高维Nitsche猜想：精确界与刚性
The Higher-Dimensional Nitsche Conjecture: Sharp Bounds and Rigidity
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中文总结 AI 辅助
该研究针对n≥3的高维调和坐标函数满射同胚h，证明了Nitsche猜想的两类精确界，揭示其临界情形的刚性，并通过概率耦合等方法完成了严格证明。
中文摘要 AI 辅助
设n≥3，h:\u0391(r,1)→\u0391(R,1)⊂ℝⁿ是一个满射同胚，其坐标函数为调和函数。我们证明了精确的Nitsche界R≤R_{n,+}(r):=nr/(n-1+rⁿ)；当h交换两个端点时，还证明了严格更强的精确界R≤R_{n,-}(r):=nr^{n-1}/(1+(n-1)rⁿ)。两种临界情形均具有刚性：等式成立时，在正交变换下对应保端点或反端点的径向调和同胚。本文未假设存在到闭环面的连续延拓、边界同胚、边界雅可比或雅可比的符号条件。证明过程将每个内部方向映射的非零次数转化为概率耦合，利用球冠重心的严格凹性，建立了正区域核下向量测度的精确收缩原理。在任一临界值处，二阶端点缺陷会迫使极限转移核取等，其等号分类产生正交耦合图；保端点情形下，剩余迹被Dirichlet-to-Neumann谱间隙锁定，反端点情形下则被端点Hölder正则性及方向映射的一致收敛锁定。
英文摘要
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 Hölder regularity and uniform convergence of the direction maps in the end-reversing case.
