设$G$是复李群，$ΛG$表示从单位圆${\mathbb S}^1$到$G$的适当类的映射群.一个从流形$M$到$ΛG$的可微映射$F$称为\emph{connection order $(_a^b)$}的可微映射，如果$F$的Maurer-Cartan形式的${\mathbb S}^1$族的回路参数$λ$中的傅立叶展开，即$F_λ^{-1} \dd F_λ$，具有$\sum_{i=a}^b α_i λ^i$的形式。几何学中的大多数可积系统都与这样的映射有关。DPW方法利用Birkhoff型分裂将一个调和映射约简到一个对称空间中，该空间可以用一个特定阶的$(_{-1}^1)$映射表示，约简成一对分别为$(_{-1}^{-1})$和$(_1^1)$阶的简单映射.相反，可以从任何一对$(_{-1}^{-1})$和$(_1^1)$映射构造这样的调和映射。这允许维尔斯特拉斯型描述调和映射到对称空间。我们扩展这种方法表明，对于一个大类的循环组，连接顺序$(_a^b)$地图，$a<0<b$，唯一地分裂成一对$(_a^{-1})$和$(_1^b)$地图。作为应用，我们证明了球面或双曲空间中具有平坦法丛的常非零曲率子流形分裂成平坦子流形对，从而将问题（至少局部地）简化为平坦的情况。为了充分扩展DPW方法来处理这个问题，需要一个更一般的岩泽型分裂的循环组，我们证明始终保持至少在本地。
英文摘要：
Let $G$ be a complex Lie group and $ΛG$ denote the group of maps from the unit circle ${\mathbb S}^1$ into $G$, of a suitable class. A differentiable map $F$ from a manifold $M$ into $ΛG$, is said to be of \emph{connection order $(_a^b)$} if the Fourier expansion in the loop parameter $λ$ of the ${\mathbb S}^1$-family of Maurer-Cartan forms for $F$, namely $F_λ^{-1} \dd F_λ$, is of the form $\sum_{i=a}^b α_i λ^i$. Most integrable systems in geometry are associated to such a map. Roughly speaking, the DPW method used a Birkhoff type splitting to reduce a harmonic map into a symmetric space, which can be represented by a certain order $(_{-1}^1)$ map, into a pair of simpler maps of order $(_{-1}^{-1})$ and $(_1^1)$ respectively. Conversely, one could construct such a harmonic map from any pair of $(_{-1}^{-1})$ and $(_1^1)$ maps. This allowed a Weierstrass type description of harmonic maps into symmetric spaces. We extend this method to show that, for a large class of loop groups, a connection order $(_a^b)$ map, for $a<0<b$, splits uniquely into a pair of $(_a^{-1})$ and $(_1^b)$ maps. As an application, we show that constant non-zero curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds, reducing the problem (at least locally) to the flat case. To extend the DPW method sufficiently to handle this problem requires a more general Iwasawa type splitting of the loop group, which we prove always holds at least locally.
We investigate the behavior of the second fundamental form of an isometric immersion of a space form with negative curvature into a space form so that the extrinsic curvature is negative. If the immersion has flat normal bundle, we prove that its second fundamental form grows exponentially.
We $δ$-approximate strictly short (e.g. constant) maps between Riemannin manifolds $f_0:X^m\to Y^N$ for $N>>m^2/2$ by $C^\infty$-smooth isometric immersions $f_δ:X^m\to Y^N$ with curvatures $curv(f_δ) < \frac {\sqrt 3}δ$, for $δ\to 0$
Using the results of \cite{P1}, we get some estimates of warping functions for isometric immersions by changing the target manifolds by some types of Riemannian manifolds: constant space forms and Hermitian symmetric spaces. And we deal with equality cases and obtain their applications. Finally, we give some open problems.
We introduce a family of variational functionals for spinor fields on a compact Riemann surface $M$ that can be used to find close-to-conformal immersions of $M$ into $\mathbb{R}^3$ in a prescribed regular homotopy class. Numerical experiments indicate that, by taking suitable limits, minimization of these functionals can also yield piecewise smooth isometric immersions of a prescribed Riemannian metric on $M$.
In this paper, we study biharmonic isometric immersions of a surface into and biharmonic Riemannian submersion from 3-dimensional Berger spheres. We obtain a classification of proper biharmonic isometric immersions of a surface with constant mean curvature into Berger 3-spheres. We also give a complete classification of proper biharmonic Hopf tori in Berger 3-sphere. For Riemannian submersions, we prove that a Riemannian submersion from Berger 3-spheres into a surface is biharmonic if and only if it is harmonic.
Minimal isometric immersions into S^2 x R and H^2 x R
对于给定的单连通黎曼曲面Sigma，我们将求Sigma到S^2 x R或H^2 x R的最小等距浸入问题与Sigma上的两个偏微分方程的系统联系起来.证明了S^2 x R或H^2 x R中的常内曲率极小曲面是全测地曲面，或者是H^2 x R中某个极限悬链面的一部分。我们还证明了，如果一个非常曲率黎曼曲面允许一个连续的单参数极小等距浸入族进入S^2 x R或H^2 x R，则所有这些浸入是相伴的。
We prove the existence, uniqueness, and $W^{1,2}$-regularity for the solution to the Pfaff system with antisymmetric $L^2$-coefficient matrix in arbitrary dimensions. Hence, we establish the equivalence between the existence of $W^{2,2}$-isometric immersions and the weak solubility of the Gauss--Codazzi--Ricci equations on simply-connected domains. The regularity assumptions of these results are sharp. As an application, we deduce a weak compactness theorem for $W^{2,2}_{\rm loc}$-immersions.
