Quantization for biharmonic maps from non-collapsed degenerating Einstein 4-manifolds

2021/04/18 by Youmin Chen, Miaomiao Zhu, Chen, Youmin +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2104.08830

openalex publication_date 2021/04/18 · openalex created_date 2021/04/26 · openalex updated_date 2026/07/28

Abstract

For a sequence of extrinsic or intrinsic biharmonic maps u_j: M_j→ N from a sequence of non-collapsed degenerating closed Einstein 4-manifolds (M_j,g_j) with bounded Einstein constants, bounded diameters and bounded L² curvature energy into a compact Riemannian manifold (N,h) with uniformly bounded biharmonic energy, we establish a compactness theory modular finitely many bubbles, which are finite energy biharmonic maps from ℝ⁴, or from ℝ⁴ / Γ for some nontrivial finite group Γ⊂ SO(4), or from some complete, noncompact, Ricci flat, non-flat ALE 4-manifold (orbifold). To achieve this, we develop a sophisticated asymptotic analysis for solutions over degenerating neck regions.

Citations

• Regularity of Einstein manifolds and the codimension 4 conjecture
• On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth
• Uniqueness of tangent cone for biharmonic map with isolated singularity

Cited by

• Yang-Mills energy quantization over non-collapsed degenerating Einstein manifolds and applications

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