弯曲映射的连续性

次黎曼指数映射的正则性与连续性性质

Regularity and Continuity properties of the sub-Riemannian exponential map

本文通过次黎曼Jacobi场和Jacobi曲线的Maslov指标，证明了次黎曼指数映射的Warner正则性与连续性性质，并由此推出三维Heisenberg群的指数映射在共轭向量附近非单射。

Bending and stretching unit vector fields in Euclidean and hyperbolic 3-space

New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harmonic maps from 3-dimensional Euclidean domains to the 2-sphere. Finally, the harmonic unit vector field tangent to a parallel family of hyperbolic geodesics is shown to be unstable, by constructing a class of compactly supported energy-decreasing variations. All examples consider…

Differentiability of the Evolution Map and Mackey Continuity

We solve the differentiability problem for the evolution map in Milnor's infinite dimensional setting. We first show that the evolution map of each $C^k$-semiregular Lie group $G$ (for $k\in \mathbb{N}\sqcup\{\mathrm{lip},\infty\}$) admits a particular kind of sequentially continuity $-$ called Mackey k-continuity. We then prove that this continuity property is strong enough to ensure differentiability of the evolution map. In particular, this drops any continuity presumptions made in this context so far. Remarkably, Mackey k-continuity arises directly from the regularity problem itself, which…

双曲3-流形中弯曲的Schwarzian界

Schwarzian Bounds on Bending in Hyperbolic 3-Manifolds

该研究建立了双曲3-流形中Schwarzian范数与弯曲范数的显式量化界，给出了满足条件的单叶映射的弯曲范数控制函数，并应用得到拟富克斯流形弯曲叶状结构的新有效界，强化了二者的解析-几何对应关系。

Lipschitz continuity of harmonic maps between Alexandrov spaces

In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally Hölder continuous. In [39], F. H. Lin proposed a challenge problem: Can the Hölder continuity be improved to Lipschitz continuity? J. Jost also asked a similar problem about Lipschitz regularity of harmonic maps between singular spaces (see Page 38 in [28]). The main theorem of this paper gives a complete resolution to it.

双曲曲面的有限弯曲等距浸入的分支点与非稠密性

Branch points and non-density for finite-bending isometric immersions of hyperbolic surfaces

该研究明确了负曲率曲面的W²,²等距浸入的分支点性质，证明其指数稳定且无法由C²等距浸入逼近，从而得出C²等距浸入在有限弯曲等距浸入中不稠密的结论。