By Stephen Shing-Taung Yau

The yearly Surveys in Differential Geometry quantity is acquired with anticipation every year because it summarizes some of the contemporary discoveries within the box. This year's quantity is devoted to Professors Calabi, Lawson, Siu, and Uhlenbeck. It includes very important contributions through their scholars and associates and displays the real paintings within the box through those nice mathematicians. the amount is acceptable for graduate scholars and researchers drawn to geometry and topology. desk of Contents: 1. Projective planes, Severi forms and spheres - M. Atiyah and J. Berndt 2. Degeneration of Einstein metrics and metrics with precise holonomy - J. Cheeger three. The min-max development of minimum surfaces - T. H. Colding and C. De Lellis four. common quantity bounds in Riemannian manifolds - C. B. Croke and M. Katz five. A Kawamata-Viehweg vanishing theorem on compact Kahler manifolds - J.-P. Demailly and T. Peternell 6. second maps in differential geometry - S. ok. Donaldson 7. neighborhood stress for cocycles - D. Fisher and G. A. Margulis eight. Einstein metrics, four-manifolds, and differential topology - C. LeBrun nine. Topological quantum box concept for Calabi-Yau threefolds and $G_2$-manifold - N. C. Leung 10. Geometric leads to classical minimum floor concept - W. H. Meeks III eleven. On international lifestyles of wave maps with serious regularity - A. Nahmod 12. Discreteness of minimum versions of Kodaira size 0 and subvarieties of moduli stacks - E. Viehweg and ok. Zuo thirteen. Geometry of the Weil-Petersson finishing touch of Teichm??ller house - S. A. Wolpert

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By definition, if Mn is Kahler, there is a parallel almost complex structure J. P -1, V'J O. Suppose Mn splits off a line isometrically, Mn R X F- 1 • If v denotes the parallel vector field tangent to the R factor of this splitting, then V'v = O. Hence, V' J(v) = 0 as well. By the de Rham decomposition theorem, it follows that the vector fields, v, J(v), are tangent to the R 2 -factor of a local isometric splitting of the metric. In addition, the factor, R2, can be naturally identified with the complex plane C.

Appendix; Review of special holonomy References o. 29 37 43 47 51 53 56 57 61 64 65 70 Introduction A a riemannian manifold, Mn, is called Einstein if for some constant, A, the metric, g, and Ricci tensor, RicM'" satisfy RicM" = Ag. 2). This paper is not intended as a general survey of the subject of the Einstein metrics. g. [Cal], [Yaul]' [Yau2]. Rather, we try to give a detailed overview of one specific line of progress due to Colding, Tian and the author (in various combinations) which has shed light on the following question: What sorts of objects arise as limits of sequences of badly behaved Einstein metrics?

Ann. Math. (2) 96 (1972), 277-295. L. Zak: Severi varieties. Math. USSR, S6. 54 (1986), 113-127. ac . ac . uk Degeneration of Einstein metrics and metrics with special holonomy Jeff Cheeger CONTENTS o. Introduction 1. Preliminaries 2. Almost rigidity and rescaling 3. The structure of limit spaces 4. Einstein limit spaces 5. Special holonomy and co dimension 4 singularities 6. Special holonomy and tangent cones 7. Integral bounds on curvature 8. Rectifiability of singular sets 9. Anti-self-duality of curvature and the structure of singular sets 10.