Preface Llu´ ıs Antoni Santal´ o Sors (1911–2001) was a Spanish mathematician that stud- ied at Hamburg under the differential geometer Wilhelm Blaschke. In 1936, Pro- fessor Blaschke initiated two young students, Santal´ o and S.S. Chern, to integral geometry. Unlike Chern, who spent long periods of his career in different places as France, China and USA, Santal´ o moved definitely to Argentina after a short pe- riod in Spain, because of the Spanish Civil War nevertheless he become recognized worldwide by his studies in Integral Geometry, Stereology, Geometric Probability and Projective Geometry (see the recent volume Luis Antonio Santal´ o Selected Works edited by A. Naveira and A. Revent´ os under Springer Verlag, for a selection of Santal´ o’s best papers and bibliography). In his honor, the Royal Mathematical Society of Spain (RSME) organizes since 2002 a yearly advanced summer School about various aspects in Mathematics. Except for the 2010 occasion, the RSME Santal´ o School had been developed within the framework of the Summer Courses of the International University Men´ endez y Pelayo in Santander, Spain. The 2010 event was held at the Uni- versity of Granada, and the chosen topic was that of Geometric Analysis. This is a rather vague term to refer to a part of mathematics whose width has been in- creasing in the last decades, and whose frontiers have trespassed areas that a priori, were not assumed to be reachable. In a very simplified manner, we can describe Geometric Analysis as the interface between Differential Geometry and Differential Equations. The primary example of this interaction could be that of the calculus of variations, where the object of study is the perturbation of a functional acting on geometric objects. The critical points of the functional are often characterized as solutions of a differential equation (the associated Euler-Lagrange equation). These critical points are tools for understanding the geometry of the manifold over which the functional is defined. Different problems coming from areas apparently far, have been solved by application of tools in Geometric Analysis: among others, we can mention the solution by R. Schoen and S. T. Yau of the positive mass conjecture, and the more recent positive answer by G. Perelman to the Poincar´ e conjecture. The main objective of the 2010 Santal´ o School was to carry out mini-courses and talks in which distinguished researchers in Geometric Analysis would explain from the basics to the some of the most up-to-date aspects of this area. The School was mainly intended for researchers in Mathematics and degree or PhD students, although everyone with mathematical interest, an inquisitive mind and strong geometrical intuition was invited to join it. This set of lecture notes was originated from the series of lectures given at the 2010 Santal´ o School on Geometric Analysis, held at the University of Granada from June 28 to July 2, 2010. The organization of this volume is as follows. The vii
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