The action of the diffeomorphism group on the space of immersions

微分几何中的淹没、浸入与平展映射

Submersions, immersions, and étale maps in diffeology

本文通过非线性方法将淹没、浸入和平展映射适应于微分几何，引入微分几何平展流形以统一处理流形与无理环面，证明相关定理并简化内部切空间与维数的计算。

On a new construction of special Lagrangian immersions in complex Euclidean space

We construct new special Lagrangian submanifolds in complex Euclidean space using a pair of minimal Legendrian submanifolds in odd-dimensional spheres and certain Lagrangian surface belonging to a family that can be considered as a generalization of the special Lagrangian surfaces in complex Euclidean plane. Our examples include those invariant under the standard action of SO(p+1)xSO(q+1) on C^n = C^(p+1) x C^(q+1), n=p+q+2.

Liouville Action for Harmonic Diffeomorphisms

In this paper, we introduce a Liouville action for a harmonic diffeomorphism from a compact Riemann surface to a compact hyperbolic Riemann surface of genus $g\ge 2$. We derive the variational formula of this Liouville action for harmonic diffeomorphisms when the source Riemann surfaces vary with a fixed target Riemann surface.

Transformations of flat Lagrangian immersions and Egoroff nets

We associate a natural $λ$-family ($λ\in \R \setminus \{0\} $) of flat Lagrangian immersions in $\C^n$ with non-degenerate normal bundle to any given one. We prove that the structure equations for such immersions admit the same Lax pair as the first order integrable system associated to the symmetric space $\frac{\U(n) \ltimes \C^n}{\OO(n) \ltimes \R^n}$. An interesting observation is that the family degenerates to an Egoroff net on $\R^n$ when $λ\to 0$. We construct an action of a rational loop group on such immersions by identifying its generators and computing their dressing actions. The ac…

Sobolev Metrics on Shape Space, II: Weighted Sobolev Metrics and Almost Local Metrics

In continuation of [3] we discuss metrics of the form $$ G^P_f(h,k)=\int_M \sum_{i=0}^pΦ_i(\Vol(f)) \g((P_i)_fh,k) \vol(f^*\g) $$ on the space of immersions $\Imm(M,N)$ and on shape space $B_i(M,N)=\Imm(M,N)/\on{Diff}(M)$. Here $(N,\g)$ is a complete Riemannian manifold, $M$ is a compact manifold, $f:M\to N$ is an immersion, $h$ and $k$ are tangent vectors to $f$ in the space of immersions, $f^*\g$ is the induced Riemannian metric on $M$, $\vol(f^*\g)$ is the induced volume density on $M$, $\Vol(f)=\int_M\vol(f^*\g)$, $Φ_i$ are positive real-valued functions, and $(P_i)_f$ are operators like s…

Minimizing immersions of a hyperbolic surface in a hyperbolic $3$-manifold

Let $(S,h)$ be a closed hyperbolic surface and $M$ be a quasi-Fuchsian 3-manifold. We consider incompressible maps from $S$ to $M$ that are critical points of an energy functional $F$ which is homogeneous of degree $1$. These "minimizing" maps are solutions of a non-linear elliptic equation, and reminiscent of harmonic maps -- but when the target is Fuchsian, minimizing maps are minimal Lagrangian diffeomorphisms to the totally geodesic surface in $M$. We prove the uniqueness of smooth minimizing maps from $(S,h)$ to $M$ in a given homotopy class. When $(S,h)$ is fixed, smooth minimizing maps …

On a class of immersions of spheres into space forms of nonpositive curvature

Let $ M^{n+1} $ ($ n \ge 2 $) be a simply-connected space form of sectional curvature $ -κ^2 $ for some $ κ\geq 0 $, and $ I $ an interval not containing $ [-κ,κ] $ in its interior. It is known that the domain of a closed immersed hypersurface of $ M $ whose principal curvatures lie in $ I $ must be diffeomorphic to the sphere $ S^n $. These hypersurfaces are thus topologically rigid. The purpose of this paper is to show that they are also homotopically rigid. More precisely, for fixed $ I $, the space $ \mathscr{F} $ of all such closed hypersurfaces is either empty or weakly homotopy equivale…

