Proceedings of Symposia in Pure Mathematics 1993; 2000 pp; hardcover Volume: 54 ISBN10: 0821814931 ISBN13: 9780821814932 List Price: US$315 Member Price: US$252 Order Code: PSPUM/54
 These three parts contain the proceedings of the AMS Summer Institute on Differential Geometry, held at the University California, Los Angeles in July 1990. This was the largest AMS Summer Institute ever, reflecting the wideranging and intense research activity in the subject. The parts contain many extensive survey articles presenting perspectives on relatively broad topics; these articles would be accessible to advanced graduate students. In addition, the authors of the research articles were encouraged to survey the relevant literature. The three parts together offer the deepest and most comprehensive survey of recent research in differential geometry available today. Readership Research mathematicians and/or advanced graduate students in differential geometry, partial differential equations and certain related areas of physics. Reviews "One can learn from perusing this volume what is going on in the rest of mathematics."  The Bulletin of Mathematics Books Table of Contents Part 1. Partial Differential Equations on Manifolds  S. T. Yau  Open problems in geometry
 F. Almgren  Questions and answers about areaminimizing surfaces and geometric measure theory
 D. Bao and T. Ratiu  On the geometrical origin and the solutions of a degenerate MongeAmpère equation
 R. W. Brockett  Differential geometry and the design of gradient algorithms
 Y.J. Chiang  Spectral geometry of \(V\)manifolds and its application to harmonic maps
 H. I. Choi and A. Treibergs  Constructing harmonic maps into the hyperbolic space
 D. DeTurck and W. Ziller  Spherical minimal immersions of spherical space forms
 J. Dorfmeister  Banach manifolds of solutions to nonlinear partial differential equations, and relations with finitedimensional manifolds
 R. Hardt  Some new harmonic maps
 J. Hass, J. T. Pitts, and J. H. Rubinstein  Existence of unstable minimal surfaces in manifolds with homology and applications to triply periodic minimal surfaces
 W.Y. Hsiang  Closed minimal submanifolds in the spheres
 G. Huisken  Local and global behaviour of hypersurfaces moving by mean curvature
 T. Ilmanen  The levelset flow on a manifold
 J. Jost  Unstable solutions of twodimensional geometric variational problems
 J. Jost and S. T. Yau  Harmonic maps and superrigidity
 A. Kasue  Harmonic functions of polynomial growth on complete manifolds
 N. Korevaar and R. Kusner  The structure of constant mean curvature embeddings in Euclidean three space
 Z. Li  Uniformization of spherical CR manifolds and the CR Yamabe problem
 P. Li  The theory of harmonic functions and its relation to geometry
 Y. Y. Li and G. Tian  Harmonic maps with prescribed singularities
 F. H. Lin  Some recent results on harmonic maps to spheres
 W. H. Meeks III  The geometry, topology, and existence of periodic minimal surfaces
 F. Morgan  Soap films and mathematics
 L. H. Mou  Uniform boundary regularity estimates for minima of certain quadratic functionals
 V. Oliker  Selfsimilar solutions and asymptotic behavior of flows of nonparametric surfaces driven by Gauss or mean curvature
 P. L. Robinson  A report on geometric quantization
 J. E. Taylor  Motion of curves by crystalline curvature, including triple junctions and boundary points
 C.L. Terng  Recent progress in submanifold geometry
 P. Tomter  Constant mean curvature surfaces in the Heisenberg group
 H. C. Wente  Complete immersions of constant mean curvature
 H. Wu  Banach manifolds of minimal surfaces in the 4sphere
 S. Zheng  On the isolatedness for the solutions of Plateau's problem
Part 2. Geometry in Mathematical Physics and Related Topics  M. Adams, C. McCrory, T. Shifrin, and R. Varley  Invariants of Gauss maps of theta divisors
 A. Banyaga  On characteristics of hypersurfaces in symplectic manifolds
 J. K. Beem  Disprisoning and pseudoconvex manifolds
 L. B. Bergery and A. Ikemakhen  On the holonomy of Lorentzian manifolds
 J.P. Bourguignon  Spinors, Dirac operators, and changes of metrics
 C. P. Boyer and B. M. Mann  The Hyperkähler geometry of the ADHM construction and quaternionic geometric invariant theory
