PhD course

**Real submanifolds of
complex space**

**Lecturer: Egmont Porten (Mid Sweden
University)**

25 January - 15 February 2018

Schedule:

1. Thursday January 25

2. Tuesday January 30

3. Wednesday January 31

4. Thursday February 1

5. Tuesday February 6

6. Thursday February 8

7. Tuesday February 13

8. Thursday February 15

All lectures will be held in Aula D'Antoni from 14:00 to 16:00

The course addresses advanced students, PhD students
and researcher.

**Program**

**Lecture 1**: General introduction, first
examples and problems.

**Lecture 2**: Boundary values of holomorphic
functions, CR-functions, removable singularities, of CR-functions:
relation of the topic to polynomial convexity and envelopes of
holomorphy.

**Lecture 3**: Surfaces in **C**^{2}, filling of
2-spheres, characteristic foliations, application to removable
singularity in strictly pseudoconvex and general boundaries in **C**^{2}.

**Lecture 4**: More envelopes of holomorphy,
example of a 2-sphere bounding two Levi flat 3-balls.

**Lecture 5**: General CR-manifolds, holomorphic discs and the
Bishop equation, Levi curvature and extension of CR-functions,
propagation along CR orbits, simplified proof of holomorphic wedge
extension from minimal CR manifolds.

**Lecture 6**: L2-methods on Riemann domains, functions with
polynomial growth towards the boundary, Dloussky’s theorem on
subvarities that are singular loci.

**Lecture 7**: Pseudoconcave CR manifolds, problems on
regularity and extension, subelliptic estimates for tangential CR
equations, essential pseudoconcave manifolds.

**Lecture 8**: Local holomorphic extension to full
neighbourhoods, relation to hypoellipticity, homogenous examples,
microlocal view and open problems.

This course is funded by a INdAM Visiting Scholarship