Let $M_Σ$ be an $n$-dimensional Thom-Mather stratified space of depth $1$. We denote by $βM$ the singular locus and by $L$ the associated link. In this paper we study the problem of when such a space can be endowed with a wedge metric of positive scalar curvature. We relate this problem to recent work on index theory on stratified spaces, giving first an obstruction to the existence of such a metric in terms of a wedge $α$-class $α_w (M_Σ)\in KO_n$. In order to establish a sufficient condition we need to assume additional structure: we assume that the link of $M_Σ$ is a homogeneous space of positive scalar curvature, $L=G/K$, where the semisimple compact Lie group $G$ acts transitively on $L$ by isometries. Examples of such manifolds include compact semisimple Lie groups and Riemannian symmetric spaces of compact type. Under these assumptions, when $M_Σ$ and $βM$ are spin, we reinterpret our obstruction in terms of two $α$-classes associated to the resolution of $M_Σ$, $M$, and to the singular locus $βM$. Finally, when $M_Σ$, $βM$, $L$, and $G$ are simply connected and $\dim M$ is big enough, and when some other conditions on $L$ (satisfied in a large number of cases) hold, we establish the main result of this article, showing that the vanishing of these two $α$-classes is also sufficient for the existence of a well-adapted wedge metric of positive scalar curvature.
In this paper we continue the study of positive scalar curvature (psc) metrics on a depth-1 Thom-Mather stratified space $M_Σ$ with singular stratum $βM$ (a closed manifold of positive codimension) and associated link equal to $L$, a smooth compact manifold. We briefly call such spaces manifolds with $L$-fibered singularities. Under suitable spin assumptions we give necessary index-theoretic conditions for the existence of wedge metrics of positive scalar curvature. Assuming in addition that $L$ is a simply connected homogeneous space of positive scalar curvature, $L=G/H$, with the semisimple …
We define and study, under suitable assumptions, the fundamental class, the index class and the rho class of a spin Dirac operator on the regular part of a spin stratified pseudomanifold. More singular structures, such as singular foliations, are also treated. We employ groupoid techniques in a crucial way; however, an effort has been made in order to make this article accessible to readers with only a minimal knowledge of groupoids. Finally, whenever appropriate, a comparison between classical microlocal methods and groupoids methods has been provided.
The goal of this paper is to investigate the topological structure of open simply-connected 3-manifolds whose scalar curvature has a slow decay at infinity. In particular, we show that the Whitehead manifold does not admit a complete metric, whose scalar curvature decays slowly, and in fact that any contractible complete 3-manifolds with such a metric is homeomorphic to $\mathbb{R}^{3}$. Furthermore, using this result, we prove that any open simply-connected 3-manifold $M$ with $π_{2}(M)=\mathbb{Z}$ and a complete metric as above, is homeomorphic to $\mathbb{S}^{2}\times \mathbb{R}$.
Using recent work of Bettiol, we show that a first-order conformal deformation of Wilking's metric of almost-positive sectional curvature on $S^2\times S^3$ yields a family of metrics with strictly positive average of sectional curvatures of any pair of 2-planes that are separated by a minimal distance in the 2-Grassmanian. A result of Smale's allows us to conclude that every closed simply connected 5-manifold with torsion-free homology and trivial second Stiefel-Whitney class admits a Riemannian metric with a strictly positive average of sectional curvatures of any pair of orthogonal 2-planes…
We show that in each dimension $4n+3$, $n\ge 1$, there exist infinite sequences of closed smooth simply connected manifolds $M$ of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension seven, and our result does also hold for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, in conjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such $M$ the moduli space of complete…
Zero mean curvature surfaces in the simply isotropic 3-space $\mathbb{I}^3$ naturally appear as intermediate geometry between geometry of minimal surfaces in $\mathbb{E}^3$ and that of maximal surfaces in $\mathbb{L}^3$. In this paper, we investigate reflection principles for zero mean curvature surfaces in $\mathbb{I}^3$ as with the above surfaces in $\mathbb{E}^3$ and $\mathbb{L}^3$. In particular, we show a reflection principle for isotropic line segments on such zero mean curvature surfaces in $\mathbb{I}^3$, along which the induced metrics become singular.
We classify simply-connected, complete Willmore surfaces with vanishing Gaussian curvature. We also study the Willmore cones and give a classification. As an application, we give a Bernstein-type theorem.
