[Paper Review] Morse theory and infinite families of harmonic maps between spheres

Kevin Corlette, Robert M. Wald|ArXiv.org|Dec 1, 1999

Geometric Analysis and Curvature Flows7 references4 citations

TL;DR

This paper applies Morse theory to prove the existence of infinite families of harmonic maps between spheres, demonstrating that reflection symmetry and a singular harmonic map of infinite index are key to generating such sequences. It generalizes prior results by Bizon and Chmaj, establishing index and convergence properties for these maps using variational and geometric methods in differential geometry and mathematical physics.

ABSTRACT

Existence of an infinite sequence of harmonic maps between spheres of certain dimensions was proven by Bizon and Chmaj. This sequence shares many features of the Bartnik-McKinnon sequence of solutions to the Einstein-Yang-Mills equations as well as sequences of solutions that have arisen in other physical models. We apply Morse theory methods to prove existence of the harmonic map sequence and to prove certain index and convergence properties of this sequence. In addition, we generalize the result of Bizon and Chmaj to produce infinite sequences of harmonic maps not previously known. The key features ``responsible'' for the existence and properties of these sequences are thereby seen to be the presence of a reflection symmetry and the existence of a singular harmonic map of infinite index which is invariant under this symmetry.

Motivation & Objective

• To establish the existence of infinite sequences of harmonic maps between spheres using Morse theory.
• To analyze the index and convergence behavior of these harmonic map sequences.
• To generalize Bizon and Chmaj's result to new infinite families of harmonic maps.
• To identify the geometric and topological conditions—specifically reflection symmetry and a singular harmonic map of infinite index—that underlie the existence of such sequences.
• To unify structural features observed in harmonic maps with those in solutions to Einstein-Yang-Mills and other physical models.

Proposed method

• Application of Morse theory to the energy functional on the space of maps between spheres.
• Identification of a singular harmonic map invariant under reflection symmetry, serving as a critical point of infinite Morse index.
• Use of variational methods to construct a sequence of harmonic maps via mountain pass geometry.
• Analysis of the Hessian and index of critical points to determine the Morse index of each map in the sequence.
• Exploitation of symmetry to reduce the problem to a finite-dimensional setting, facilitating index computation.
• Generalization of the construction to new dimensions and map families by extending symmetry and index arguments.

Experimental results

Research questions

• RQ1 What geometric and topological conditions give rise to infinite families of harmonic maps between spheres?
• RQ2 How do reflection symmetries influence the existence and index properties of harmonic maps?
• RQ3 What role does a singular harmonic map of infinite index play in generating such sequences?
• RQ4 Can Morse theory be systematically applied to prove the existence and structural properties of these harmonic map sequences?
• RQ5 How do these harmonic map families relate to known solutions in mathematical physics, such as Bartnik-McKinnon or Einstein-Yang-Mills solutions?

Key findings

• An infinite sequence of harmonic maps between spheres exists in certain dimensions, proven via Morse theory.
• The existence of such sequences is linked to a reflection-symmetric singular harmonic map of infinite Morse index.
• Each map in the sequence has finite but increasing Morse index, confirming a pattern of instability growth.
• Convergence properties of the sequence are established, showing the maps approach the singular harmonic map in a controlled manner.
• The method generalizes Bizon and Chmaj's result, producing previously unknown infinite families of harmonic maps.
• The structural features of the sequence mirror those seen in physical models like Einstein-Yang-Mills, suggesting a deeper geometric universality.