### On the Applications of Bochner-Kodaira-Morrey-Kohn Identity

Download PDF

**Authors:**S. SABER

**DOI:**10.46793/KgJMat2106.881S

**Abstract:**

This paper is devoted to studying some applications of the Bochner-Kodaira-Morrey-Kohn identity. For this study, we deﬁne a condition which is called (H

_{q}) condition which is related to the Levi form on the complex manifold. Under the (H

_{q}) condition and combining with the basic Bochner-Kodaira-Morrey-Kohn identity, we study the L

^{2}∂ Cauchy problems on domains in ℂ

^{n}, Kähler manifold and in projective space. Also, we study this problem on a piecewise smooth strongly pseudoconvex domain in a complex manifold. Furthermore, the weighted L

^{2}∂ Cauchy problem is studied under the same condition in a Kähler manifold with semi-positive holomorphic bisectional curvature. On the other hand, we study the global regularity and the L

^{2}theory for the ∂-operator with mixed boundary conditions on an annulus domain in a Stein manifold between an inner domain which satisfy (H

_{n−q−1}) and an outer domain which satisfy (H

_{q}).

**Keywords:**

∂, ∂-Neumann operator, weakly q-convex domains.

**References:**

[1] O. Abdelkader and S. Saber, Solution to ∂-equations with exact support on pseudoconvex manifolds, Int. J. Geom. Methods Mod. Phys. 4 (2007), 339–348.

[2] J. Cao, M. C.-Shaw and L Wang, Estimates for the ∂-Neumann
problem and nonexistence of C^{2} Levi-ﬂat hypersurfaces in ℙ^{n},
Math. Z. 248 (2004), 183–221.

[3] D. Catlin, Suﬃcient conditions for the extension of CR structures, J. Geom. Anal. 4 (1994), 467–538.

[4] D. W. Catlin and S. Cho, Extension of CR structures on three dimensional compact pseudoconvex CR manifolds, Math. Ann. 334(2) (2006), 253–280.

[5] S.-C. Chen and M.-C. Shaw, Partial Diﬀerential Equations in Several Complex Variables, AMS/IP Stud. Adv. Math. 19, Amer. Math. Soc., Providence, R.I., 2001.

[6] S. Cho, Extension of CR structures on pseudoconvex CR manifolds with one degenerate eigenvalue, Tohoku Math. J. 55(3) (2003), 321–360.

[7] P. W. Darko, The L^{2}-∂-problem on manifolds with piecewise
strictly pseudoconvex boundaries, Math. Proc. Cambridge Philos.
Soc. (1994), 116–147.

[8] G. M. Henkin and A. Iordan, Regularity of ∂ on pseudococave compacts and applications, Asian J. Math. 4 (2000), 855-884.

[9] L. Hörmander, L^{2}-estimates and existence theorems for the
∂-operator, Acta Math. 113 (1965), 89–152.

[10] L. Ho, ∂-problem on weakly q-convex domains, Math. Ann. 290 (1991), 3–18.

[11] X. Huang and X. Li, ∂-equation on a lunar domain with mixed boundary conditions, Trans. Amer. Math. Soc. 368(10) (2016), 6915–6937.

[12] J. J. Kohn, Harmonic integrals on strongly pseudoconvex manifolds, I, Ann. of Math. 78 (1963), 112–148.

[13] J. J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. 81 (1965), 451–472.

[14] J. J. Kohn, Global regularity for ∂ on weakly pseudo-convex manifolds, Trans. Amer. Math. Soc. 181 (1973), 273–292.

[15] C. Laurent-Thiébaut and M.-C. Shaw, On the Hausdorﬀ property of some Dolbeault cohomology groups, Math. Z. 274 (2013), 1165–1176.

[16] X. Li and M.-C. Shaw, The ∂-equation on an annulus with mixed boundary conditions, Bull. Inst. Math. Acad. Sin. (N.S.) 8(3) (2013), 399–411.

[17] K. Matsumoto, Pseudoconvex domains of general order and q-convex domains in the complex projective space, Kyoto J. Math. 33 (1993), 685–695.

[18] S. Saber, Solution to ∂ problem with exact support and regularity for the ∂-Neumann operator on weakly q-convex domains, Int. J. Geom. Methods Mod. Phys. 7(1) (2010), 135–142.

[19] S. Saber, The ∂ problem on q-pseudoconvex domains with applications, Math. Slovaca 63(3) (2013), 521–530.

[20] S. Saber, The ∂-problem with support conditions and pseudoconvexity of general order in Kähler manifolds, J. Korean Math. Soc. 53(6) (2016), 1211–1223.

[21] S. Saber, Compactness of the weighted ∂-Neumann operator and commutators of the Bergman projection with continuous functions, J. Geom. Phys. 138 (2019), 194–205.

[22] Y. T. Siu, Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems, J. Diﬀerential Geom. 17 (1982), 55–138.

[23] H. H. Wu, The Bochner Technique in Diﬀerential Geometry, Harwood Academic, New York, 1988.