Gromov's compactness theorem for pseudo holomorphic curves

Author:
Rugang Ye

Journal:
Trans. Amer. Math. Soc. **342** (1994), 671-694

MSC:
Primary 58E12; Secondary 53C23

MathSciNet review:
1176088

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a complete proof for Gromov's compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary.

**[El]**Yakov Eliashberg,*Filling by holomorphic discs and its applications*, Geometry of low-dimensional manifolds, 2 (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 151, Cambridge Univ. Press, Cambridge, 1990, pp. 45–67. MR**1171908****[Fe]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[Ga]**L. Z. Gao,*Convergence of Riemannian manifolds, Ricci pinching and*-*curvature pinching*, preprint.**[Gi]**Mariano Giaquinta,*Multiple integrals in the calculus of variations and nonlinear elliptic systems*, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR**717034****[Gro]**M. Gromov,*Pseudo holomorphic curves in symplectic manifolds*, Invent. Math.**82**(1985), 307-347.**[Grü]**Michael Grüter,*Regularity of weak 𝐻-surfaces*, J. Reine Angew. Math.**329**(1981), 1–15. MR**636440**, 10.1515/crll.1981.329.1**[Grü-Hi-Ni]**Michael Grüter, Stefan Hildebrandt, and Johannes C. C. Nitsche,*On the boundary behavior of minimal surfaces with a free boundary which are not minima of the area*, Manuscripta Math.**35**(1981), no. 3, 387–410. MR**636464**, 10.1007/BF01263271**[Ha-Wi]**Philip Hartman and Aurel Wintner,*On the local behavior of solutions of non-parabolic partial differential equations*, Amer. J. Math.**75**(1953), 449–476. MR**0058082****[Hi]**Stefan Hildebrandt,*Nonlinear elliptic systems and harmonic mappings*, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) Science Press, Beijing, 1982, pp. 481–615. MR**714341****[Ho-Spru]**D. Hoffman and J. Spruck,*Sobolev and isoperimetric inequalities for submanifolds in Riemannian manifolds*, Comm. Pure Appl. Math., 1970.**[Jo1]**Jürgen Jost,*Two-dimensional geometric variational problems*, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1991. A Wiley-Interscience Publication. MR**1100926****[Jo2]**Jürgen Jost,*On the regularity of minimal surfaces with free boundaries in Riemannian manifolds*, Manuscripta Math.**56**(1986), no. 3, 279–291. MR**856366**, 10.1007/BF01180769**[Jo3]**Jürgen Jost,*Continuity of minimal surfaces with piecewise smooth free boundaries*, Math. Ann.**276**(1987), no. 4, 599–614. MR**879539**, 10.1007/BF01456989**[Mc]**Dusa McDuff,*Elliptic methods in symplectic geometry*, Bull. Amer. Math. Soc. (N.S.)**23**(1990), no. 2, 311–358. MR**1039425**, 10.1090/S0273-0979-1990-15928-2**[Oh]**Yong-Geun Oh,*Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions*, Comm. Pure Appl. Math.**45**(1992), no. 1, 121–139. MR**1135926**, 10.1002/cpa.3160450106**[Pan]**P. Pansu,*Pseudo-holomorphic curves in symplectic manifolds*, preprint, Ecole Polytechnique, Palaiseau, 1986.**[Par-Wo]**T. H. Parker and J. G. Wolfson,*A compactness theorem for Gromov's moduli space*, preprint, 1991.**[Sa-U]**J. Sacks and K. Uhlenbeck,*The existence of minimal immersions of 2-spheres*, Ann. of Math. (2)**113**(1981), no. 1, 1–24. MR**604040**, 10.2307/1971131**[Sc]**Richard M. Schoen,*Analytic aspects of the harmonic map problem*, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), Math. Sci. Res. Inst. Publ., vol. 2, Springer, New York, 1984, pp. 321–358. MR**765241**, 10.1007/978-1-4612-1110-5_17**[Wo]**J. G. Wolfson,*Gromov’s compactness of pseudo-holomorphic curves and symplectic geometry*, J. Differential Geom.**28**(1988), no. 3, 383–405. MR**965221****[Ye1]**Rugang Ye,*Regularity of a minimal surface at its free boundary*, Math. Z.**198**(1988), no. 2, 261–275. MR**939540**, 10.1007/BF01163295**[Ye2]**-,*On minimal surfaces of higher topology*, preprint, Stanford, 1988.**[Ye3]**-,*An isoperimetric inequality for Riemannian submanifolds with free boundary*, in preparation.**[Ye4]**-,*Existence, regularity and finiteness of minimal surfaces with free boundary*, preprint no. 1, SFB256, Bonn, 1987.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58E12,
53C23

Retrieve articles in all journals with MSC: 58E12, 53C23

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1176088-1

Article copyright:
© Copyright 1994
American Mathematical Society