Contents of the Current Volume
Volume 1
metric Geometry and minimal SubmanifoldS
Foreword by Shing-Tung Yau XV
Preface by the Editors XXI
Commentary by Richard Hamilton XXIII
Commentary by William H Meeks III XXV
Commentary by Richard Schoen XXXI
Commentary by Leon Simon XXXVII
Articles
“On the fundamental group of compact manifolds of non-positive curvature”，
Annals of Mathematics 93 (1971) 579–585 1
(with H Lawson)， “Compact manifolds of nonpositive curvature”， J Differential
Geom 7 (1972)， 211–2289
“Some global theorems on non-complete surfaces”， Comment Math Helv 48
(1973)， 177–187 27
(with H Lawson)， “Scalar curvature， non-abelian group actions， and the degree of
symmetry of exotic spheres”， Comment Math Helv 49 (1974)， 232–24439
“Submanifolds with constant mean curvature”， Amer J Math 96 (1974)， no 2，
346–36653
“Submanifolds with constant mean curvature II”， Amer J Math 97 (1975)， no 1，
76–10075
Contents of the Current Volume
“Curvature preserving diffeomorphisms”， Annals of Mathematics (2) 100
(1974)， 121–130 101
“Non-existence of continuous convex functions on certain Riemannian manifolds”，
Math Ann 207 (1974) 269–270 111
(with S-Y Cheng)， “Hypersurfaces with constant scalar curvature”， Math Ann 225
(1977)， no 3， 195–204 113
(with R Schoen and L Simon)， “Curvature estimates for minimal hypersurfaces”，
Acta Math 134 (1975)， no 3-4， 275–288 123
(with S-Y Cheng)， “Maximal space-like hypersurfaces in the Lorentz-Minkowski
spaces”， Annals of Mathematics 104 (1976)， no 3， 407–419 137
“Remarks on the group of isometries of a Riemannian manifold”， Topology 16
(1977)， no 3， 239–247 151
(with W H Meeks)， “The equivariant Dehn’s lemma and loop theorem”， Comment
Math Helv 56 (1981) 225–239 161
(with R Schoen)， “On the structure of manifolds with positive scalar curvature”，
Manuscripta Math 28 (1979)， no 1-3， 159–183 177
(with W H Meeks)， “The equivariant loop theorem for three-dimensional
manifolds and a review of the existence theorems for minimal surfaces”，
Pure Appl Math 112 (1984) 153–163 191
(with W H Meeks)， “Topology of three-dimensional manifolds and the embedding
problems in minimal surface theory”， Annals of Mathematics 112
(1980) 441–484 203
(with W H Meeks)， “The classical Plateau problem and the topology of three-
dimensional manifolds: The embedding of the solution given by Douglas-Morrey
and an analytic proof of Dehn’s lemma”， Topology 21 (1982) 409–442 247
(with W H Meeks and L Simon)， “Embedded minimal surfaces， exotic spheres，
and manifolds with positive Ricci curvature”， Annals of Mathematics 116 (1982)， 621–
659 281
(with R Schoen)， “Complete three-dimensional manifolds with positive Ricci
curvature and scalar curvature”， Annals of Mathematics Studies 102 (Seminar
on Differential Geometry) 209–228， Princeton University Press， 1982 321
Volume 1 Metric Geometry and Minimal Submanifolds
(with WH Meeks)， “The existence of embedded minimal surfaces and the problem
of uniqueness”， Math Z 179 (1982)， 151–168 341
(with W H Meeks)， “Group actions on R3”， Pure Appl Math 112 (1984)，
167–179 359
Appendices
Photographs 373
Acknowledgements 383
Contents of Volumes 2， 3， 4 and 5
Volume 2
metric Geometry and Harmonic functionS
Foreword by Shing-Tung Yau XV
Preface by the Editors XXI
Commentary by Peter Li XXIII
Articles
(with M Friedman)， “Homotopically trivial symmetries of Haken manifolds are
toral”， Topology 22 (1983) 179–189 1
(with S-Y Cheng)， “Complete affine hypersurfaces I The completeness of affine
