Einstein manifolds with zero Ricci curvature (Q2771288)

This paper contains a survey about Ricci-flat Einstein manifolds. Starting from the original motivation for studying such manifolds, the author discusses several solved and unsolved problems arising from current developments in string theories.NEWLINENEWLINENEWLINEA fundamental problem is the existence of such manifolds. In the context of Kähler geometry this leads to the famous Calabi Conjecture for compact manifolds which was eventually solved by the author. The so-called Calabi-Yau manifolds resulting from this solution are an important ingredient in understanding the mathematics behind string theories. This leads to several new interesting and beautiful developments in mathematics. By considering moduli spaces of Calabi-Yau manifolds new geometries, so-called special geometries, were discovered. Another important consequence of these developments has been the discovery of mirror symmetry and quantum cohomology. The wish for finding a method for constructing mirror manifolds leads to the Strominger-Yau-Zaslow Conjecture and involves special Lagrangian geometries. The classification of Calabi-Yau 3-manifolds is closely related to the construction of manifolds with \(G_2\)- and Spin(7)-holonomy.NEWLINENEWLINENEWLINEThis article is a well-written survey about these recent beautiful developments that relate various different areas of mathematics to each other. Many references are given helping the reader to find the relevant literature.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00021].