Stable proper biharmonic maps in Euclidean spheres

2025/07/09 by Branding, Volker, Siffert, Anna · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.06708

Abstract

We construct an explicit family of stable proper weak biharmonic maps from the unit ball B^m, m≥ 5, to Euclidean spheres. To the best of the authors knowledge this is the first example of a stable proper weak biharmonic map from at compact domain. To achieve our result we first establish the second variation formula of the bienergy for maps from the unit ball into a Euclidean sphere. Employing this result, we examine the stability of the proper weak biharmonic maps q:B^m→\mathbbS^m^ℓ, m,ℓ∈ℕ with ℓ≤ m, which we recently constructed in \citeBS25 and thus deduce the existence of an explicit family of stable proper biharmonic maps to Euclidean spheres.

Citations

• A family of triharmonic maps to spheres in all dimensions greater than two
• Remarks on constructing biharmonic and conformal biharmonic maps to spheres
• Biharmonic and biconservative hypersurfaces in space forms
• Energy convexity of intrinsic bi-harmonic maps and applications I: spherical target
• Rotationally symmetric biharmonic maps between models
• Best Constants in the Hardy-Rellich Inequalities and Related Improvements

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