Monotonicity Formulae and Holomorphicity of Harmonic Maps between K\"ahler manifolds

TL;DR

This paper develops monotonicity formulae for partial energies of harmonic and pluriharmonic maps between Kähler manifolds, leading to new holomorphicity and Liouville theorems, and explores holomorphic extension of CR maps.

Contribution

It introduces stress-energy tensors for partial energies and establishes monotonicity formulae, enabling new results on holomorphicity and extension of maps between Kähler manifolds.

Findings

Monotonicity formulae for partial energies of harmonic and pluriharmonic maps.

Holomorphicity and Liouville type results derived from monotonicity inequalities.

Investigation of holomorphic extension of CR maps using stress-energy tensors.

Abstract

In this paper, we introduce the stress-energy tensors of the partial energies E'(f) and E"(f) of maps between Kaehler manifolds. Assuming the domain manifolds poss some special exhaustion functions, we use these stress-energy tensors to establish some monotonicity formulae of the partial energies of pluriharmonic maps into any Kaehler manifolds and harmonic maps into Kaehler manifolds with strongly semi-negative curvature respectively. These monotonicity inequalities enable us to derive some holomorphicity and Liouville type results for these pluriharmonic maps and harmonic maps. We also use the stress-energy tensors to investigate the holomorphic extension problem of CR maps.