Generic uniqueness of least area planes in hyperbolic space

Least Area Planes in Hyperbolic 3-Space are Properly Embedded

We show that if P is an embedded least area (area minimizing) plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded.

Number of Least Area Planes in Gromov Hyperbolic 3-Spaces

We show that for a generic simple closed curve C in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exist a unique least area plane P in X with asymptotic boundary C. This result has interesting topological applications for constructions of canonical 2-dimensional objects in 3-manifolds.

Least area Spherical Catenoids in Hyperbolic Three-Dimensional Space

For a family of spherical minimal catenoids C_a in the hyperbolic 3-space, there exist two constants 0<a_c<a_l such that the following are true: (1) C_a is an unstable minimal surface with index one if a<a_c, (2) C_a is a stable minimal surface if a>=a_c, and (3) C_a is a least area minimal surface in the sense of Meeks-Yau if a>=a_l.

Properly Embedded Least Area Planes in Gromov Hyperbolic 3-Spaces

We show that for any simple closed curve in the sphere at infinity of a Gromov hyperbolic 3-space with cocompact metric, there exist a properly embedded least area plane in the space spanning the given curve. This gives a positive answer to a conjecture of Gabai. Soma has already proven this conjecture earlier. Our technique here is simpler and more general, and it can be applied to many similar settings.

关于粗中值结构的唯一性

On uniqueness of coarse median structures

该研究证明浓密双曲空间乘积的粗中值结构唯一，且该唯一性在相对双曲性下封闭，进而发现存在具唯一粗中值结构的非双曲裤图，补充了粗几何领域的相关结论。

Closed Minimal Surfaces in Cusped Hyperbolic Three-manifolds

Motivated by classical theorems on minimal surface theory in compact hyperbolic three-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic three-manifold of finite volume. We prove any closed immersed incompressible surface can be deformed to a closed immersed least area surface within its homotopy class in any cusped hyperbolic three-manifold. Our techniques highlight how special structures of these cusped hyperbolic three-manifolds prevent any least area minimal surface going too deep into the cusped region.