2. Juli 2025
Jonas Hirsch (Universität Leipzig)
A Nash-Kuiper theorem for isometric embeddings beyond Borisov’s
exponent
joint work with Wentao Cao and Dominik Inauen
For any given short embedding from an n-dimensional region into (n + 1)-dimensional Euclidean space, and for any Hölder exponent α < (n2− n + 1)−1, a
C1,α isometric embedding is built within any C0 neighbourhood of the given short embedding through convex integration, which refines the classical Nash-Kuiper theorem and extends the flexibility of C1,α isometric embedding beyond Borisov’s exponent. Notably, when n = 2, we attain the Onsager exponent 1/3 for isometric embeddings. This convex integration scheme is performed through new construction and leveraging iterative “integration by parts” to effectively transfer large-scale errors to smaller ones.
In my talk, I would like to give some ideas for the “integration by parts” procedure. Furthermore, I will highlight the differences between the schemes that were previously used.