The stable converse soul question (SCSQ) asks whether, given a real vector bundle \(E\) over a compact manifold, some stabilization \(E\times\R^k\) admits a metric with non-negative (sectional) curvature. We extend previous results to show that the SCSQ has an affirmative answer for all real vector bundles over any simply connected homogeneous manifold with positive curvature, except possibly for the Berger space \(B^{13}\). Along the way, we show that the same is true for all simply connected homogeneous spaces of dimension at most seven, for arbitrary products of simply connected compact ran…
We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally Kähler if and only if the determinant of the self-dual Weyl curvature is positive.
In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined by Heintze, Liu and Olmos. We will also prove a splitting theorem which reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results of Liu and Terng for the mean curvature flow of isoparametric submanifolds in spheres…
We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic geometries are the study of a three-dimensional space equipped with a rank 2 metric of index zero and one, respectively. We show that the one-parameter subgroups of isotropic rigid motions lead to seven types of invariant surfaces, which then generalizes the study of revolution and helicoidal surfaces in Euclidean and Lorentzian spaces to the context of singular metrics. After computing the two fundamental forms of these surfaces and their Ga…
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。
In this work, we are interested in the differential geometry of surfaces in simply isotropic $\mathbb{I}^3$ and pseudo-isotropic $\mathbb{I}_{\mathrm{p}}^3$ spaces, which consists of the study of $\mathbb{R}^3$ equipped with a degenerate metric such as $\mathrm{d}s^2=\mathrm{d}x^2\pm\mathrm{d}y^2$. The investigation is based on previous results in the simply isotropic space [B. Pavković, Glas. Mat. Ser. III $\mathbf{15}$, 149 (1980); Rad JAZU $\mathbf{450}$, 129 (1990)], which point to the possibility of introducing an isotropic Gauss map taking values on a unit sphere of parabolic type and of…
We prove that simply connected H-surfaces with bounded area and free boundary in a domain necessarily concentrate at a critical point of the mean curvature of the boundary of this domain.
We consider the extrinsic geometry of surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the metric is degenerate, a surface normal cannot be unequivocally defined based on metric properties only. To understand the contrast between distinct choices of an isotropic Gauss map, here we study surfaces with a Gauss map whose coordinates are eigenfunctions of the surface Laplace-Beltrami operator. We take into account two choices, the so-called minimal and parabolic normals, and show that when applied to simply isotropic invariant surface…
Let $\mathcal{M}_{0}^n$ be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if $M\in \mathcal{M}_{0}^n$, then $M$ is equivariantly diffeomorphic to the free linear quotient by a torus of a product of spheres of dimensions greater than or equal to three. As an immediate consequence, we prove the Maximal Symmetry Rank Conjecture for all $M\in \mathcal{M}_{0}^n$. Finally, we show the Maximal Symmetry Rank Conjecture for simply-connected, non-negatively curved manifolds holds for dimens…
We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension $4k \geq 12$, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to nonnegative curvature.
We give a complete classification of left invariant para-Kähler structures on four-dimensional simply connected Lie groups up to an automorphism. As an application we discuss some curvatures properties of the canonical connection associated to these structures as flat, Ricci flat and existence of Ricci solitons.
Let $M$ be a topological spherical space form, i.e. a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.
Let $X$ be a compact Kähler manifold. We prove that if $X$ admits a smooth Hermitian metric $ω$ with quasi-positive second Chern-Ricci curvature $\mathrm{Ric}^{(2)}(ω)$, then $X$ is projective and rationally connected. In particular, $X$ is simply connected.
We prove two injectivity theorems for the geodesic ray transform on two-dimensional, complete, simply connected Riemannian manifolds with non-positive Gaussian curvature, also known as Cartan-Hadamard manifolds. The first theorem is concerned with bounded non-positive curvature and the second with decaying non-positive curvature.
We present a rigidity theorem for the action of the mapping class group $π_0(\mathrm{Diff}(M))$ on the space $\mathcal{R}^+(M)$ of metrics of positive scalar curvature for high dimensional manifolds $M$. This result is applicable to a great number of cases, for example to simply connected $6$-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the $2$-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected $7$-dimensional $\mathr…
Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth embeddings of the ball with arbitrary small volume.
