Instability of rational and polynomial convexity

TL;DR

This paper demonstrates that rational and polynomial convexity of totally real submanifolds are generally unstable under small Hölder continuous perturbations, contrasting with their known stability under $C^1$ perturbations.

Contribution

It establishes the instability of rational and polynomial convexity under $C^eta$-small perturbations for any $eta<1$, extending previous stability results.

Findings

Rational and polynomial convexity are unstable under Hölder perturbations.

Stability holds only under $C^1$-small perturbations.

The result complements prior work on stability under smoother perturbations.

Abstract

It is shown that rational and polynomial convexity of totally real submanifolds is in general unstable under perturbations that are $C^\alpha$-small for any H\"older exponent $\alpha<1$. This complements the result of L{\o}w and Wold that these properties are $C^1$-stable.