Limit Sets of Poincaré Embeddings

Thank you for the introduction, thank to all the organizors.

Before I get started, I would like to say that I am both extremely honoured to give a talk at this workshop "The h-principle and beyond" and a bit disappointed not to be able to meet you in person. However, I remain happy that we can still be together through the magic of internet. Today, I would like to talk about c^1 isometric embeddings of the Hyperbolic space into E^3, and more specifically about limit sets of such embeddings.


Most of you are pretty aware of is a isometric map, I recall the definition for the others, to be crystal clear.

You may think of the Janet dimension as a dimensional barrier : above  this dimension, local isometric embeddings are abundant. It was discovered by Nash that this barrier can be broken provided you allow your regularity to be drastically lowered.

It was Nash that discovered that our intuition is misleading...

For the moment, it is all that we need to know about isometric embeddings

I now would like to introduce (to you) the next notion needed to understand the title of this talk, namely the limit set

I would like to take advantage of this corollary to introduce the Poincaré Disk Model of the Hyperbolic space.

The Hyperbolic space has constant Gauss curvature equal to -1 and it is well-known that surfaces with negative Gauss curvature can not be immersed in E^3

It was pointed out by De Lellis that the 1959's result of Kuiper extends to the case where the limit set is not void. (personnal name or denomination)

Here is a bizarre (an odd) consequence of this theorem, I don't know if it was noticed before but it is definitely funny. Here is what's all about.

It is therefore possible to have embeddings of the hyperbolic space with a void limit set or a limit set reduced to a point. What about a limit set which is curve ? 

Here, we address the following question : ... 

I say "we" because what I will present is a work of the Hévéa Team, that is a joint work with the following people.

The extra information here is that you can choose your map to be beta Hölder, the maximum of possible regularity.

This is not straightforward but this is not mega-hard too. I don't want to go into the details, just show you quickly a possible solution to convince you that I am not cheating.

alpha_c is a truncated taylor expansion of a primitive of the conformal factor of the hyperbolic metric.

That was for the two first points. The third one is less much direct (diiraïcte) and requires to dive into the way Nash and Kuiper build C1 isometric embeddings

Everything works as if we were in the compact case

I ask the specialists for a little kindness here, for some benevolence.

sorry for having stammered, I resume
I got confused, I'll take it back

I have decided that until his release I will end each of my presentations with a slide about him