metrics”， Comm Pure Appl Math 39 (1986) 839–866 13
(with R Schoen)， “The structure of manifolds with positive scalar curvature”， in
Directions in Partial Differential Equations， 235–242 Madison， Wisconsin:
Academic Press， 1987 41
(with L Z Gao)， “The existence of negatively Ricci curved metrics on three-
manifolds”， Inventiones Mathematicae 85 (1986) 637–65249
(with R Schoen)， “Conformally flat manifolds， Kleinian groups and scalar
curvature”， Inventiones Mathematicae 92 (1988) 47–71 65
“Uniformization of geometric structures”， Proc Sympos Pure Math 48 (1988)
265–27491
“Harmonic functions on complete Riemannian manifolds”， Comm Pure Appl
Math 28 (1975)， 201–228 101
(with S-Y Cheng)， “Differential equations on Riemannian manifolds and their
geometric applications”， Comm Pure Appl Math 28 (1975) 333–354 129
Contents of Volumes 2， 3， 4 and 5
“Isoperimetric constants and the first eigenvalue of a compact Riemannian
manifold”， Ann Sci Ecole Norm Sup (4) 8 (1975)， no 4， 487–507 151
“Some function-theoretic properties of complete Riemannian manifold and their
applications to geometry”， Indiana Univ Math J 25 (1976)， no 7， 659–670 173
“Erratum to ‘Some function-theoretic properties of complete Riemannian
manifold and their applications to geometry’”， Indiana Univ Math J 31
(1982)， no 4， 607 185
(with R Schoen)， “Harmonic maps and the topology of stable hypersurfaces and
manifolds with non-negative Ricci curvature， Comment Math Helv 51 (1976)，
no 3， 333–341 187
“A general Schwarz lemma for Khler manifolds”， Amer J Math 100 (1978)， no 1，
197–203 197
(with R Schoen)， “On univalent harmonic maps between surfaces”， Inventiones
Mathematicae 44 (1978)， no 3， 265–278 205
“On the heat kernel of a complete Riemannian manifold”， J Math pures et appl
(9) 57 (1978)， no 2， 191–201 219
(with R Schoen)， “Compact group actions and the topology of manifolds with non-
positive curvature”， Topology 18 (1979)， no 4， 361–380 231
(with R Schoen)， “Corrections to: ‘Compact group actions and the topology of
manifolds with non-positive curvature’”， Topology 18 (1979)， no 4， 483 251
(with P Li)， “A new conformal invariant and its applications to the Willmore
conjecture and the first eigenvalue of compact surfaces”，
Inventiones Mathematicae 69 (1982) 269–291 253
(with P Li and R Schoen)， “On the isoperimetric inequality for minimal surfaces”，
Ann Scuola Norm Sup Pisa Cl Sci (4) 11 (1984) 237–244 277
(with J Jost)， “The strong rigidity of locally symmetric complex manifolds of rank
one and finite volume”， Math Ann 275 (1986) 291–304 285
(with J Jost)， “Harmonic maps and Khler geometry”， in Prospects in Complex
Geometry， pp 340–370， Lecture Notes in Math 1468， Springer， 1991 299
Volume 3 Eigenvalues and General Relativity
“Problem Section”， Annals of Mathematics Studies 102 (Seminar on Differential
Geometry)， 669–706， Princeton University Press， 1982 331
Appendices
Photographs 369
Acknowledgements 373
Volume 3
eiGenValueS and General relatiVity
Foreword by Shing-Tung Yau XV
Preface by the Editors XXI
Articles
(with P Li)， “Estimates of eigenvalues of a compact Riemannian manifold， in
Geometry of the Laplace operator， pp 205–239， Proc Sympos Pure Math 36，
American Mathematical Society， 19801
(with R Schoen and S Wolpert)， “Geometric bounds on the low eigenvalues of a
compact surface: Geometry of the Laplace operator”， pp 279–285， Proc
Sympos Pure Math 36， American Mathematical Society， 198037
