UC Santa Cruz

This thesis forms part of a collaborative project to establish the existence of nontrivial smooth extrinsic biharmonic maps from ℝ4 into 𝕊𝑛 through a min–max construction based on biharmonic replacement. After applying stereographic projection and removable singularity, such maps correspond to extrinsic quasi-biharmonic maps from 𝕊4 into 𝕊𝑛 . This approach is particularly relevant when 𝑛 ≥ 5, since 𝜋4(𝕊𝑛 ) = 0, and hence every map from 𝕊4 to 𝕊𝑛 is null-homotopic. As a result, classical methods such as the direct method in calculus of variations or heat-flow methods within a nontrivial homotopy class cannot be used to construct nontrivial critical points. This project aims to generalize Colding and Minicozzi’s min-max framework for harmonic maps to the case of biharmonic maps. In this thesis, we establish a compactness result for almost biharmonic maps with bounded energy which forms the essential analytical part of the broader min–max framework. In our setting, almost biharmonicity is formulated through local biharmonic replacements rather than a perturbed Euler–Lagrange equation. Under a uniform energy bound, we prove that the min-max sequence arising from this biharmonic replacement procedure bubble-converges subsequentially to a finite collection of quasi-biharmonic maps from 𝕊4 into 𝕊𝑛.