Oscillations and Spirals in Two Problems of Global Analysis

We present our works on two problems in global analysis (i.e.,analysis on manifolds): One concerns the compactness of the space of smooth $d$-dimensional immersed hypersurfaces with uniformly $L^d$-bounded second fundamental forms, and the other concerns the validity of W^{2,p}$-elliptic estimates for the Laplace--Beltrami operator on open manifolds. We construct explicit counterexamples to both problems. The onstructions involve rapid oscillations and wild spirals, with motivations derived from physical phenomena.