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Differential Geometry
Edited by: Robert E. Greene, University of California, Los Angeles, CA, and S. T. Yau, Harvard University, Cambridge, MA

Proceedings of Symposia in Pure Mathematics
1993; 2000 pp; hardcover
Volume: 54
ISBN-10: 0-8218-1493-1
ISBN-13: 978-0-8218-1493-2
List Price: US$322
Member Price: US$257.60
Order Code: PSPUM/54
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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 wide-ranging 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.


Research mathematicians and/or advanced graduate students in differential geometry, partial differential equations and certain related areas of physics.


"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 area-minimizing surfaces and geometric measure theory
  • D. Bao and T. Ratiu -- On the geometrical origin and the solutions of a degenerate Monge-Ampère 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 finite-dimensional 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 level-set flow on a manifold
  • J. Jost -- Unstable solutions of two-dimensional 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 -- Self-similar 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 4-sphere
  • 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 -- Non-abelian vortices and a new Yang-Mills-Higgs energy
  • E. Calabi and H. Gluck -- What are the best almost-complex structures on the 6-sphere?
  • 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 Yang-Mills currents
  • L. D. Drager and R. L. Foote -- Vector bundles over homogeneous spaces and complete, locally symmetric spaces
  • T. A. Drumm -- Margulis space-times
  • 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 Cheeger-Gromoll splitting theorem and its application to general relativity
  • P. Gauduchon -- Weyl structures on self-dual 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 -- Self-dual 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 -- CR-geometry 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 non-self-dual Yang-Mills connections on \(S^4\) with arbitrary Chern number
  • A. Sengupta -- The Yang-Mills measure for the two-sphere
  • 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 Calabi-Yau three-folds fibered over smooth complex surfaces
  • G. Tian -- Degeneration of Kähler-Einstein 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 higher-dimensional quasi-Fuchsian 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 distance-geometry 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 -- Curve-straightening
  • 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 -- Gromov-Hausdorff 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
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