 S. B. Bradlow  Nonabelian vortices and a new YangMillsHiggs energy
 E. Calabi and H. Gluck  What are the best almostcomplex structures on the 6sphere?
 L. A. Cordero, M. Fernández, and A. Gray  The failure of complex and symplectic manifolds to be Kählerian
 K. Corlette  Nonabelian Hodge theory
 X. Dai  Geometric invariants and their adiabatic limits
 A. Derdzinski  Geometry of elementary particles
 D. DeTurck, H. Goldschmidt, and J. Talvacchia  Existence of connections with prescribed YangMills currents
 L. D. Drager and R. L. Foote  Vector bundles over homogeneous spaces and complete, locally symmetric spaces
 T. A. Drumm  Margulis spacetimes
 D. Duncan and E. Ihrig  Incomplete flat homogeneous geometries
 P. E. Ehrlich and G. G. Emch  Geodesic and causal behavior of gravitational plane waves: astigmatic conjugacy
 J. H. G. Fu  Curvature of singular spaces via the normal cycle
 R. O. Fulp  The nonintegrable phase factor and gauge theory
 G. J. Galloway  The Lorentzian version of the CheegerGromoll splitting theorem and its application to general relativity
 P. Gauduchon  Weyl structures on selfdual conformal manifolds
 J. F. Glazebrook and F. W. Kamber  Chiral anomalies and Dirac families in Riemannian foliations
 S. G. Harris  What is the shape of space in a spacetime?
 A. D. Helfer  The kinematics of the gravitational field
 S. Helgason  Support theorems in integral geometry and their applications
 O. Hijazi  Killing spinors and eigenvalues of the Dirac operator
 G. R. Jensen and M. Rigoli  Einstein metrics on circle bundles
 T. Jiang and S. S.T. Yau  Topological and differentiable structures of the complement of an arrangement of hyperplanes
 H. Kim  The relationship between the moduli spaces of vector bundles on \(K3\) surfaces and Enriques surfaces
 C. LeBrun and Y. S. Poon  Selfdual manifolds with symmetry
 E. H. Lieb  Remarks on the Skyrme model
 E. B. Lin  Geometric settings for quantum systems with isospin
 J. Lott  Heat kernels on covering spaces and topological invariants
 Q.K. Lu  The heat kernels of symmetric spaces
 J. J. Millson  CRgeometry and deformations of isolated singularities
 L. K. Norris  Generalized symplectic geometry on the frame bundle of a manifold
 D. H. Phong  Complex geometry and string theory
 L. Sadun and J. Segert  Constructing nonselfdual YangMills connections on \(S^4\) with arbitrary Chern number
 A. Sengupta  The YangMills measure for the twosphere
 L. M. Sibner  Examples of nonminimal critical points in gauge theory
 N. K. Stanton  Spectral invariants of pseudoconformal manifolds
 M. Stern  \(L_2\)cohomology and index theory of noncompact manifolds
 W.W. Sung  On CalabiYau threefolds fibered over smooth complex surfaces
 G. Tian  Degeneration of KählerEinstein manifolds. I
 Y. L. L. Tong  Flat connections on products of determinant bundles
 M. Troyanov  Surfaces Riemanniennes à singularités simples
 S. T. Yau and F. Zheng  Remarks on certain higherdimensional quasiFuchsian domains
 S. Zucker  \(L^p\)cohomology: Banach spaces and homological methods on Riemannian manifolds
Part 3. Riemannian Geometry  R. E. Greene  Some concepts and methods in Riemannian geometry
 K. Abe and A. Haas  Isometric immersions of \(H^n\) into \(H^n+1\)
 S. B. Alexander, I. D. Berg, and R. L. Bishop  The distancegeometry of Riemannian manifolds with boundary
