A removable singularity property for CR mappings between real analytic hypersurfaces

TY - JOUR

T1 - A removable singularity property for CR mappings between real analytic hypersurfaces

AU - Huang, Xiaojun

N1 - Funding Information: real analytic CR submanifold of holomorphic dimension 0 ([14]). .As is well-known, f (C') is thus also contained in a real analytic strongly pseudoconvex hypersurface. By a result proved in [26] [16], it thus follows that f extends locally bihomorphically across U (unless f is constant). Notice that WL(M1) must have measure zero in M1. Once f is known to be extendible across an open dense subset of ,lfl, Proposition 3.1 indicates that f extends as a multiple valued holomorphic map across p. Using Theorem D(b) of [21], it follows that f is locally finite to one. Again by the results of Bell-Catlin and Malgrange. we conclude that f extends holomorphically across p. I Proof of Corollary 2: Notice that Theorem 1 implies that f extends across all strongly pseudoconvex points of MI. Now, using a finiteness result of [12] or [7] for smooth CR mappings. it follows easily that for any z E All, we can make f-'( f (z)) cc MI, by shrinking MI suitably around z. Therefore. the results of Bell-Catlin give the result in Corollary 2. 1 Proof of Corollary 3: It suffices for us to verify that y c Q, holds only at finitely many z E y. Suppose not. Then without loss of generality, we can assume that y c Q, for all z E y. Notice that Q, is always tangential to ~ ~ ' " ' ~ 2 1 1at I. Hence. we can assume without loss of generality that MI near p = 0 E -, is defined by an equation of the form: v = p(z',fi,u), where z = (2u, + zv), and y = {(XI + z0,0,. . . ,O)}. Notice that for t = (XI , 0 . . . , 0) E y, the assumption that y C H, for all such t shows that p(xl , . . . , 0; x;, . . . : 0) = 0 for all real xl and 2; close to the origin. This implies that M1 contains the holomorphic curve {tz = ... = zn = O), contradicting the D'Angelo finiteness of M1. I Acknowledgement: This paper was written when the author was visiting the Department of Mathematics. Wuhan University. China in the late spring semester of 1997. The author acknowledges a financial support for this visit from a Chinese Natural Science Foundation. He is very grateful to the department, in particular, to Professors Jinyuan Du and Jianke Lu for their hospitality during the author's stay at Wuhan. The author also acknowledges the support from an NSF grant and an NSF Postdoctoral Fellowship.

PY - 2000

Y1 - 2000

UR - http://www.scopus.com/inward/record.url?scp=0038397521&partnerID=8YFLogxK

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U2 - 10.1080/03605300008821514

DO - 10.1080/03605300008821514

M3 - Article

AN - SCOPUS:0038397521

VL - 25

SP - 299

EP - 317

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 1-2

ER -