The construction of $\epsilon$-splitting map

TL;DR

This paper constructs an epsilon-splitting map on a geodesic ball with non-negative Ricci curvature and almost maximal volume, introducing new techniques to find directional points and estimate errors without compactness arguments.

Contribution

It presents a novel method for constructing epsilon-splitting maps without compactness, using induction and stratified almost Gou-Gu Theorem for directional points.

Findings

Successfully constructs epsilon-splitting maps under specified conditions.

Introduces new technical methods for directional point selection.

Provides error estimates ensuring directional points determine distinct directions.

Abstract

For a geodesic ball with non-negative Ricci curvature and almost maximal volume, without using compactness argument, we construct an $\epsilon$-splitting map on a concentric geodesic ball with uniformly small radius. There are two new technical points in our proof. The first one is the way of finding $n$ directional points by induction and stratified almost Gou-Gu Theorem. The other one is the error estimates of projections, which guarantee the $n$ directional points we find really determine $n$ different directions.