An important question in geometry and analysis is to know when two
k-forms f and g are equivalent through a change of variables. The
problem is therefore to find a map φ so that it satisfies the pullback
equation: φ*(g) = f.
In more physical terms, the question under consideration can be seen as
a problem of mass transportation. The problem has received considerable
attention in the cases k = 2 and k = n, but much less when 3 k
n-1. The present monograph provides the first comprehensive study of the
equation.
The work begins by recounting various properties of exterior forms and
differential forms that prove useful throughout the book. From there it
goes on to present the classical Hodge-Morrey decomposition and to give
several versions of the Poincaré lemma. The core of the book discusses
the case k = n, and then the case 1k n-1 with special attention on
the case k = 2, which is fundamental in symplectic geometry. Special
emphasis is given to optimal regularity, global results and boundary
data. The last part of the work discusses Hölder spaces in detail; all
the results presented here are essentially classical, but cannot be
found in a single book. This section may serve as a reference on Hölder
spaces and therefore will be useful to mathematicians well beyond those
who are only interested in the pullback equation.
The Pullback Equation for Differential Forms is a self-contained and
concise monograph intended for both geometers and analysts. The book may
serve as a valuable reference for researchers or a supplemental text for
graduate courses or seminars.
