On boundary regularity of holomorphic correspondences
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Let D be an arbitrary domain in $\mathbb{C}^n$, n > 1, and $M{\subset}{\partial}D$ be an open piece of the boundary. Suppose that M is connected and ${\partial}D$ is smooth real-analytic of finite type (in the sense of D'Angelo) in a neighborhood of $\bar{M}$. Let f : $D{\rightarrow}\mathbb{C}^n$ be a holomorphic correspondence such that the cluster set $cl_f$(M) is contained in a smooth closed real-algebraic hypersurface M' in $\mathbb{C}^n$ of finite type. It is shown that if f extends continuously to some open peace of M, then f extends as a holomorphic correspondence across M. As an application, we prove that any proper holomorphic correspondence from a bounded domain D in $\mathbb{C}^n$ with smooth real-analytic boundary onto a bounded domain D' in $\mathbb{C}^n$ with smooth real-algebraic boundary extends as a holomorphic correspondence to a neighborhood of $\bar{D}$.
Abstract
Let D be an arbitrary domain in $\mathbb{C}^n$, n > 1, and $M{\subset}{\partial}D$ be an open piece of the boundary. Suppose that M is connected and ${\partial}D$ is smooth real-analytic of finite type (in the sense of D'Angelo) in a neighborhood of $\bar{M}$. Let f : $D{\rightarrow}\mathbb{C}^n$ be a holomorphic correspondence such that the cluster set $cl_f$(M) is contained in a smooth closed real-algebraic hypersurface M' in $\mathbb{C}^n$ of finite type. It is shown that if f extends continuously to some open peace of M, then f extends as a holomorphic correspondence across M. As an application, we prove that any proper holomorphic correspondence from a bounded domain D in $\mathbb{C}^n$ with smooth real-analytic boundary onto a bounded domain D' in $\mathbb{C}^n$ with smooth real-algebraic boundary extends as a holomorphic correspondence to a neighborhood of $\bar{D}$.
참고문헌 (25)
- B. Ayed, "Analytic continuation of holomorphic mappings", C. R. Math. Acad. Sci. Paris, 347(17-18), pp.1011~1016 (2009)
- B. Ayed, "A Local extension of proper holomorphic maps between some unbounded do- mains in Cn", Bull. Belg. Math. Soc. Simon Stevin, 17(3), pp.513~534 (2010)
- M. S. Baouendi, "Germs of CR maps between real analytic hyper- surfaces", Invent. Math, 93(3), pp.481~500 (1998)
- M. S. Baouendi, "Images of real hypersurfaces under holomorphic mappings", J. Differential Geom, 36(1), pp.75~88 (1992)
- F. Berteloot, "On the continuous extension of holomorphic correspon- dences", Ann. Scuola Norm. Sup. Pisa Cl. Sci, 24(4), pp.747~766 (1997)
- E. M. Chirka, "Complex Analytic Sets" (1989), Kluwer Academic Publishers
- S. Chern, "Real hypersurfaces in complex manifolds", Acta Math, 133, pp.219~271 (1974)
- K. Diederich, "Proper holomorphic mappings between real-analytic pseudoconvex domains in Cn", Math. Ann, 282(4), pp.681~700 (1988)
- K. Diederich, "Analytic sets extending the graphs of holomorphic map- pings", J. Geom. Anal, 14(2), pp.231~239 (2004)
- K. Diederich, "Proper holomorphic maps in dimension 2 extend", Indiana Univ. Math. J, 44(4), pp.1089~1126 (1995)
- K. Diederich, "Regularity of continuous CR maps in arbitrary dimension", Mich. Math. J, 51(1), pp.111~140 (2003)
- K. Diederich, "Proper holomphic mapping between real-analytic pseudo- convex domains in Cn", Math. Ann, 282(4), pp.681~700 (1988)
- K. Diederich, "A rejection principle for degenerate real hypersurfaces", Duke Math. J, 47(4), pp.835~845 (1980)
- J. Merker, "On wedge extendability of CR-meromorphic functions", Math. Z, 241(3), pp.485~512 (2002)
- S. Pinchuk, "On holomorphic maps of real-analytic hypersurfaces", Math. USSR Sb, 34, pp.503~519 (1978)
- S. Pinchuk, "Analytic continuation of holomorphic mappings and the problem of holomorphic classi cation of multidimensional domains", 박사, pp.1980
- S. Pinchuk, "Analytic sets and the boundary regularity of CR mappings", Proc. Amer. Math. Soc, 129(9), pp.2623~2632 (2001)
- R. Shafikov, "Analytic continuation of germs of holomorphic mappings", Mich. Math. J, 47(1), pp.133~149 (2000)
- R. Shafikov, "Analytic continuation of holomorphic correspondences and equivalence of do- mains in Cn", Invent. Math, 152(3), pp.665~682 (2003)
- R. Shafikov, "On boundary regularity of proper holomorphic mappings", Math. Z, 242(3), pp.517~528 (2002)
- R. Shafikov, "A local extension theorem for proper holomorphic mappings in C2", J. Geom. Anal, 13(4), pp.697~714 (2003)
- R. Shafikov, "Extension of holomorphic maps between real hypersurfaces of different dimen- sion", Ann. Inst. Fourier, Grenoble, 57(6), pp.2063~2080 (2007)
- R. Shafikov, "Boundary regularity of correspondences in Cn", Proc. Indian Acad. Sci. (Math. Sci.), 116(1), pp.59~70 (2006)
- K. Verma, "Boundary regularity of correspondences in C2", Math. Z, 231(2), pp.253~299 (1999)
- S. M. Webster, "On the mapping problem for algebraic real hypersurfaces", Invent. Math, 43(1), pp.53~68 (1977)