Introduction .................................................. vii
1. Harmonic, pluriharmonic, holomorphic maps and basic
Hermitian and Kahlerian geometry ............................. 1
1.1. The general setting ..................................... 1
1.2. The complex case ........................................ 6
1.3. Hermitian bundles ...................................... 10
1.4. Complex geometry via moving frames ..................... 12
1.5. Weitzenbock-type formulas .............................. 17
2. Comparison Results .......................................... 27
2.1. Hessian and Laplacian comparison ....................... 27
2.2. Volume comparison and volume growth .................... 40
2.3. A monotonicity formula for volumes ..................... 58
3. Review of spectral theory ................................... 63
3.1. The spectrum of a self-adjoint operator ................ 63
3.2. Schrodinger operators on Riemannian manifolds .......... 69
4. Vanishing results ........................................... 83
4.1. Formulation of the problem ............................. 83
4.2. Liouville and vanishing results ........................ 84
4.3. Appendix: Chain rule under weak regularity ............. 99
5. A finite-dimensionality result ............................. 103
5.1. Peter Li's lemma ...................................... 107
5.2. Poincare-type inequalities ............................ 110
5.3. Local Sobolev inequality .............................. 114
5.4. L2 Caccioppoli-type inequality ........................ 117
5.5. The Moser iteration procedure ......................... 118
5.6. A weak Harnack inequality ............................. 121
5.7. Proof of the abstract finiteness theorem .............. 122
6. Applications to harmonic maps .............................. 127
6.1. Harmonic maps of finite Lp-energy ..................... 127
6.2. Harmonic maps of bounded dilations and a Schwarz-
type lemma ............................................ 136
6.3. Fundamental group and harmonic maps ................... 141
6.4. A generalization of a finiteness theorem of Lemaire ... 143
7. Some topological applications .............................. 147
7.1. Ends and harmonic functions ........................... 147
7.2. Appendix: Further characterizations of parabolicity ... 165
7.3. Appendix: The double of a Riemannian manifold ......... 171
7.4. Topology at infinity of submanifolds of C-H spaces .... 172
7.5. Line bundles over Kahler manifolds .................... 178
7.6. Reduction of codimension of harmonic immersions ....... 179
8. Constancy of holomorphic maps and the structure of
complete Kähler manifolds .................................. 183
8.1. Three versions of a result of Li and Yau .............. 183
8.2. Plurisubharmonic exhaustions .......................... 199
9. Splitting and gap theorems in the presence of
a Poincare-Sobolev inequality .............................. 205
9.1. Splitting theorems .................................... 205
9.2. Gap theorems .......................................... 223
9.3. Gap Theorems, continued ............................... 229
A. Unique continuation ........................................ 235
В. Lp-cohomology of non-compact manifolds ..................... 251
B.l. The Lp de Rham cochain complex: reduced and
unreduced cohomologies ................................ 251
B.2. Harmonic forms and L2-cohomology ...................... 260
B.3. Harmonic forms and Lp≠2-cohomology ..................... 262
B.4. Some topological aspects of the theory ................ 265
Bibliography .................................................. 269
Index ......................................................... 281
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