Minimal Surfaces in $\widetilde{PSL_2(\mathbb{R})}$

On the asymptotic Plateau problem in ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$

We prove some non-existence results for the asymptotic Plateau problem of minimal and area minimizing surfaces in the homogeneous space ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$ with isometry group of dimension 4, in terms of their asymptotic boundary. Also, we show that a properly immersed minimal surface in ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$ contained between two bounded entire minimal graphs separated by vertical distance less than $\sqrt{1+4τ^2}π$ have multigraphical ends. Finally, we construct simply connected minimal surfaces with finite total curvature which are not graphs and a …

Harmonic maps for Hitchin representations

Let $(S,g_0)$ be a hyperbolic surface, $ρ$ be a Hitchin representation for $PSL(n,\mathbb R)$, and $f$ be the unique $ρ$-equivariant harmonic map from $(\widetilde S, \widetilde g_0)$ to the corresponding symmetric space. We show its energy density satisfies $e(f)\geq 1$ and equality holds at one point only if $e(f)\equiv 1$ and $ρ$ is the base $n$-Fuchsian representation of $(S,g_0)$. In particular, we show given a Hitchin representation $ρ$ for $PSL(n,\mathbb R)$, every $ρ$-equivariant minimal immersion $f$ from a hyperbolic plane $\mathbb H^2$ into the corresponding symmetric space $X$ is d…

$Nil^{4}$中Codazzi超曲面的分类及极小超曲面注记

Classification of Codazzi and note on minimal hypersurfaces in $Nil^{4}$

本文研究李群$(Nil^{4},\tilde{g})$中的超曲面问题，完成了Codazzi超曲面的分类，刻画了一类极小超曲面并给出实例，丰富了该空间的超曲面理论。

Unstable minimal surfaces in $\mathbb{R}^n$ and in products of hyperbolic surfaces

We prove that every unstable equivariant minimal surface in $\mathbb{R}^n$ produces a maximal representation of a surface group into $\prod_{i=1}^n\textrm{PSL}(2,\mathbb{R})$ together with an unstable minimal surface in the corresponding product of closed hyperbolic surfaces. To do so, we lift the surface in $\mathbb{R}^n$ to a surface in a product of $\mathbb{R}$-trees, then deform to a surface in a product of closed hyperbolic surfaces. We show that instability in one context implies instability in the other two.

通过极小曲面将Donaldson的超凯勒结构扩展到$\boldsymbol{\rm PSL}(2,\boldsymbol{\rm C})$中

Extending Donaldson's HyperKähler Structure via Minimal Surfaces into $\mathrm{PSL}(2,\mathbb{C})$

该研究构造了$\boldsymbol{\rm PSL}(2,\boldsymbol{\rm C})$中极小浸入的形变空间，定义了映射类群不变的复超凯勒结构，其限制对应多种已知几何结构且轨迹全测地，还探讨了相关$\boldsymbol{\rm C}^*$-作用的哈密顿性质。

Minimal surfaces and symplectic structures of moduli spaces

Given a closed surface S of genus at least 2, we compare the symplectic structure of Taubes' moduli space of minimal hyperbolic germs with the Goldman symplectic structure on the character variety X(S, PSL(2,C)) and the affine cotangent symplectic structure on the space of complex projective structures CP(S) given by the Schwarzian parametrization. This is done in restriction to the moduli space of almost-Fuchsian structures by involving a notion of renormalized volume, used to relate the geometry of a minimal surface in a hyperbolic 3-manifold to the geometry of its ideal conformal boundary.

Minimal orbifolds and (a)symmetry of piecewise locally symmetric manifolds

We show that if $g$ is a Riemannian metric on a closed piecewise locally symmetric manifold $M$, then the lift of $g$ to the universal cover $\widetilde{M}$ has a discrete isometry group. We also show that the index $[\Isom(\widetilde{M}): π_1(M)]$ is bounded by a constant independent of $g$.

Cylinders as Left Invariant CMC Surfaces in $\operatorname{Sol}_3$ and $E(κ,τ)$-Spaces Diffeomorphic to $\mathbb{R}^3$

In the present paper we give a geometric proof for the existence of cylinders with constant mean curvature $H>H(X)$ in certain simply connected homogeneous three-manifolds $X$ diffeomorphic to $\mathbb{R}^3$, which always admit a Lie group structure. Here, $H(X)$ denotes the critical value for which constant mean curvature spheres in $X$ exist. Our cylinders are generated by a simple closed curve under a one-parameter group of isometries, induced by left translations along certain geodesics. In the spaces $\operatorname{Sol}_3$ and $\widetilde{\operatorname{PSL}}_2(\mathbb{R})$ we establish ex…

Curvature of almost quaternion-Hermitian manifolds

We study the decomposition of the Riemannian curvature R tensor of an almost quaternion-Hermitian manifold under the action of its structure group Sp(n)Sp(1). Using the minimal connection, we show that most components are determined by the intrinsic torsion ξand its covariant derivative \widetilde\nablaξand determine relations between the decompositions of ξ\otimesξ, \widetilde\nablaξand R. We pay particular attention to the behaviour of the Ricci curvature and the q-Ricci curvature.