The space of Cauchy-Riemann structures on 3-D compact contact manifolds

We study the action of the group of contact diffeomorphisms on CR deformations of compact three-dimensional CR manifolds. Using anisotropic function spaces and an anisotropic structure on the space of contact diffeomorphisms, we establish the existence of local transverse slices to the action of the contact diffeomorphism group in the neighbourhood of a fixed embeddable strongly pseudoconvex CR structure.

Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds

A recent preprint of Csikós, Pyber and Szabó (arXiv:1411.7524) proves that the diffeomorphism group of $T^2\times S^2$ is not Jordan. The purpose of this paper is to generalize the arguments of Csikós, Pyber and Szabó in order to obtain many other examples of compact manifolds whose diffeomorphism group fails to be Jordan. In particular we prove that for any $ε>0$ there exist manifolds admitting effective actions of arbitrarily large $p$-groups $Γ$ all of whose abelian subgroups have at most $|Γ|^ε$ elements. Finally, we also recover some results on nonexistence of effective actions of compact…

Sobolev metrics on shape space of surfaces

Let $M$ and $N$ be connected manifolds without boundary with $\dim(M) < \dim(N)$, and let $M$ compact. Then shape space in this work is either the manifold of submanifolds of $N$ that are diffeomorphic to $M$, or the orbifold of unparametrized immersions of $M$ in $N$. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: $$ G^P_f(h,k) = \int_{M} \g(P^f h, k)\, \vol(f^*\g)$$ where $\g$ is some fixed metric on $N$, $f^*\g$ is the induced metric on $M$, $h,k \in Γ(f^*TN)$ are tangent vectors at $f$ to the spac…

The Group of Contact Diffeomorphisms for Compact Contact Manifolds

For a compact contact manifold it is shown that the anisotropic Folland-Stein function spaces form an algebra. The notion of anisotropic regularity is extended to define the space of Folland-Stein contact diffeomorphisms, which is shown to be a topological group under composition and a smooth Hilbert manifold. These results are used in a subsequent paper to analyze the action of the group of contact diffeomorphisms on the space of CR structures on a compact, three dimensional manifold.

Almost local metrics on shape space of hypersurfaces in n-space

This paper extends parts of the results from [P.W.Michor and D. Mumford, \emph{Appl. Comput. Harmon. Anal.,} 23 (2007), pp. 74--113] for plane curves to the case of hypersurfaces in $\mathbb R^n$. Let $M$ be a compact connected oriented $n-1$ dimensional manifold without boundary like the sphere or the torus. Then shape space is either the manifold of submanifolds of $\mathbb R^n$ of type $M$, or the orbifold of immersions from $M$ to $\mathbb R^n$ modulo the group of diffeomorphisms of $M$. We investigate almost local Riemannian metrics on shape space. These are induced by metrics of the foll…

On the adjoint action of the group of symplectic diffeomorphisms

We study the action of Hamiltonian diffeomorphisms of a compact symplectic manifold ($X,ω$) on $C^\infty(X)$ and on functions $C^\infty(X)\to \mathbb R$. We describe various properties of invariant convex functions on $C^\infty(X)$. Among other things we show that continuous convex functions $C^\infty(X)\to \mathbb R$ that are invariant under the action are automatically invariant under so called strict rearrangements and they are continuous in the sup norm topology of $C^\infty(X)$; but this is not generally true if the convexity condition is dropped.

