The Soul Conjecture in Alexandrov Geometry in dimension 4

TL;DR

This paper proves the Soul Conjecture for 4-dimensional Alexandrov spaces with non-negative curvature, establishing that such spaces with positive curvature around a point have a zero-dimensional soul.

Contribution

It provides the first proof of the Soul Conjecture in four-dimensional Alexandrov geometry, extending known results to higher dimensions.

Findings

Soul of the space is a point under given conditions

Validates the conjecture in 4D Alexandrov spaces with positive curvature

Advances understanding of geometric structure in Alexandrov spaces

Abstract

In this paper, we prove the Soul Conjecture in Alexandrov geometry in dimension $4$, i.e. if $X$ is a complete non-compact $4$-dimensional Alexandrov space of non-negative curvature and positive curvature around one point, then a soul of $X$ is a point.