and Hausdor measures . . . . . . . . . . . . . . 182 Distributions 212.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Operations on distributions . . . . . . . . . . . . . . . . . . . . . 232.3 Convolution and regularization . . . . . . . . . . . . . . . . . . . 252.4 Laplacian and subharmonic functions . . . . . . . . . . . . . . . . 273 Currents 313.1 Di erential forms and currents . . . . . . . . . . . . . . . . . . . . 313.2 Operations on currents and Poincare’s lemma . . . . . . . . . . . 343.3 Convolution and regularization . . . . . . . . . . . . . . . . . . . 374 Currents on manifolds 394.1 Di erentiable manifolds . . . . . . . . . . . . . . . . . . . . . . . 394.2 Vector bundles . . . . . . . . . . . . . . . . . . . ...