(with Paul C Yang) “Eigenvalues of the Laplacian of compact Riemann surfaces
and minimal submanifolds”， Ann Scuola Norm Sup Pisa Cl Sci (4) 7
(1980) 55–6345
(with J Cheeger)， “A lower bound for the heat kernel”， Comm Pure Appl Math
34 (1981) 465–48055
(with S-Y Cheng and P Li)， “On the upper estimate of the heat kernel of a
complete Riemannian manifold”， Amer J Math 103 (1981) 1021–1063 71
(with P Li)， “On the Schrdinger equation and the eigenvalue problem”， Comm
Math Phys 88 (1983) 309–318 115
(with S-Y Cheng and P Li)， “Heat equations on minimal submanifolds and their
applications”， Amer J Math 106 (1984) 1033–1065 125
Contents of Volumes 2， 3， 4 and 5
(with I M Singer， B Wong and Stephen S-T Yau)， “An estimate of the gap of the
first two eigenvalues in the Schrdinger operator”， Ann Scuola Norm Sup
Pisa Cl Sci (4) 12 (1985)， 319–333 159
(with P Li)， “On the parabolic kernel of the Schrdinger operator”， Acta Math
156 (1986) 153–201 175
(with Y Y Lu)， “Eigenvalues of the Laplacian through boundary integral equations”，
SIAM J Matrix Anal Appl 12 (1991) 597–609 225
(with R Schoen)， “Positivity of the total mass of a general space-time”， Phys Rev
Lett 43 (1979)， no 20， 1457–1459 239
(with R Schoen)， “Existence of incompressible minimal surfaces and the topology
of three-dimensional manifolds with nonnegative scalar curvature”， Annals of
Mathematics (2) 110 (1979) 127–142 243
(with R Schoen)， “On the proof of the positive mass conjecture in general
relativity”， Comm Math Phys 65 (1979)， no 1， 45–76 259
“The total mass and the topology of an asymptotically flat space-time”， in The
Chern Symposium 1979， pp 255–259 Springer， 1980 291
(with R Schoen， “The energy and the linear momentum of space-times in general
relativity”， Comm Math Phys 79 (1981) 47–51 297
(with R Schoen)， “Proof of the positive mass theorem， II”， Comm Math Phys 79
(1981) 231–260 303
(with R Schoen)， “Proof that the Bondi mass is positive”， Phys Rev Lett 48
(1982) 369–371 333
(with R Schoen)， “The existence of a black hole due to condensation of matter”，
Comm Math Phys 90 (1983) 575–579 337
(with D Christodoulou)， “Some remarks on the quasi-local mass”， 9–14， Contemp
Math 71， American Mathematical Society， 1988343
(with J Smoller， A Wasserman and B McLeod)， “Smooth static solutions of the
Einstein/Yang-Mills equations”， Comm Math Phys 143 (1991) 115–147 349
Volume 4 Khler Geometry I
Appendices
Photographs 383
Acknowledgements 389
Volume 4
KHler Geometry i
Foreword by Shing-Tung Yau XIII
Preface by the Editors XIX
Commentary by Duong H Phong XXI
Commentary by Ngaiming Mok XXV
Articles
“The role of partial differential equations in differential geometry”， in Proceedings
of the International Congress of Mathematicians， 237–250 Helsinki: Acad
Sci Fennica， 1980 1
(with J-P Bourguignon)， “Sur les métriques riemanniennes á courbure de Ricci
nulle sur le quotient d’une surface K3”， C R Acad Sci Paris Sér A-B 277
(1973)， A1175–A1177 15
(with S-Y Cheng)， “On the regularity of the solution of the n-dimensional
Minkowski problem”， Comm Pure Appl Math 29 (1976)， no 5， 495–51619
“Intrinsic measures of compact complex manifolds”， Math Ann 212 (1975)，
317–32941
(with Yum-Tong Siu)， “On the structure of complete simply-connected Khler
manifolds with nonpositive curvature”， Proc Nat Acad Sci USA 73 (1976)，
no 4， p 100855
“Parallelizable manifolds without complex structure”， Topology 15 (1976)， no 1，
51–53 57
