---
tag: S4842D7MAE
kind: concept
name: Levi-flat hypersurface
msc: [32V40, 32V05, 53C12]
specializes: [ZC78JKZX9S]
sources:
  - url: https://en.wikipedia.org/wiki/CR_manifold
---

## Summary

A Levi-flat hypersurface is a real hypersurface in a [[complex manifold|JSGWAN2RJV]] whose [[Levi form|80MTNXWW80]] vanishes at every point.
For a smooth hypersurface this means that it is filled, near each point, by complex hypersurfaces stacked in a one-parameter family.
Levi-flat hypersurfaces are the opposite extreme to strongly pseudoconvex ones, and CR functions on them need not extend holomorphically to either side.

## Explanations

### L1 human

A real hypersurface in $\C^n$ has one real dimension less than the space.
Most such hypersurfaces bend in the complex directions, and the Levi form measures this bending.
A Levi-flat hypersurface does not bend at all in these directions.
The model is $\set{z \in \C^2 : \operatorname{Im} z_1 = 0}$, which is the union of the complex lines $\set{z_1 = t}$ for real $t$.
So it looks like a stack of complex lines indexed by a real number, like the pages of a book.