Thomas Schick
For non-compact manifolds with boundary we prove that bounded geometry defined by coordinate-free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geometry, and we study the change of geodesic coordinate maps.
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A surgery classification theory is introduced for manifolds of bounded geometry up to quasi-isometry. The Borel conjecture for this theory is proven for flat Euclidean space.
In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compact set has nonnegative Ricci curvature and nonnegative mean curvature (of the boundary) then its first relative L^2-cohomology vanishes (this in particular answers a question of Roe). We prove the Hodge-de Rham-theorem for L^2-cohomology of oriented boundary-manifolds of bounded g…
We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.
Bounded volume class and Cheeger isoperimetric constant for negatively curved manifolds
研究负曲率流形中有界体积类与Cheeger等周常数的关系，证明有界体积类消失当且仅当Cheeger常数为正，解决Kim和Kim的猜想部分。
Verifying a conjecture of Gromov we establish a generalized Margulis Lemma for manifolds with lower Ricci curvature bound. Among the various applications are finiteness results for fundamental groups of compact $n$-manifolds with upper diameter and lower Ricci curvature bound modulo nilpotent normal subgroups.
We study noncompact, complete, finite volume, Riemannian 4-manifolds $M$ with sectional curvature $-1<K<0$. We prove that $π_1 M$ cannot be a 3-manifold group. A classical theorem of Gromov says that $M$ is homeomorphic to the interior of a compact manifold $\M$ with boundary $\partial\barM$. We show that for each $π_1$-injective boundary component $C$ of $\M$, the map $i_*$ induced by inclusion $i\colon C\rightarrow \M$ has infinite index image $i_*(π_1 C)$ in $π_1 \M$. We also prove that $M$ cannot be homotoped to be contained in $\partial\M$.
We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between Kähler geometry of the complex hyperbolic space and the contact structure at its infinity (the one-point compactification of the Heisenberg group), in particular an established structural theorem for discrete group actions on nilpotent Lie groups.
A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extended to a leafwise version of the Novikov differential complex.
Let $M$ be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential $2$-form $ξ\inΩ^2(M)$ defines a bounded cocycle $c_ξ\in C_b^2(M)$ by integrating $ξ$ over straightened $2$-simplices. In particular Barge and Ghys proved that, when $M$ is a closed hyperbolic surface, $Ω^2(M)$ injects this way in $H_b^2(M)$ as an infinite dimensional subspace. We show that any class of the form $[c_ξ]$, where $ξ$ is an exact differential 2-form, belongs to the radical of the cup product on the graded algebra $H_b^\bullet(M)$.
A 4-manifold is constructed with some curious metric properties; or maybe it is many 4-manifolds masquerading as one, which would explain why it looks curious. Anyway, knots in the 3-sphere with complete finite volume hyperbolic metrics on their complements play a role in this story.
We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed manifolds with (sectional curvature) local bounded covering geometry, to manifolds with (global) Ricci boundedcovering geometry.
In this work, we (partially) generalize two classical tools in study of collapsed manifolds with bounded sectional curvature: a (singular) fibration theorem by Fukaya (1987) and Cheeger-Fukaya-Gromov (1992), and the stability for isometric compact Lie group actions on manifolds by Palais (1961) and Grove-Karcher (1973), to manifolds with local bounded Ricci covering geometry. Our two generalized results have been used in a recent work of Xiaochun Rong in generalizing Gromov's almost flat manifolds theorem to maximally collapsed manifolds with local bounded Ricci covering geometry.
In this paper we study the extension of holomorphic canonical forms on complete d-bounded Kahler manifolds by using L2 analytic methods and L2 Hogde theory, which generalizes some classical results to noncompact cases.
In this note, we prove that for a complete noncompact three dimensional Riemannian manifold with bounded sectional curvature, if it has uniformly positive scalar curvature, then there is a uniform lower bound on the injectivity radius.
We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor having finite $L^2$ norm in dimension 4.
We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary.