(with S-Y Cheng)， “On the regularity of the Monge–Ampère equation”， Comm
Pure Appl Math 30 (1977)， no 1， 41–6861
Contents of Volumes 2， 3， 4 and 5
“Calabi’s conjecture and some new results in algebraic geometry”， Proc Nat Acad
Sci USA 74 (1977)， no 5， 1798–179989
(with Y-T Siu)， “Complete Khler manifolds with nonpositive curvature of faster
than quadratic decay”， Annals of Mathematics (2) 105 (1977)， no 2， 225–264 91
“On the Ricci curvature of a compact Khler manifold and the complex Monge-
Ampére equation I”， Comm Pure Appl Math 31 (1978)， no 3， 339–411 131
“Métriques de Khler-Einstein sur les variétés ouvertés”， Astérisque 58 (1978)
163-167 205
(with S-Y Cheng)， “On the existence of a complete Khler metric on noncompact
complex manifolds and the regularity of Fefferman’s equation”， Comm Pure
Appl Math 33 (1980) 507–544 211
(with N Mok and Y-T Siu)， “The Poincaré-Lelong equation on complete Khler
manifolds”， Compositio Math 44 (1981) 183–218 265
(with S-Y Cheng)， “The real Monge–Ampère equation and affine flat structures”，
in Proceedings of the 1980 Beijing Symposium on Differential Geometry and
Differential Equations， 339–370 Beijing: Science Press， 1982 301
(with Y-T Siu)， “Compactification of negatively curved complete Khler manifolds
of finite volume”， Annals of Mathematics Studies 102 (Seminar on Differential
Geometry)， 363–380， Princeton University Press， 1982 333
(with N Mok)， “Completeness of the Khler-Einstein metric on bounded domains
and the characterization of domains of holomorphy by curvature conditions”，
Proc Sympos Pure Math 39， Part 1 (1983)， 41–59 351
Appendices
Photographs 371
Acknowledgements 375
Volume 5 Khler Geometry II
Volume 5
KHler Geometry ii
Foreword by Shing-Tung Yau XIII
Preface by the Editors XIX
Articles
(with K Uhlenbeck)， “On the existence of Hermitian-Yang-Mills connections in
stable vector bundles”， Comm Pure Appl Math 39 (1986)， supplement，
S257–S293 1
(with J Jost)， “A strong rigidity theorem for a certain class of compact complex
analytic surfaces”， Math Ann 271 (1985) 143–15239
(with S Y Cheng)， “Inequality between Chern numbers of singular Khler surfaces
and characterization of orbit space of discrete group of SU (2， 1)”， Contemp
Math 49 (1986)， 31–4449
“Nonlinear analysis in geometry”， Enseign Math (2) 33 (1987) no 1-2， 109–158 63
(with J Li)， “Hermitian-Yang-Mills Connection on Non-Khler Manifolds”， in
Mathematical Aspects of String Theory， 560–573， World Scientific， 1987
DOI: 101142/9789812798411_0027113
(with G Tian)， “Existence of Khler-Einstein metrics on complete Khler manifolds
and their applications to algebraic geometry”， in Mathematical Aspects of
String Theory， 574–628， World Scientific， 1987
DOI: 101142/9789812798411_0028 127
(with G Tian)， “Three-dimensional algebraic manifolds with C1 = 0 and χ = –6”，
in Mathematical Aspects of String Theory， 543–559， World Scientific， 1987
DOI: 101142/9789812798411_0026 183
(with G Tian)， “Khler-Einstein metrics on complex surfaces with C1 > 0”， Comm
Math Phys 112 (1987) 175–203 201
(with K Uhlenbeck)， “A note on our previous paper: ‘On the existence of Hermitian
Yang-Mills connections in stable vector bundles’”， [Comm Pure Appl Math
39 (1986)， S257–S293]， Comm Pure Appl Math 42 (1989) 703–707 231
Contents of Volumes 2， 3， 4 and 5
(with B R Green， A Shapere and C Vafa)， “Stringy cosmic strings and non-
compact Calabi-Yau manifolds”， Nuclear Phys B 337 (1990)， no 1， 1–36 237
(with J Li and F Zheng)， “A simple proof of Bogomolov’s theorem on class II0
