Regularity of the boundaries of analytic sets

E. M. Chirka

Sbornik. Mathematics, Tome 45 (1983) no. 3, pp. 291-335

Résumé

In this article the author studies the boundary behavior of a one-dimensional complex analytic set $A$ in a neighborhood of a totally real manifold $M$ in $\mathbf C^n$ with smoothness greater than 1. He proves that the limit points of $A$ on $M$ form a set of locally finite length and that near almost every limit point the closure of $A$ is either a manifold with boundary (with smoothness corresponding to $M$) or a union of two manifolds with boundary. He investigates the structure of the tangent cone to $A$ at the limit points and proves a theorem concerning the boundary regularity of holomorphic discs “glued” to $M$. Bibliography: 22 titles.

Bibliographie

[1] Ganning P., Poccu X., Analiticheskie funktsii mnogikh kompleksnykh peremennykh, Mir, M., 1969 | MR

[2] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[3] Levin B. Ya., Raspredelenie kornei tselykh funktsii, Gostekhizdat, M., 1956

[4] Malgranzh B., Idealy differentsiruemykh funktsii, Mir, M., 1968

[5] Pinchuk S. I., “Granichnaya teorema edinstvennosti dlya golomorfnykh funktsii neskolkikh kompleksnykh peremennykh”, Matem. zametki, 15:2 (1974), 205–212 | Zbl

[6] Privalov I. I., Granichnye svoistva analiticheskikh funktsii, Gostekhizdat, M., L., 1950

[7] Kharvi R., Golomorfnye tsepi i ikh granitsy, Mir, M., 1979 | MR

[8] Khenkin G. M., Chirka E. M., “Granichnye svoistva golomorfnykh funktsii neskolkikh kompleksnykh peremennykh”, Sovremennye problemy matematiki, 4, VINITI, M., 1975, 13–142

[9] Chirka E. M., “Ob ustranimykh osobennostyakh analiticheskikh mnozhestv”, DAN SSSR, 248:1 (1979), 47–50 | MR | Zbl

[10] Chirka E. M., “Priblizhenie golomorfnymi funktsiyami na gladkikh mnogoobraziyakh v $\mathbf C^n$”, Matem. sb., 78 (120) (1969), 101–123 | Zbl

[11] Shabat B. V., Vvedenie v kompleksnyi analiz, T. 2, Nauka, M., 1976 | MR

[12] Alexander H., “Continuing $\mathrm l$-dimensional analytic sets”, Math. Ann., 191:2 (1971), 143–144 | DOI | MR | Zbl

[13] Alexander H., “Polynomial approximation and hulls in sets of finite linear measure in $\mathbf C^n$”, Amer. J. Math., 93:1 (1971), 65–74 | DOI | MR | Zbl

[14] Becker J., “Continuing analytic sets across $\mathbf R^n$”, Math. Ann., 195:2 (1972), 103–106 | MR | Zbl

[15] Bishop E., “Conditions for the analyticity of certain sets”, Michigan Math. J., 11:4 (1964), 289–304 | DOI | MR | Zbl

[16] Bishop E., “Differentiable manifolds in complex Euclidean space”, Duke Math. J., 32:1 (1965), 1–21 | DOI | MR | Zbl

[17] Harvey R., Lawson B., “On boundaries of complex analytic varieties. I”, Ann. Math., 102:2 (1975), 233–290 | DOI | MR

[18] Harvey F. R., Wells R. O. Jr., “Zero-sets of non-negative strictly plurisubharmonic functions”, Math. Ann., 201:2 (1973), 165–170 | DOI | MR | Zbl

[19] Hörmander L., Wermer J., “Uniform approximation on compact sets in $\mathbf C^n$”, Math. Scand., 23:1 (1968), 5–21 | MR | Zbl

[20] Shiffman B., “On the continuation of analytic curves”, Math. Ann., 184:4 (1970), 268–274 | DOI | MR | Zbl

[21] Shiffman B., “On the continuation of analytic sets”, Math. Ann., 185:1 (1970), 1–12 | DOI | MR | Zbl

[22] Stolzenberg G., “Uniform approximation on smooth curves”, Acta Math., 115:3-4 (1965), 185–198 | MR