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Explore a quantitative stability result for geodesic spheres in rank-one symmetric spaces of non-compact type, including real, complex, quaternionic, and octonionic hyperbolic spaces. Examine how these manifolds possess negatively pinched sectional curvature distributed according to their underlying algebraic structure, and discover how radial vector fields induce natural comparison maps with real hyperbolic space through exponential coordinates with explicit control of tangential deformations. Learn the key techniques used to demonstrate that geodesic spheres are isoperimetric among all sets sharing suitable central symmetry, and understand the proof that geodesic spheres maintain uniform quantitative stability under small volume-preserving C1-perturbations. Investigate the rescaling argument that provides quantitative proof showing geodesic spheres as unique isoperimetric regions for small volumes in rank one symmetric spaces, with particular focus on applications in complex hyperbolic space geometry.
