Isoparametric theory and its applications

TL;DR

This survey reviews recent advances in isoparametric theory, highlighting solutions to longstanding conjectures, new examples in differential geometry, and applications to eigenvalues, Willmore submanifolds, and exotic spheres.

Contribution

It compiles recent progress in isoparametric theory, including solutions to Yau's conjecture, counterexamples to Leung's conjectures, and new geometric constructions.

Findings

Confirmed Yau's conjecture for the first eigenvalue in isoparametric cases

Provided counterexamples to Yau's 76th problem and Leung's conjectures

Constructed new examples of Willmore submanifolds and isoparametric functions on exotic spheres

Abstract

This is a survey on the recent progress in several applications of isoparametric theory, including an affirmative answer to Yau's conjecture on the first eigenvalue of Laplacian in the isoparametric case, a negative answer to Yau's 76th problem in his Problem Section, new examples of Willmore submanifolds in spheres, a series of examples to Besse's problem on the generalization of Einstein condition, isoparametric functions on exotic spheres, counterexamples to two conjectures of Leung, as well as surgery theory on isoparametric foliation.