Second order differentiation formula on $RCD(K,N)$ spaces

TL;DR

This paper establishes a second order differentiation formula along geodesics in finite-dimensional RCD(K,N) spaces, using approximation by entropic interpolations and new estimates for these interpolations.

Contribution

It introduces a novel second order differentiation formula in RCD(K,N) spaces and develops new estimates for entropic interpolations, advancing the understanding of geometric analysis in these spaces.

Findings

Proved second order differentiation formula in RCD(K,N) spaces

Developed new estimates for entropic interpolations

Enhanced techniques for approximation of geodesics

Abstract

We prove the second order differentiation formula along geodesics in finite-dimensional $RCD(K,N)$ spaces. Our approach strongly relies on the approximation of $W_2$-geodesics by entropic interpolations and, in order to implement this approximation procedure, on the proof of new (even in the smooth setting) estimates for such interpolations.