Austere and arid properties for PF submanifolds in Hilbert spaces

TL;DR

This paper extends the concepts of austere and arid submanifolds to infinite-dimensional Hilbert spaces, exploring their properties, relations, and examples within the context of proper Fredholm submanifolds.

Contribution

It introduces and analyzes austere and arid properties for PF submanifolds in Hilbert spaces, providing examples and discussing classification problems.

Findings

Existence of infinite-dimensional austere PF submanifolds

Existence of infinite-dimensional arid PF submanifolds

Discussion of minimal orbits in hyperpolar PF actions

Abstract

Austere submanifolds and arid submanifolds constitute respectively two different classes of minimal submanifolds in finite dimensional Riemannian manifolds. In this paper we introduce these two notions into a class of proper Fredholm (PF) submanifolds in Hilbert spaces, discuss their relation and show examples of infinite dimensional austere PF submanifolds and arid PF submanifolds in Hilbert spaces. We also mention a classification problem of minimal orbits in hyperpolar PF actions on Hilbert spaces.