On the Morse index of harmonic maps and minimal immersions

TL;DR

This paper investigates the Morse index of harmonic maps and minimal immersions from compact Riemannian manifolds into homogeneous strongly harmonic manifolds, revealing their extremal properties on certain eigenspaces.

Contribution

It provides new results on the Morse index behavior of these maps under conformal variations and identifies conditions for them to be global maxima in eigenspaces.

Findings

Morse index varies along conformal vector fields.

Harmonic maps are global maxima on specific eigenspaces.

Results apply to maps into homogeneous strongly harmonic manifolds.

Abstract

In this paper we are concerned with harmonic maps and minimal immersions defined on compact Riemannian manifolds and with values in homogenous strongly harmonic manifolds. We show some results on the Morse index by varying these maps along suitable conformal vector fields. We obtain also that they are global maxima on some subspaces of the eigenspaces corresponding to the nonvanishing eigenvalues of the Laplacian operator on the target manifolds.