Canonical extension of submanifolds and foliations in noncompact symmetric spaces

TL;DR

This paper introduces a method to extend submanifolds and foliations from boundary components to entire noncompact symmetric spaces, preserving key geometric properties and leading to new examples of inhomogeneous isoparametric hypersurfaces.

Contribution

It presents a novel extension technique for submanifolds and foliations in noncompact symmetric spaces, enabling the construction of previously unknown inhomogeneous hypersurfaces.

Findings

Extension preserves minimality and isoparametric properties.

First examples of inhomogeneous isoparametric hypersurfaces in higher-rank spaces.

Method applicable to various geometric structures in symmetric spaces.

Abstract

We propose a method to extend submanifolds, singular Riemannian foliations and isometric actions from a boundary component of a noncompact symmetric space to the whole space. This extension method preserves minimal submanifolds, isoparametric foliations and polar actions, among other properties. One of the several applications yields the first examples of inhomogeneous isoparametric hypersurfaces in noncompact symmetric spaces of rank at least two.