 S. B. Alexander and R. J. Currier  Hypersurfaces and nonnegative curvature
 S. J. Altschuler  Shortening space curves
 M. T. Anderson  Degeneration of metrics with bounded curvature and applications to critical metrics of Riemannian functionals
 A. Basmajian  The geometry of totally geodesic hypersurfaces in hyperbolic manifolds
 R. Brooks, P. Perry, and P. P. V  Finiteness of diffeomorphism types of isospectral manifolds
 P. Buser, B. Colbois, and J. Dodziuk  Small eigenvalues of the Laplacian on negatively curved manifolds
 R. D. Canary  Geometrically tame hyperbolic \(3\)manifolds
 I. Chavel and E. A. Feldman  Isoperimetric constants and large time heat diffusion in Riemannian manifolds
 I. Chavel and L. Karp  Large time behavior of solutions of the heat equation
 H. Chen  Manifolds with \(2\)nonnegative curvature operator
 D. DeTurck, H. Gluck, C. Gordon, and D. Webb  The geometry of isospectral deformations
 I. Dimitrić  Quadric representation of a submanifold and spectral geometry
 H. Donnelly  Embedded eigenvalues for asymptotically flat surfaces
 P. Eberlein, U. Hamenstädt, and V. Schroeder  Manifolds of nonpositive curvature
 F. T. Farrell and L. E. Jones  Topological rigidity for compact nonpositively curved manifolds
 K. Fukaya and T. Yamaguchi  Almost nonnegatively curved manifolds
 P. Ghanaat  Local structure of framed manifolds
 S. Gigena  Constant affine mean curvature hypersurfaces of decomposable type
 P. B. Gilkey  Heat equation asymptotics
 R. E. Greene and H. Wu  Nonnegatively curved manifolds which are flat outside a compact set
 D. Gromoll  Spaces of nonnegative curvature
 K. Grove  Critical point theory for distance functions
 D. Gurarie  Spectral geometry in higher ranks: Closed geodesics and flat tori
 C.K. Han  Ellipticity of local isometric embeddings
 J. Hass and P. Scott  Curve flows on surfaces and intersections of curves
 N. Hingston  Curve shortening, equivariant Morse theory, and closed geodesics on the \(2\)sphere
 D. Kalish  Morse theory for geodesics
 M. Kuranishi  On some metrics on \(S^2\times S^2\)
 A. Linnér  Curvestraightening
 Z.D. Liu  Ball covering property and nonnegative Ricci curvature outside a compact set
 C. M. Margerin  General conjugate Loci are not closed
 Y. Otsu  Collapsing of Riemannian manifolds and their excesses
 P. P. V  GromovHausdorff convergence of metric spaces
 E. A. Ruh  Cartan connections
 A. G. Setti  Eigenvalue estimates for the Laplacian with lower order terms on a compact Riemannian manifold
 J.P. Sha and D. Yang  Positive Ricci curvature on compact simply connected \(4\)manifolds
 Z. Shen and G. Wei  Volume growth and finite topological type
 K. Shiohama  Recent developments in sphere theorems
 T. Shioya  On the excess of open manifolds
 U. Simon and C. P. Wang  Local theory of affine \(2\)spheres
 R. J. Spatzier  Riemannian manifolds with completely integrable geodesic flows
 Y. Suyama  Differentiable structure on spheres and curvature
 Z. I. Szabo  Spectral theory for operator families on Riemannian manifolds
 P. Tondeur  Riemannian foliations and tautness
 J. Tysk  Eigenvalue problems for manifolds with singularities
 G. Walschap  Some rigidity aspects of Riemannian fibrations
 J.Y. Wu  Hausdorff convergence and sphere theorems
 R. J. Zimmer  Automorphism groups and fundamental groups of geometric manifolds
