Non-constancy and multiplicity of half-harmonic maps from intervals into the circle

2026/07/22 by Ali Hyder, Luca Martinazzi
#math.AP

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Abstract

We study one-dimensional half-harmonic maps from the real line into the circle with prescribed exterior data. We show that, for every positive integer k, if two disjoint intervals are sufficiently close, there exist at least k distinct non-constant half-harmonic maps with constant exterior data. More generally, we establish a multiplicity result for boundary data with energy below the critical threshold 2π by introducing a local relative degree and proving corresponding degree-jump estimates. Working on a single interval and for boundary data arising as traces of finite Blaschke products, we investigate the existence and non-existence of energy minimizers in prescribed degree classes.

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