If Pi: M -> B is an onto smooth maximal rank map between complete Riemannian manifolds M and B with bounded geometry, we prove sufficient conditions for M to be roughly isometric to the Riemannian product FxB, where F is a fiber of M.
We study the Yamabe problem on open manifolds of bounded geometry and show that under suitable assumptions there exist Yamabe metrics, i.e. conformal metrics of constant scalar curvature. For that, we use weighted Sobolev embeddings.
In this paper, we study the evolution of L2 p-forms under Ricci flow with bounded curvature on a complete non-compact or a compact Riemannian manifold. We show that under curvature pinching conditions on such a manifold, the L2 norm of a smooth p-form is non-increasing along the Ricci flow. The L^{\infty} norm is showed to have monotonicity property too.
Let (M,g) be a complete noncompact riemannian manifold with bounded geometry and parallel Ricci curvature. We show that some operators, "affine" relatively to the Ricci curvature, are locally invertible, in some classical Sobolev spaces, near the metric g.
We compute the curvature tensor of the tangent bundle of a Riemannian manifold endowed with a natural metric and we get some relationships between the geometry of the base manifold and the geometry of the tangent bundle.
We obtain a partial parallelism of the complex structure on Kähler Finsler manifolds. As applications, we prove Synge-Tsukamoto theorem and Bonnet-Myers theorem for positively curved Kähler Finsler manifolds. Moreover, we generalize a comparison theorem due to Ni-Zheng by introducing the notion of orthogonal Ricci curvature to Kähler Finsler geometry.
In this paper, we introduce the trans-para-Sasakian manifolds and we study their geometry. These manifolds are an analogue of the trans-Sasakian manifolds in the Riemannian geometry. We shall investigate many curvature properties of these manifolds and we shall give many conditions under which the manifolds are either $η-$Einstein or Einstein manifolds.
We consider a spinorial Yamabe-type problem on open manifolds of bounded geometry. The aim is to study the existence of solutions to the associated Euler-Lagrange-equation. We show that under suitable assumptions such a solution exists. As an application, we prove that existence of a solution implies the conformal Hijazi inequality for the underlying spin manifold.
We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.
We show that for a noncollapsing sequence of closed, connected, oriented Riemannian manifolds with Ricci curvature uniformly bounded from below and diameter uniformly bounded above, Gromov-Hausdorff convergence essentially agrees with intrinsic flat convergence.
We construct bi-Lipschitz embeddings into Euclidean space for manifolds and orbifolds of bounded diameter and curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. Our results also apply for bounded subsets of complete Riemannian manifolds, and complete flat and elliptic orbifolds. Our approach is based on analysing the structure of a bounded curvature manifold at various scales by specializing methods from collapsing theory to a certain class of model spaces.
We establish some important inequalities under a lower weighted Ricci curvature bound on Finsler manifolds. Firstly, we establish a relative volume comparison of Bishop-Gromov type. As one of the applications, we obtain an upper bound for volumes of the Finsler manifolds. Further, when the S-curvature is bounded on the whole manifold, we obtain a theorem of Bonnet-Myers type on Finsler manifolds. Finally, we obtain a sharp Poincaré-Lichnerowicz inequality by using integrated Bochner inequality, from which we obtain a sharp lower bound for the first eigenvalue on the Finsler manifolds.
We propose a new approach to the study of compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary or positive Ricci curvature and convex boundary. Several conjectures are formulated. Some partial results that support these conjectures are established.
We consider a complete biharmonic submanifold $ϕ:(M,g)\rightarrow (N,h)$ in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant $c$. Assume that the mean curvature is bounded from below by $\sqrt c$. If (i) $\int_M (|{\bf H}|^2-c)^{p}dv_g<\infty$, for some $0<p<\infty$, or (ii) the Ricci curvature of $M$ is bounded from below, then the mean curvature is $\sqrt c$. Furthermore, if $M$ is compact, then we obtain the same result without the assumption (i) or (ii).