大形变框架下有限维李群与微分同胚群的作用解耦

Decoupling actions of finite-dimensional Lie groups and of groups of diffeomorphisms in the large deformation framework

本文针对计算解剖学中LDDMM框架的形变建模问题，通过引入有限维李群与微分同胚群的半直积结构，提出两类形变联合优化的配准模型，实现两类形变作用的解耦，提升了配准精度。

紧致带角流形上光滑正密度空间的所有微分同胚不变张量场的确定

Determination of all diffeomorphism invariant tensor fields on the space of smooth positive densities on a compact manifold with corners

本文将闭流形上Fisher-Rao度量唯一性定理推广至紧致带角流形，并完整刻画了正概率密度空间上的所有微分同胚不变张量场。

Equivariant homotopy and deformations of diffeomorphisms

We present a way of constructing and deforming diffeomorphisms of manifolds endowed with a Lie group action. This is applied to the study of exotic diffeomorphisms and involutions of spheres and to the equivariant homotopy of Lie groups.

Obstructions to smooth group actions on 4-manifolds from families Seiberg-Witten theory

Let $X$ be a smooth, compact, oriented $4$-manifold. Building upon work of Li-Liu, Ruberman, Nakamura and Konno, we consider a families version of Seiberg-Witten theory and obtain obstructions to the existence of certain group actions on $X$ by diffeomorphisms. The obstructions show that certain group actions on $H^2(X , \mathbb{Z})$ preserving the intersection form can not be lifted an action of the same group on $X$ by diffeomorphisms. Using our obstructions, we construct numerous examples of group actions which can be realised continuously but can not be realised smoothly for any differenti…

Hitchin harmonic maps are immersions

In 1992, Hitchin used his theory of Higgs bundles to construct an important family of representations of the fundamental group of a closed, oriented surface of genus at least two into the split real form of a complex adjoint simple Lie group. These Hitchin representations comprise a component of the space of conjugacy classes of all such representations and are deformations of the irreducible Fuchsian representations which uniformize the surface. For any choice of marked complex structure on the surface and any Hitchin representation, we show that the corresponding equivariant harmonic map fro…

Differentiable conjugacy for groups of area preserving circle diffeomorphisms

We study groups of circle diffeomorphisms whose action on the cylinder $\mathcal C=\mathbb S^1\times \mathbb S^1\setminus Δ$ preserves a volume form. We first show that such a group is topologically conjugate to a subgroup of $\rm{PSL}(2,\mathbb R)$, then discuss the existence of a differentiable conjugacy.

Equivariant hyperbolic diffeomorphisms and representation coverings

Let G be a compact Lie group and X be a compact smooth G-manifold with finitely many G-fixed points. We show that if X admits a G-equivariant hyperbolic diffeomorphism having a certain convergence property, there exists an open covering of X indexed by the G-fixed points so that each open set is G-stable and G-equivariantly diffeomorphic to the tangential G-representation at the corresponding G-fixed point. We also show that the converse is also true in case of holomorphic torus actions

Weakly hyperbolic actions of Kazhdan groups on tori

In this paper, we investigate the ergodic and rigidity properties of weakly hyperbolic group actions. Motivated by classical theorems describing Anosov diffeomorphisms, we obtain two main results: First, all C^2 volume preserving weakly hyperbolic actions on closed manifolds are ergodic. This result generalizes Anosov's classical theorem on ergodicity of Anosov diffeomorphisms to the wider class of weakly hyperbolic group actions. To state the second main result, suppose that A denotes a weakly hyperbolic action of a higher rank lattice (such as SL(n,Z) for n>2) by C^2 volume preserving diffeo…

On the projective derivative cocycle for circle diffeomorphisms

We study the projective derivative as a cocycle of Möbius transformations over groups of circle diffeomorphisms. By computing precise expressions for this cocycle, we obtain several results about reducibility and almost reducibility to a cocycle of rotations. We also introduce an extension of this cocycle to the diagonal action on the 3-torus for which we generalize the previous results.

关于双曲空间中等变浸入的高斯映射

On the Gauss map of equivariant immersions in hyperbolic space

本文完整刻画了双曲空间中等变浸入的高斯映射的可实现性条件，对主曲率在(-1,1)内的情形给出局部条件及基于Maslov类和紧支撑Hamilton辛同胚群作用的两种整体刻画。