surfaces with b2 = 01”， Illinois J Math 34 (1990) 217–220 273
(with S-Y Lu)， “Holomorphic curves in surfaces of general type”， Proc Nat Acad
Sci USA 87 (1990) 80–82 277
(with G Tian)， “Complete Khler manifolds with zero Ricci curvature I”， J Amer
Math Soc 3 (1990) 579–609 281
(with B R Greene and S-S Roane)， “Geometric singularities and spectra of
Landau-Ginzburg models”， Comm Math Phys 142 (1991) 245–259 313
(with F Zheng)， “On projective manifolds covered by space C”， in International
Symposium in Memory of Hua Loo Keng， Vol II (Beijing， 1988)， 323–332，
Springer， 1991 329
(with F Zheng)， “Negatively 1/4-pinched Riemannian metric on a compact Khler
manifold”， Inventiones Mathematicae 103 (1991) 527–535 339
(with G Tian)， “Complete Khler manifolds with zero Ricci curvature II”，
Inventiones Mathematicae 106 (1991) 27–60 349
Appendices
Curriculum Vitae of Shing-Tung Yau 383
List of Former Students 387
List of Publications by Shing-Tung Yau， 1970–1991 389
Acknowledgements 397
內容試閱：
Preface by the Editors
One of the greatest mathematicians in the world， Shing-Tung Yau has earned
numerous honors， including the Fields Medal， the mathematical equivalent of
the Nobel Prize. The focus of much of his work has been in the areas of differential
geometry， algebraic and K.hler geometry， general relativity， and string theory.
Yau has also been an advisor and mentor to a vast number of mathematicians
including the editors of this volume. His influence in the development and establishment
of these research areas cannot be overstated.
These five volumes constitute the first part of the Selected Works of Shing-
Tung Yau， which reproduce part of his published mathematical work from 1971
to 1991. It is the period where numerous groundbreaking works of Yau were
produced， in geometric analysis， in K.hler geometry and in general relativity.
These volumes reflect his achievements in this period. We divide the volumes
into subjects on metric geometry， minimal submanifolds， harmonic functions， eigenvalues，
general relativity， and K.hler geometry. We include commentaries by
experts on the mathematical content in each volume， and personal comments on
some of the development of the ideas for these papers. The order of the appearance
of each paper in the volumes is that best reflecting the development of the
subjects. We thank Pengfei Guan， Bong Lian， Kefeng Liu， Leon Simon， Valentino
Tosatti， Jiaping Wang， Xu-Jia Wang， and Mu-Tao Wang for their help in suggesting
the papers to be included in these volumes. We thank Shing-Tung Yau for his
help in the final selection of the papers， and in the ordering of the appearance of
the papers. We hope that the works and the commentaries in these volumes will
bring to readers anew his insight， and his perspective on the subject over time，
as it certainly did to us， the editors of the volumes. The volumes will be of great
interest to research mathematicians， theoretical physicists， and graduate students
in these areas.
We thank International Press and the generosity of the journal publishers of
the papers， without which these volumes would not have been possible. Finally，
we take this opportunity to express our deepest gratitude to Shing-Tung Yau for
his inspiration and generosity which continues to be a driving force behind our
mathematical work.
Huai-Dong Cao， Jun Li， and Richard Schoen
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