Nowhere Locally Metric-Minimising Saddle Surfaces

Abstract

We study the relation between the saddle property and local metric minimality for smooth surfaces in Euclidean spaces. Anton Petrunin and Stephan Stadler proved that every smooth strictly saddle surface in $\mathbb{R}^3$ is locally metric-minimising and expressed the expectation that this phenomenon fails in $\mathbb{R}^4$. We confirm this expectation and prove a stronger density result: Every smooth strictly saddle embedding of a closed disc into $\mathbb{R}^4$ can be approximated arbitrarily closely in the $C^\infty$-topology by smooth strictly saddle embeddings that are nowhere locally metric-minimising. In particular, nowhere locally metric-minimising saddle embeddings exist in $\mathbb{R}^d$ for every $d \geq 4$. The main ingredient is the construction of non-trivial infinitesimal isometric deformations of saddle immersions in $\mathbb{R}^4$ supported in arbitrarily small discs. We also provide an explicit twisted quadratic surface that is saddle in a neighbourhood of the origin but fails to be locally metric-minimising there, by deriving a necessary condition for metric minimality.