Proceedings of Symposia in Pure Mathematics Volume 54 (1993), Part 1 Open Problems in Geometry SHING TUNG YAU Since the publication [Y] of open problems for the special year in differen- tial geometry in 1979, there has been a lot of progress in geometry. Some old problems were resolved and some new problems were created. In 1984, the author was approached by Princeton University Press to update the book [Y]. The students and the author spent a month to do the job. Before the revised version was sent to the press, the editors of the series changed their mind and declined to reprint the book. Therefore the updated version has never been published. I take the opportunity of this conference to write some more open problems that have come across my mind. I hope to make comments about my previous problem section on a different occasion. I wish to thank my friends S. Bando, Y. Kawamata, P. Li, S. Nishikawa, and R. Schoen for com- ments. I also would like to express by deep gratitude to Laura Schlesinger for her patience in typing this manuscript even after I made substantial changes many times. I. Metric geometry 1. Find a general method to construct canonical metrics on a compact Riemannian manifold. All the known canonical metrics are obtained by the variational principle. In the past, geometers studied the critical points of the functional defined by the total scalar curvature, the L2-norm of the Rie- mannian curvature or these functional coupled with the Yang-Mills fields. If there is a compact group acting on the manifold, one can reduce the varia- tional problem to a lower-dimensional problem which may be easier to solve. The last approach was used by relativities in looking for an exact solution for the Einstein equation. A possible fruitful approach to find canonical metrics is to use singular perturbation method. For example, if two compact man- ifolds admit an Einstein metric with scalar curvature of the same sign, can 1991 Mathematics Subject Classification. Primary 51-02. Research supported by the National Science Foundation. A revised version of this paper will be submitted for publication elsewhere. ©1993 American Mathematical Society 0082-0717/93 $1.00+ $.25 per page 1
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