# Gauss maps of the Ricci-mean curvature flow

## TL;DR

This paper studies the evolution of Gauss maps under Ricci-mean curvature flow, extending previous harmonic map results to a coupled flow setting and deriving their heat flow equations.

## Contribution

It derives the evolution equation for Gauss maps in Ricci-mean curvature flow and proves they satisfy a specific harmonic map heat flow in codimension one.

## Key findings

- Gauss maps satisfy the vertically harmonic map heat flow equation in codimension one.
- Evolution equations for Gauss maps under Ricci-mean curvature flow are established.
- Extension of harmonic map properties to coupled Ricci-mean curvature flows.

## Abstract

In this paper, we investigate the Gauss maps of a Ricci-mean curvature flow. A Ricci-mean curvature flow is a coupled equation of a mean curvature flow and a Ricci flow on the ambient manifold. Ruh and Vilms proved that the Gauss map of a minimal submanifold in a Euclidean space is a harmonic map, and Wang extended this result to a mean curvature flow in a Euclidean space by proving its Gauss maps satisfy the harmonic map heat flow equation. In this paper, we deduce the evolution equation for the Gauss maps of a Ricci-mean curvature flow, and as a direct corollary we prove that the Gauss maps of a Ricci-mean curvature flow satisfy the vertically harmonic map heat flow equation when the codimension of submanifolds is 1.