Sigmundur Gudmundsson
We construct new explicit proper biharmonic functions on the $3$-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $H^2\times\rn$ and $S^2\times\rn$.
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In this work we construct a variety of new complex-valued proper biharmonic maps and (2,1)-harmonic morphisms on Riemannian manifolds with non-trivial geometry. These are solutions to a non-linear system of partial differential equations depending on the geometric data of the manifolds involved.
For any positive natural number $r\in\N^+$ we construct new explicit proper $r$-harmonic functions on the celebrated $3$-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $\H^2\times\rn$ and $\s^2\times\rn$.
We find analogues of the Willmore functional for each of the Thurston geometries with 4-dimensional isometry group such that the CMC-spheres in these geometries are critical points of these functionals.
We classify pairs $(M,G)$ where $M$ is a $3$--dimensional simply connected smooth manifold and $G$ a Lie group acting on $M$ transitively, effectively with compact isotropy group.
Homogeneous Hypersurfaces in 4-dimensional Thurston Geometries with 4-dimensional Isometry Group
通过分类Lie子代数及其轨道，完整描述了4维Thurston几何中具有4维等距群的齐次超曲面，并研究了轨道叶状结构的几何性质。
This is a survey on the global theory of constant mean curvature surfaces in Riemannian homogeneous 3-manifolds. These ambient 3-manifolds include the eight canonical Thurston 3-dimensional geometries, i.e. R3, H3, S3, H2 \times R, S2 \times R, the Heisenberg space Nil3, the universal cover of PSL2(R) and the Lie group Sol3. We will focus on the problems of classifying compact CMC surfaces and entire CMC graphs in these spaces. A collection of important open problems of the theory is also presented.
Thurston geometries in dimension four from a Riemannian perspective
本文从黎曼几何视角综述四维瑟斯顿几何，重点描述非乘积且等距群为4维或5维的 $\mathrm{Sol}^4_0$、$\mathrm{Sol}^4_1$、$\mathrm{Sol}^4_{m,n}$ 和 $\mathrm{Nil}^4$ 空间，并给出其全部不变黎曼度量。
Biharmonic Riemannian submersions from a 3-dimensional BCV space
本文给出了3维BCV空间中真双调和Riemannian浸没的完整分类，证明其仅在两种映射情况下存在并构造了无限族解，推广了先前对3维空间形式的研究。
Homogeneous hypersurfaces of the four-dimensional Thurston geometries $\mathrm{Sol}_1^4$, $\mathrm{Sol}_{m,n}^4$ and $\mathrm{Nil}^4$
本文分类了等距群为四维的四维Thurston几何 $\mathrm{Sol}_1^4$, $\mathrm{Sol}_{m,n}^4$ 和 $\mathrm{Nil}^4$ 中的齐次超曲面。
We prove that a totally umbilical biharmonic surface in any $3$-dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if and only if it is a part of $S^2(1/\sqrt{2})$ in $S^3$. We also give complete classifications of constant mean curvature proper biharmonic surfaces in 3-dimensional geometries and in 3-dimensional Bianchi-Cartan-Vranceanu spaces, and a complete classifications of proper biharmonic Hopf cylinders in 3-dimensional Bianchi-Cartan-Vranceanu spaces.
We describe algorithms that produce accurate real-time interactive in-space views of the eight Thurston geometries using ray-marching. We give a theoretical framework for our algorithms, independent of the geometry involved. In addition to scenes within a geometry $X$, we also consider scenes within quotient manifolds and orbifolds $X / Γ$. We adapt the Phong lighting model to non-euclidean geometries. The most difficult part of this is the calculation of light intensity, which relates to the area density of geodesic spheres. We also give extensive practical details for each geometry.
Uniqueness of the Bonnet problem in Thurston geometries
在Thurston几何中，研究Bonnet问题的唯一性，通过建立Bonnet配对的唯一性，探讨等距浸入连续变形的条件，发现仅极小曲面和主曲率恒定曲面满足要求。
Complément to the Thurston 3D-Geometrization
本文通过引入“区域李生成几何”概念，使仿射粘合尊重度量几何解释，利用Thurston几何化定理解决了一个45年前关于庞加莱猜想的疑问/方法。
In this paper, we study harmonic Riemannian submersions from 3-dimensional geometries using the ( generalized) integrability data associated to an orthonormal frame natural to a Riemannian submersion. We give complete classifications of harmonic Riemannian submersions from Thurston's 3-dimensional geometries, 3-dimensional BCV spaces and Berger sphere into a surface. We also give some explicit constructions of these harmonic Riemannian submersions.
We discuss existence and classification of totally umbilic surfaces in the model geometries of Thurston and the Berger spheres. We classify such surfaces in $H^2 \times R$, $S^2 \times R$ and the Sol group. We prove nonexistence in the Berger spheres and in the remaining model geometries other than the space forms.
We derive the Weierstrass (or spinor) representation for surfaces in three-dimensional Lie groups Nil, \tilde{SL}_2, and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. By using the spectral properties of the corresponding Dirac operators we find analogs of the Willmore functional for surfaces in these geometries.
We classify the $5$-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 2 of 3) classifies those in which the linear isotropy representation is either irreducible or trivial. The $5$-dimensional geometries with irreducible isotropy are the irreducible Riemannian symmetric spaces, while those with trivial isotropy are simply-connected solvable Lie groups of the form $\mathbb{R}^3 \rtimes \mathbb{R}^2$ or $N \rtimes \mathbb{R}$ where $N$ is nilpotent.
We classify the 5-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 3 of 3) classifies those in which the linear isotropy representation is nontrivial but reducible. Most of the resulting geometries are products. Some interesting examples include a countably infinite family of inequivalent geometries diffeomorphic to $S^3 \times S^2$; an uncountable family in which only a countable subfamily admits compact quotients; and the non-maximal geometry SO(4)/SO(2) realized by two distinct maximal geometries.
This paper is the first of a 3-part series that classifies the 5-dimensional Thurston geometries. The present paper (part 1 of 3) summarizes the general classification, giving the full list, an outline of the method, and some illustrative examples. This includes phenomena that have not appeared in lower dimensional geometries, such as an uncountable family of geometries.
In the present paper we study $S^2 \times R$ and $H^2 \times R$ geometries, which are homogeneous Thurston 3-geometries. We analyse the interior angle sums of geodesic triangles in both geometries and prove, that in $S^2 \times R$ space it can be larger or equal than $π$ and in $H^2 \times R$ space the angle sums can be less or equal than $π$. In our work we will use the projective model of $S^2 \times R$ and $H^2 \times R$ geometries described by E. Molnár in \cite{M97}.
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