The number of constant mean curvature isometric immersions of a surface
在经典曲面理论中，只有很少的已知曲面例子允许非平凡等距变形，更少的是非单连通的。本文研究具有常中曲率H的定向非单连通曲面的浸入x：M \to R ^[3]的等距变形问题。我们证明了M的所有具有常数中曲率H的等距浸入的空间是R^3的模同余，有限或圆。当它是一个圆时，对于浸入x，M中的每个周期都有消失的力，当H不为0时，也有消失的扭矩。本文将极小曲面的刚性结果推广到常中曲率曲面。此外，我们确定封闭的向量值1-形式的周期给力和扭矩。
We prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric immersion in a Euclidean space via an eigenmap, then the eigenmap is a locally bi-Lipschitz embedding map to a sphere, which generalizes a fundamental theorem of Takahashi in submanifold theory to a non-smooth setting. Applications of these results include a topological sphere theorem and topological finiteness theorems, which are new even for closed Riemannian m…
We prove uniqueness of smooth isometric immersions within the class of negatively curved corrugated two-dimensional immersions embedded into $\mathbb{R}^3$. The main tool we use is the relative entropy method employed in the setting of differential geometry for the Gauss-Codazzi system. The result allows us to compare also two solutions to the Gauss-Codazzi system that correspond to a smooth and a $C^{1,1}$ isometric immersion of not necessarily the same metric and prove continuous dependence of their second fundamental forms in terms of the metric and initial data in $L^2$.
In this article, we study constant mean curvature isometric immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and we classify these isometric immersions when the surface has constant intrinsic curvature. As applications, we use the sister surface correspondence to classify the constant mean curvature surfaces with constant intrinsic curvature in the $3-$dimensional homogenous manifolds $\mathbb{E}(κ, τ)$ and we use the Torralbo-Urbano correspondence to classify the parallel mean curvature surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{H}^2 \time…
In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equations occupy a particularly special place amongst pss equations, since a series of recent papers [2, 7, 11, 12, 13], on several classes of pss equations, seemed to suggest that only the sine-Gordon equation had the above property. Explicit examples are given, which include the sh…
In this article, we study minimal isometric immersions of Kähler manifolds into product of two real space forms. We analyse the obstruction conditions to the existence of pluriharmonic isometric immersions of a Kähler manifold into those spaces and we prove that the only ones into $\mathbb{S}^{m-1}\times\mathbb{R}$ and $\mathbb{H}^{m-1}\times \mathbb{R}$ are the minimal isometric immersions of Riemannian surfaces. Futhermore, we show that the existence of a minimal isometric immersion of a Kähler manifold $M^{2n}$ into $\mathbb{S}^{m-1}\times\mathbb{R}$ and $\mathbb{S}^{m-k}\times \mathbb{H}^k…
A Riemannian manifold endowed with $k\ge2$ complementary pairwise orthogonal distributions is called a Riemannian almost $k$-product manifold. In the article, for the first time, we study the following problem: find a relationship between intrinsic and extrinsic invariants of a Riemannian almost $k$-product manifold isometrically immersed in another Riemannian manifold. For such immersions, we establish an optimal inequality that includes the mixed scalar curvature and the square of the mean curvature. Although Riemannian curvature tensor belongs to intrinsic geometry, a special part called th…
We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields on the manifold, satisfying a certain non-linear equation involving the Riemannian curvature tensor of $M$. Setting $k=1$, we proceed to recover the fundamental theorem of hypersurfaces. In the case of manifolds of positive sectional curvature and $n\geq 3$, we reduce the solvability of the Gauss and Codazzi equations to the cancelation of a set of obstructio…
On any compact Riemann surface there always exists a singular non-CSC (constant scalar curvature) extremal K$\ddot{a}$hler metric which is called a non-CSC HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper, by moving frames, we show that any non-CSC HCMU metric can not be isometrically minimal immersed into 3-dimensional real space forms even locally. In general, any non-CSC HCMU metric can not be isometrically immersed into 3-dimensional real space forms with constant mean curvature (CMC).
We study curvature invariants of a sub-Riemannian manifold (i.e., a manifold with a Riemannian metric on a non-holonomic distribution) related to mutual curvature of several pairwise orthogonal subspaces of the distribution, and prove geometrical inequalities for a sub-Riemannian submanifold. As applications, inequalities are proved for submanifolds with mutually orthogonal distributions that include scalar and mutual curvature. For compact submanifolds, inequalities are obtained that are supported by known integral formulas for almost-product manifolds.
Isometric immersions of constant curvature one metrics on the 2-sphere with two conical singularities into Euclidean 3-space: A partial solution to the Gálvez-Hauswirth-Mira problem
