Schwarz Lemma for VT harmonic map

TL;DR

This paper extends the Schwarz Lemma to VT harmonic maps, establishing conditions under which these maps are distance and volume decreasing, based on eigenvalues and curvature bounds.

Contribution

It generalizes the Schwarz Lemma for V harmonic maps to VT harmonic maps, incorporating new conditions involving eigenvalues and curvature bounds.

Findings

Established Schwarz Lemma for VT harmonic maps.

Proved distance decreasing property under certain eigenvalue conditions.

Demonstrated volume decreasing property with curvature bounds.

Abstract

In this paper, we obtained Schwarz Lemma of $ VT $ harmonic map including distance decreasing property and volume decreasing property under some conditions about the eigenvalue of $ du^{+} \circ du $, $ T $ and the lower bound of $ Ric_f^N $ or $ Ric_V^N $. We generalized Schwarz lemma of $ V $ harmonic map.