By Yau S.-T. (ed.)

ISBN-10: 1571460691

ISBN-13: 9781571460691

The Surveys in Differential Geometry are vitamins to the magazine of Differential Geometry, that are released by means of foreign Press. They comprise major invited papers combining unique examine and overviews of the most up-tp-date learn in particular components of curiosity to the growing to be magazine of Differential Geometry neighborhood. The survey volumes function carrying on with references, inspirations for brand new study, and introductions to the range of issues of curiosity to differential geometers. those supplementations are released each year seeing that 1999. This quantity arises out of the convention subsidized by means of the magazine of Differential Geometry and held at Harvard college to honor the 4 mathematicians who based Index conception. loads of geometers accumulated for this old celebration which integrated a variety of tributes and memories with a view to be released in a separate quantity. The 4 founders of the Index concept - Michael Atiyah, Raoul Bott, Frederich Hirzebruch, and Isadore Singer - are resources of suggestion, mentors and academics for the opposite audio system and contributors on the convention. The larger-than-usual dimension of this quantity derives at once from the great recognize and admiration for the honorees. desk of Contents: 1. Projective planes, Severi types and spheres - M. Atiyah and J. Berndt 2. Degeneration of Einstein metrics and metrics with distinct 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. okay. Donaldson 7. neighborhood tension 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 ends up in classical minimum floor concept - W. H. Meeks III eleven. On international life 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 crowning glory of Teichmüller area - S. A. Wolpert

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26 Exercise. , S’ = {(z,y) E R2 : x2 +y2 = 1)). Then F2(S1) is a Moebius band; that is, Fz(S’) is homeomorphic to the quotient space obtained from [0, l] x [0, l] by identifying the point (0,~) with the point (1,l - y) for each 9 E [0, 11. Remark. Some of the results in the exercises above are in [6]. If you read [6], you should be aware of two errors: (c) on p. 3 on p. 162 (cf. 5 of [7, pp. 40-411 and [8 or 91, respectively). You should also be careful to remember the standing assumption on p.

Polon. Sci. Cl. R. 19 (1958), 668. 3. 4. F. , Berlin, 1927. K. Kuratowski, Topology, Vol. I, Acad. , 1966. 5. 6. K. Kuratowski, Topology, Vol. II, Acad. , 1968. Ernest Michael, Topologies on spacesof subsets,Trans. Amer. Math. Sot. 71 (1951), 152-182. REFERENCES 7. 8. 9. 29 Sam B. , Vol. , 1978. E. Smithson, First countable hyperspaces, Proc. Amer. Math. Sot. 56 (1976), 325-328. Daniel E. Wulbert, Subsets of first countable spaces, Proc. Amer. Math. Sot. 19 (1968), 1273-1277. This Page Intentionally Left Blank II.

If Y is an infinite, discrete space, then the Vietoris topology for CL(Y) does not have a countable base. Let ,0 be a base for the Vietoris topology for CL(Y). Note that Proof. each A E CL(Y) is open in Y. A C (A). Thus, it follows easily that (1) A = USA for each A E CL(Y). Now, it follows immediately from (1) that if A, A’ E CL(Y) such that A # A’, then DA # a,&. jA is a one-toone function from CL(Y) into p. Hence, A RESULT ABOUT METRIZABILITY c-4 ICW)l OF CL(X) 13 I IPI- Since Y is a discrete space, CL(Y) = {A c Y : A # 0}; thus, since Y is an infinite set, CL(Y) is uncountable.

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