Monatsh Math (2017) 184:171–173
© Springer-Verlag GmbH Austria 2017
Noguchi, J., Winkelmann J., Nevanlinna Theory in Several Complex Variables and
Diophantine Approximation (Grundlehren der mathematischen Wissenschaften Vol.
350). XIV, 416 pp., Springer, Tokyo Heidelberg New York Dordrecht London, 2014.
This intriguing monograph describes the state of the art of large parts of Nevanlinna
theory in several complex variables. After reviewing the one-dimensional case the
authors show the First Main Theorem for meromorphic mappings from the multidi-
mensional complex space into a compact complex manifold with respect to holomor-
phic line bundles and coherent ideal sheaves of the structure sheaf. Then they cover
several cases in which the Second Main Theorem has been established, including a
version for maps to abelian and semi-abelian varieties, which is due to the authors and
K. Yamanoi. Further chapters are devoted to applications to Kobayashi hyperbolicity,
Nevanlinna theory over function ﬁelds (including an analogue of the abc-conjecture),
and Diophantine approximation over number ﬁelds in the spirit of Vojta’s dictionary.
As can be seen from this description this volume is too advanced for newcomers to the
ﬁeld. It will, however, be an important reference for graduate students and researchers
in Nevanlinna theory for many years.
Ch. BAXA, Wien
Dacorogna, B.: Introduction to the Calculus of Variations 3rd Edition. X, 311 pp.,
Imperial College Press, London, 2015. £4500.
This book is undoubtedly a classic, providing a well-crafted introduction to all the
main aspects of the calculus of variations. A brief historical introduction is followed
by a chapter on some basics from analysis (Hölder spaces, Lebesgue- and Sobolev
spaces). Then the classical calculus of variations (Euler–Lagrange, Hamiltonian for-
mulation, Hamilton–Jacobi) is laid out in chapter 2. This is followed by two chap-
ters on direct methods, devoted to existence and regularity. As the book focuses on
inner-mathematical applications, it includes chapters on minimal surfaces and on the
isoperimetric inequality. The ﬁnal chapter provides solutions of all the exercises. The