Speaker: Prof.Dominik.inauen ，Leipzig University

Inviter: 邱国寰副研究员

Title: Rigidity and Flexibility of Isometric Embeddings

Language: Chinese

Time & Venue: 2024.01.14 10:00-11:00 思源楼S525

Abstract: The problem of embedding abstract Riemannian manifolds isometrically (i.e., preserving distances) into Euclidean space arises from a fundamental question: are abstract Riemannian manifolds and submanifolds of Euclidean space the same? Interestingly, the behavior of such embeddings varies drastically depending on whether they have a low (i.e., C^1) or high (i.e.,C^2) regularity. For instance, the famous Nash–Kuiper theorem shows thatC^1isometric embeddings of the standard 2-sphere can exist inside arbitrarily small balls inR^3. In contrast, in the C^2 category there is (up to translation and rotation) just one isometric embedding, namely the standard inclusion. This dichotomy raises a question analogous to the Onsager conjecture in fluid dynamics: is there a sharp regularity threshold that distinguishes between these contrasting behaviors?In my talk I will review some known results and present arguments for why the Hölder exponent 1/2 can be seen as a critical exponent in the problem.