Existence of nontrival n-harmonic maps via min-max methods

Area of Science:

• Differential Geometry
• Topology

Background:

• Harmonic maps are fundamental objects in geometry and topology.
• Understanding the existence and behavior of harmonic maps is crucial for studying geometric structures.

Purpose of the Study:

• To establish the existence of nontrivial n-harmonic maps from the n-sphere to a closed manifold N.
• To investigate the relationship between n-harmonic maps and p-harmonic maps in the limit p approaches n.

Main Methods:

• Utilizing min-max constructions.
• Analyzing bubbling limits of p-harmonic maps as p approaches n from above.

Main Results:

• Existence of nontrivial n-harmonic maps is proven for n >= 3 and manifolds N with non-trivial homotopy groups pi_{n+k}(N).
• For k >= 1, these n-harmonic maps arise as singular limits of p-harmonic maps (p > n) in min-max procedures.

Conclusions:

• The study provides a constructive proof for the existence of n-harmonic maps under specific topological assumptions.
• It links the theory of p-harmonic maps to the existence of n-harmonic maps through limiting behavior.