Geometry and Topology Seminar —— Rigidity and Flexibility of Isometric Embeddings
报告人：Dominik Inauen(University of Leipzig)
时间：2025-01-10 10:40-11:40
地点：智华楼225(四元厅)
【摘要】
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 that $C^1$ isometric embeddings of the standard 2-sphere can exist inside arbitrarily small balls in $\mathbb{R}^3$. 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.
