The Higher-Dimensional Nitsche Conjecture: Sharp Bounds and Rigidity
Let $n\ge3$ and let $h:\A(r,1)\to\A(R,1)\subset\mathbb R^n$ be an onto homeomorphism with harmonic coordinate functions. We prove the sharp Nitsche bound \[ R\le R_{n,+}(r):=\frac{nr}{n-1+r^n}, \] and, when $h$ interchanges the two ends, the strictly stronger sharp bound \[ R\le R_{n,-}(r):=\frac{nr^{n-1}}{1+(n-1)r^n}. \] Both critical cases are rigid: equality forces, up to an orthogonal transformation, the corresponding end-preserving or end-reversing radial harmonic homeomorphism. No continuous extension to the closed annulus, boundary homeomorphism, boundary Jacobian, or sign condition on the Jacobian is assumed. The proof converts the nonzero degree of each interior direction map into a probability coupling and establishes a sharp contraction principle for vector measures under positive zonal kernels, using the strict concavity of spherical-cap barycenters. At either critical value, a second-order endpoint defect forces equality for a limiting transfer kernel, whose equality classification yields an orthogonal coupling graph. The remaining trace is locked by a Dirichlet-to-Neumann spectral gap in the end-preserving case and by endpoint H\"older regularity and uniform convergence of the direction maps in the end-reversing case.