Constancy of the index for gradient mappings

TL;DR

This paper proves that the index of gradient mappings of certain functions remains constant under specific conditions, confirming a conjecture from 1992 and extending the result to quasiregular gradient mappings.

Contribution

It establishes the local constancy of the index for gradient mappings with positively determinant Hessians, generalizing previous conjectures and results.

Findings

Index is locally constant for gradient mappings with positive Hessian determinant.

The result applies to quasiregular gradient mappings.

Confirms a conjecture by vere1k from 1992.

Abstract

We show that if the Hessian of a $C^{1,1}$ function has uniformly positive determinant almost everywhere then its index is locally constant, as conjectured by \v{S}ver\'ak in 1992. We deduce this result as a consequence of a more general theorem valid for quasiregular gradient mappings.