[论文解读] Quasiregular values from generalized manifold with controlled geometry

Deguang Zhong|arXiv (Cornell University)|Mar 10, 2026

Geometric Analysis and Curvature Flows被引用 0

一句话总结

该论文将 Reshetnyak 定理推广至广义 n-流形上具有受控几何的 quasiregular 值，并扩展至欧几里得空间，超越了先前的欧几里得结果。

ABSTRACT

The main aim of this paper is to establish the Reshetnyak's theorem for quasiregualr values from generalized $n$-manifold with suitable controlled geometry to Euclidean space $\mathbb{R}^{n}.$ This generalizes a previous result due to Kangasniemi and Onninen on the setting of Euclidean space [A single-point Reshetnyak's theorem, Trans. Amer. Math. Soc., 378(2025): 3105-3128].

研究动机与目标

• Motivate extending Reshetnyak’s theorem to generalized n-manifolds with controlled geometry.
• Develop a framework of Newtonian spaces on metric measure spaces suitable for generalized manifolds.
• Establish Hölder regularity and Lusin’s condition (N) for generalized finite distortion maps.
• Prove discreteness, local positivity of index, and openness for quasiregular values in this setting.

提出的方法

• Use Newtonian spaces N^{1,n} on generalized n-manifolds with controlled geometry.
• Introduce generalized finite distortion and quasiregular value inequalities with K and Σ terms.
• Apply Hölder regularity and Gehring-type arguments to obtain higher integrability and regularity.
• Develop a local degree/index framework and local openness results for values.
• Derive a Reshetnyak-type theorem for quasiregular values mapping to Euclidean space.

实验结果

研究问题

• RQ1 Can Reshetnyak’s theorem for quasiregular values be extended from Euclidean spaces to generalized n-manifolds with controlled geometry?
• RQ2 What regularity, integrability, and local topological properties (discreteness, positivity of local index, openness) hold for generalized finite distortion mappings?
• RQ3 Under what conditions on the distortion function K and Σ do Hölder continuity and Lusin’s condition (N) persist in this generalized setting?
• RQ4 How do concepts of local degree and Jacobian transfer from generalized manifolds to Euclidean targets?
• RQ5 What are the key corollaries (e.g., totally disconnected sets) that accompany the main theorem in this generalized framework?

主要发现

• A Reshetnyak-type theorem is established for quasiregular values from generalized n-manifolds with controlled geometry to R^n.
• Under Σ in local L^p with p>1 and K in local L^p with p>n, the preimage of y0 is discrete and the local index is positive at preimages.
• Neighborhoods of preimages map into neighborhoods of y0, giving a form of local openness.
• f is locally Hölder continuous under the stated integrability assumptions for K and Σ.
• Lusin’s condition (N) holds locally for such mappings, and several corollaries follow (e.g., Hölder regularity, index positivity).
• The framework supports a sequence of results including a Jacobian-degree formula and totally disconnectedness for values of finite distortion.