scholarly journals COMPLEX SUBMANIFOLDS OF ALMOST COMPLEX EUCLIDEAN SPACES

2009 ◽  
Vol 61 (4) ◽  
pp. 401-405 ◽  
Author(s):  
A. J. Di Scala ◽  
L. Vezzoni
Keyword(s):  
Euclidean Spaces ◽  
Almost Complex ◽  
Complex Submanifolds

Realizing homology classes by almost-complex submanifolds

1980 ◽  
Vol 172 (2) ◽  
pp. 195-201
Author(s):  
Elmer Rees ◽  
Emery Thomas
Keyword(s):  
Almost Complex ◽  
Complex Submanifolds

Almost complex submanifolds of nearly Kähler manifolds

2020 ◽  
Vol 115 (3) ◽  
pp. 353-358
Author(s):  
Limiao Lin ◽  
Luc Vrancken ◽  
Anne Wijffels
Keyword(s):  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Almost Complex ◽  
Complex Submanifolds ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds

scholarly journals On embeddings of almost complex manifolds in almost complex Euclidean spaces

2016 ◽  
Vol 101 ◽  
pp. 19-26 ◽  
Author(s):  
Antonio J. Di Scala ◽  
Naohiko Kasuya ◽  
Daniele Zuddas
Keyword(s):  
Complex Manifolds ◽  
Euclidean Spaces ◽  
Almost Complex Manifolds ◽  
Almost Complex

scholarly journals ANALYSIS OF CONTACT CAUCHY–RIEMANN MAPS II: CANONICAL NEIGHBORHOODS AND EXPONENTIAL CONVERGENCE FOR THE MORSE–BOTT CASE

2017 ◽  
Vol 231 ◽  
pp. 128-223
Author(s):  
YONG-GEUN OH ◽  
RUI WANG
Keyword(s):  
Exponential Decay ◽  
A Priori ◽  
Asymptotic Convergence ◽  
Decay Estimates ◽  
Pseudoholomorphic Curves ◽  
Contact Homology ◽  
Interval Method ◽  
Almost Complex ◽  
Complex Submanifolds ◽  
Cauchy Riemann

This is a sequel to the papers Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817; Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3). In Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817), the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,\unicode[STIX]{x1D706},J)$. In Oh and Wang (Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3), they used the connection to establish a priori$W^{k,p}$-coercive estimates for maps $w:\dot{\unicode[STIX]{x1D6F4}}\rightarrow M$ satisfying $\overline{\unicode[STIX]{x2202}}^{\unicode[STIX]{x1D70B}}w=0$, $d(w^{\ast }\unicode[STIX]{x1D706}\circ j)=0$without involving symplectization. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus $Q$ foliated by closed Reeb orbits of a Morse–Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,\unicode[STIX]{x1D706},J)$, with $\unicode[STIX]{x1D706}$ a Morse–Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes.We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse–Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois (A Morse–Bott approach to contact homology, Ph.D. dissertation, Stanford University, 2002) (resp. by Bao (On J-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends, Pacific J. Math. 278(2) (2015), 291–324)), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse–Bott case is an essential ingredient in the setup of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).

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