Title: Analytic characterization of monotone Hopf-harmonics Show Chinese title

Title: 单调霍普夫调和的解析特征

Abstract: We study solutions of the inner-variational equation associated with the Dirichlet energy in the plane, given homeomorphic Sobolev boundary data. We prove that such a solution is monotone if and only if its Jacobian determinant does not change sign. These solutions, called monotone Hopf-harmonics, are a natural alternative to harmonic homeomorphisms. Examining the topological behavior of a solution (not a priori monotone) on the trajectories of Hopf quadratic differentials plays a sizable role in our arguments. Show Chinese abstract

Abstract: 我们研究了平面中与Dirichlet能量相关的内变分方程的解，在同胚Sobolev边界数据下。我们证明，这样的解是单调的当且仅当其雅可比行列式不改变符号。这些解被称为单调Hopf调和映射，是调和同胚的自然替代方案。检查一个解（并非先验单调）在Hopf二次微分轨迹上的拓扑行为在我们的论证中起到了重要作用。