Several complex variables and analytic spaces CR manifolds Theorem

Lewy extension theorem

Builds on Levi form CR function Holomorphic function of several variables

Statement

Let $n \ge 2$ and let $M$ be a smooth real hypersurface in $\C^n$ . Let $p \in M$ , and let $\rho$ be a smooth real function on a neighbourhood $V$ of $p$ with $M \cap V = \set{\rho = 0}$ and $d\rho \ne 0$ on $M \cap V$ . Assume that the Levi form of $\rho$ at $p$ , restricted to the complex tangent space of $M$ at $p$ , has at least one positive eigenvalue. Then there is a neighbourhood $U \subset V$ of $p$ with the following property. Let $f$ be a CR function of class $C^1$ on $M \cap V$ . Then there is a unique continuous function $F$ on $U \cap \set{\rho \le 0}$ with $F = f$ on $M \cap U$ that is holomorphic on $U \cap \set{\rho < 0}$ .

Remarks

The theorem is due to Hans Lewy. One proof fills the side $\set{\rho < 0}$ near $p$ with small analytic discs whose boundaries lie in $M$ . The values of $f$ on the boundary circles define $F$ by the Cauchy integral, and the CR equations make $F$ holomorphic. If the Levi form at $p$ has eigenvalues of both signs, extensions exist on both sides. Then $f$ is the restriction of a function holomorphic on a full neighbourhood of $p$ . For the unit sphere with $\rho = \abs{z}^2 - 1$ , the Levi form is positive, so CR functions extend locally into the ball. The hypothesis on the Levi form is needed. On the Levi-flat hypersurface $\set{\operatorname{Im} z_1 = 0}$ , the function $f(z) = g(\operatorname{Re} z_1)$ is CR for every real $C^1$ function $g$ . If $f$ extended holomorphically to one side, the Schwarz reflection principle in $z_1$ would make $g$ real-analytic, so a $g$ that is $C^1$ but not real-analytic gives no extension.

Going deeper

For a student

The Lewy extension theorem is a local form of the Hartogs phenomenon for CR functions. It turns questions about CR functions into questions about holomorphic functions on one-sided neighbourhoods. It is the starting point of the extension theory of CR functions, which treats submanifolds of higher codimension as well. A global analogue for the boundary of a bounded domain with connected boundary is the Hartogs–Bochner extension theorem.

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CR function concept
Holomorphic function of several variables concept
Levi form concept
Lewy extension theorem statement

Tag	`V2YCGHD719`
URI	`m2://V2YCGHD719`
Kind	statement, theorem
Writing	W0 stub
MSC	32V25, 32D15, 32V10
Status	This copy of the site has no build data. Run `m2 status V2YCGHD719`.
About	CR function
Uses	Levi form, CR function, Holomorphic function of several variables