**Contemporary Mathematics**

Volume: 644;
2015;
359 pp;
Softcover

MSC: Primary 31; 32; 53; 35;

**Print ISBN: 978-1-4704-1464-1
Product Code: CONM/644**

List Price: $112.00

AMS Member Price: $89.60

MAA Member Price: $100.80

**Electronic ISBN: 978-1-4704-2631-6
Product Code: CONM/644.E**

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# Analysis, Complex Geometry, and Mathematical Physics: In Honor of Duong H. Phong

Share this page *Edited by *
*Paul M. N. Feehan; Jian Song; Ben Weinkove; Richard A. Wentworth*

This volume contains the proceedings of the Conference on Analysis,
Complex Geometry and Mathematical Physics: In Honor of Duong H. Phong,
which was held from May 7–11, 2013, at Columbia University, New
York. The conference featured thirty speakers who spoke on a range of
topics reflecting the breadth and depth of the research interests of
Duong H. Phong on the occasion of his sixtieth birthday. A common
thread, familiar from Phong's own work, was the focus on the interplay
between the deep tools of analysis and the rich structures of geometry
and physics.

Papers included in this volume cover topics such as the complex
Monge-Ampère equation, pluripotential theory, geometric partial
differential equations, theories of integral operators, integrable
systems and perturbative superstring theory.

#### Readership

Graduate students and research mathematicians interested in partial differential equations, complex analysis, and mathematical physics.

# Table of Contents

## Analysis, Complex Geometry, and Mathematical Physics: In Honor of Duong H. Phong

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface ix10 free
- Scientific program xi12 free
- Conference Photographs xxi22 free
- Bergman kernel and pluripotential theory 128 free
- On the convergence of the Sasaki-Ricci flow 1138
- The signed mean curvature measure 2350
- Topics in two-loop superstring perturbation theory 3360
- 1. Introduction 3360
- 2. String perturbation theory 3562
- 3. Superstrings 3663
- 4. Independence of left and right chiralities 3764
- 5. Matching left and right moduli spaces 3966
- 6. The super period matrix at genus 2 3966
- 7. The chiral measure in terms of 𝜗-functions 4067
- 8. Chiral amplitudes 4168
- 9. Modular properties 4269
- 10. Supersymmetry and non-renormalization theorems 4370
- 11. The four-point amplitudes 4370
- 12. Remarks on higher genus 4572
- 13. Perturbative supersymmetry breaking 4673
- 14. Superstrings on \ZZ₂×\ZZ₂ Calabi-Yau orbifolds 4774
- 15. Modular orbits of twists 4976
- 16. Structure of the two-loop vacuum energy 5077
- 17. Contribution from the interior of super moduli space 5178
- 18. A new modular identity for 𝑆𝑝(4,ℤ)/ℤ₄ 5279
- 19. Contribution from the boundary of super moduli space 5380
- 20. Orbits under the modular subgroup 𝑆𝐿(2,\ZZ)₁×𝑆𝐿(2,\ZZ)₂ 5582
- 21. Heterotic 𝐸₈×𝐸₈ versus 𝑆𝑝𝑖𝑛(32)/\ZZ₂ 5683
- References 5784

- A rigidity theorem for hypersurfaces in higher dimensional space forms 6188
- Continuous approximation of quasiplurisubharmonic functions 6794
- Discrete holomorphicity and Ising model operator formalism 79106
- Stable Higgs bundles and Hermitian-Einstein metrics on non-Kähler manifolds 117144
- Weak solutions to the complex Monge-Ampère equation on Hermitian manifolds 141168
- Introduction 141168
- 1. Basic properties of 𝜔-psh functions in the Hermitian setting 143170
- 2. The modified comparison principle 144171
- 3. The domination principle 146173
- 4. The Dirichlet problem in a bounded domain in \coⁿ 147174
- 5. Existence of continuous solutions on a compact Hermitian manifold 149176
- References 156183

- Uniform asymptotic expansionon Riemann surfaces 159186
- Scaling asymptotics of heat kernels of line bundles 175202
- Parabolic frequency monotonicity and a theorem of Hardy-Pólya-Szegö 203230
- Topology of Dolbeault cohomology groups 211238
- 1. Introduction 211238
- 2. The 𝐿² \db problem in complex manifolds 213240
- 3. On the Hausdorff property for Dolbeault cohomology groups for domains in ℂⁿ 216243
- 4. Non-closed range property for \db on pseudoconvex domains in complex manifolds 219246
- 5. Holomorphic functions on domains with Levi-flat boundary 220247
- References 223250

- Uniformly bounded orthonormal sections of positive line bundles on complex manifolds 227254
- Poisson Equations, Uniformization, and Geometrical Optics 241268
- 1. Introduction 241268
- 2. Solving \Dg𝑢=𝑓 on compactly perturbed planes 244271
- 3. Asymptotic solutions to \Dg𝑢=𝑓 247274
- 4. Poisson integral on a complete manifold with compact boundary 250277
- 5. Solving \Dg𝑢=𝑓 on a complete Riemannian manifold 254281
- 6. Rougher boundary values 256283
- Appendix A. Harnack estimates, Liouville theorems, and uniformization 258285
- References 260287

- Non-Kähler Calabi-Yau manifolds 261288
- The point source inverse back-scattering problem 279306
- Local Regularity of the Complex Monge-Ampère Equation 291318
- Notes on holomorphic string and superstring theory measures of low genus 307334
- 1. Introduction 307334
- 2. The bosonic string in genus 1 and 2 308335
- 3. Superstrings in genus 1 and 2 320347
- 4. Superstring amplitude at a separating degeneration 334361
- 5. Superstring amplitude at a nonseparating degeneration 342369
- Appendix A. Mumford isomorphism with a marked point 351378
- Appendix B. Comparison to the result of D’Hoker and Phong 353380
- Appendix C. Super Mumford form with Ramond punctures 355382
- References 358385

- Back Cover Back Cover1388