Constant mean curvature $k$-noids in homogeneous manifolds

For each $k\geq2$, we construct two families of surfaces with constant mean curvature $H$ for $H\in[0,1/2]$ in $Σ(κ)\times\R$ where $κ+4H^2\leq0$. The surfaces are invariant under $2π/k$-rotations about a vertical fiber of $Σ(κ)\times\R$, have genus zero, and a finite number of ends. The first family generalizes the notion of $k$-noids: It has $k$ ends, one horizontal and $k$ vertical symmetry planes. The second family is less symmetric and has two types of ends. Each surface arises as the conjugate (sister) surface of a minimal graph in a homogeneous 3-manifold. The domain of the graph is non…

Embeddedness of proper minimal submanifolds in homogeneous spaces

We prove the three embeddedness results as follows. $({\rm i})$ Let $Γ_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of $Γ_{2m+1}<2mπ$. In particular, the total curvature of $Γ_5<4π$ and thus any minimal surface $Σ\subset \mathbb{R}^n$ bounded by $Γ_5$ is embedded. Let $Γ_5$ be a piecewise geodesic Jordan curve with $5$ vertices in $\mathbb{H}^n$. Then any minimal surface $Σ\subset \mathbb{H}^n$ bounded by $Γ_5$ is embedded. If $Γ_5$ is in a geodesic ball of radius $\fracπ{4}$ in $\mathbb{S}^n_+$, …

Eigenvalues and Entropy of a Hitchin representation

We show that the critical exponent of a representation in the Hitchin component of $PSL(d,\mathbb{R})$ is bounded above, the least upper bound being attained only in the Fuchsian locus. This provides a rigid inequality for the area of a minimal surface on $ρ\backslash X,$ where $X$ is the symmetric space of $PSL(d,\mathbb{R}).$ The proof relies in a construction useful to prove a regularity statement: if the Frenet equivariant curve of $ρ$ is smooth, then $ρ$ is Fuchsian.

Area-minimizing Cones over Products of Grassmannian Manifolds

This paper is the continuation of the previous one \cite{Cui2021}, where we re-proved the area-minimization of cones over Grassmannians of $n$-planes $G(n,m;\mathbb{F})(\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H})$, Cayley plane $\mathbb{O}P^2$ from the point view of Hermitian orthogonal projectors, and gave area-minimizing cones associated to oriented real Grassmannians $\widetilde{G}(n,m;\mathbb{R})$. In this paper, we make a further step on showing that the cones, of dimension no less than $\mathbf{8}$, over minimal products of $G(n,m;\mathbb{F})$ are area-minimizing. Moreover, those cones …

单模半直积中的垂直不变极小曲面

Vertically invariant minimal surfaces in unimodular semidirect products

本文研究形如$\mathbb{R}^2\rtimes_A \mathbb{R}$的单模度量李群中，在垂直轴左平移群下不变的极小曲面，并给出了$\widetilde{E}(2)$中此类曲面的新实例。

极小曲面熵与平均面积比

Minimal Surface Entropy and Average Area Ratio

本文在闭可双曲化三维流形上建立了极小曲面熵与平均面积比的精确关系，证明在数量曲率≥-6的度量中双曲度量最大化前者，从而解决Gromov猜想，证明采用带手术的Ricci流与PSL(2,R)不变层流测度。

非紧型对称空间中的不稳定极小曲面

Unstable minimal surfaces in symmetric spaces of non-compact type

本文证明了亏格至少为3的闭曲面在分裂实半单李群作用下存在不稳定极小映射，并利用高能极小映射指数的新下界，在特定参数下推翻了Labourie猜想。

关于度量Kollár-Pardon问题

On the metric Kollár-Pardon problem

本文研究度量Kollár-Pardon问题，在万有覆盖可定义且拉回度量可定义的条件下，证明基本群拟等距于局部齐性空间，并由此证明光滑非球面簇情形的Kollár-Pardon猜想。

在 $\mathbb{H}^2\times\mathbb{R}$ 中常平均曲率曲面的构造及 Krust 性质

A construction of constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$ and the Krust property

本文构造了 $\mathbb{H}^2\times\mathbb{R}$ 中一族常平均曲率曲面，推广了鞍形塔和 $k$-noids，并证明 Krust 性质对任何 $H>0$ 不成立。

Maximal compact tori in the Hamiltonian groups of 4-dimensional symplectic manifolds

We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if $M$ is a simply connected symplectic 4-manifold with $b_{2}\geq 3$, and if $\widetilde{M}_δ$ denotes a blow-up of $M$ of small enough capacity $δ$, then the rational cohomology algebra of the Hamiltonian group of $\widetilde{M}_δ)$ is not finitely generated. Both results are based on the fact that…

闭双曲三维流形中的几何与拓扑随机曲面

Geometrically and topologically random surfaces in a closed hyperbolic three manifold

本文研究闭双曲三维流形中随机极小曲面的极限测度，发现拓扑极限测度在全测地子曲面存在时完全疤痕，而几何极限测度则不然。