Mean Curvature in the Light of Scalar Curvature

TL;DR

This paper explores conjectures related to mean convex domains in Euclidean and scalar-curvature-bounded spaces, providing theoretical results that motivate further research in geometric analysis.

Contribution

It formulates new conjectures on mean convex domains and proves initial theorems supporting these conjectures in various geometric contexts.

Findings

Proved initial theorems supporting the conjectures.

Formulated several new conjectures on mean convex domains.

Extended considerations to spaces with scalar curvature bounds.

Abstract

We formulate several conjectures on mean convex domains in the Euclidean spaces, as well as in more general spaces with lower bonds on their scalar curvatures, and prove a few theorems motivating these conjectures.