We show that if a bounded domain in complex Euclidean space with $\mathcal{C}^{1,1}$ boundary covers a compact manifold, then the domain is biholomorphic to the unit ball.
Geometry of Bounded Frechet Manifolds
本文发展了有界Fréchet流形的几何，证明其二阶切丛成为向量丛当且仅当被赋予线性联络，并据此证明了向量场积分曲线的存在性与唯一性。
We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the main result of the fourth author [Mar12] in the compact case. The proof uses Ricci flow with surgery as well as arguments involving performing infinite connected sums with control on the geometry.
Sharp Sobolev and Moser-Trudinger inequalities on noncompact Riemannian manifolds with Ricci curvature bounded below
本文研究了非紧致黎曼流形上具有下界曲率的Sobolev和Moser-Trudinger不等式，得出了最佳常数。
Kołodziej-Tosatti's conjecture on compact Hermitian manifold with bounded mass property
本文在具有有界质量性质的紧 Hermitian 流形上证明了 Kołodziej-Tosatti 关于 nef (1,1) 类 Morse 型积分的猜想，并据此肯定回答 Demailly-Păun 与 Tosatti-Weinkove 猜想。
We characterize functions which are growth types of Riemannian manifolds of bounded geometry.
Rigidity for Einstein manifolds under bounded covering geometry
本文证明了有界覆盖几何下爱因斯坦流形的三个刚性结果：几乎平坦必平坦，非零几乎最大体积熵导致双曲，一致局部重绕几乎最大体积导致等距于空间形式。
Determining the associated metrics we get a local classification of contact metric three manifolds.
Physical reasons suggested in \cite{Ha-Ha} for the \emph{Quantum Gravity Problem} lead us to study \emph{type-changing metrics} on a manifold. The most interesting cases are \emph{Transverse Riemann-Lorentz Manifolds}. Here we study the conformal geometry of such manifolds.
In this paper, we systematically investigate the geometry and topology of manifolds with integral radial curvature bounds, and obtain many interesting and important conclusions.
We study non-collapsed Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This extends a result of Donaldson-Sun, who considered non-collapsed limits of polarized Kähler manifolds with two-sided Ricci curvature bounds.
This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-fold
In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. In three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. In higher dimensional manifold, we obtain a volume growth with a stronger condition.
The Blaschke rolling theorem in Riemannian manifolds of bounded curvature
本文将经典 Blaschke 滚动定理推广到有界截面曲率的任意维黎曼流形中的凸区域，所得结果具有精确性，且在常曲率模型空间中亦属首次。
Lower bounds on the normal injectivity radius of hypersurfaces and bounded geometries on manifolds with boundary
本文首次对超曲面法向注入半径进行点wise下界估计，并在任意带有边界的流形上构造具有有界几何的度量，同时证明边界可变为凸。
We study the geometry induced on the local orbit spaces of Killing vector fields on (Riemannian) $G$-manifolds, with an emphasis on the cases $G={\rm Spin}(7)$ and $G=G_2$. Along the way, we classify the harmonic morphisms with one-dimensional fibres from $G_2$-manifolds to Einstein manifolds.
Geometry of Invariant Almost Semi Kähler Submanifolds of Flag Manifolds
该文研究旗流形的不变近半Kähler子流形几何，证明了作为其近Hermite子流形的齐性空间必为极小子流形，若这类子流形也是旗流形则是全测地的。
In this paper, we investigate the geometry of Einstein-type equation on a Riemannian manifold, unifying various particular geometric structures recently studied in the literature, such as critical point equation and vacuum static equation. We show various rigidity results of Einstein-type manifolds under assumptions of several curvature conditions.
We extend a result of Patodi for closed Riemannian manifolds to the context of closed contact manifolds by showing the condition that a manifold is an $η$-Einstein Sasakian manifold is spectrally determined. We also prove that the condition that a Sasakian space form has constant $ϕ$-sectional curvature $c$ is spectrally determined.
We prove that for any complete three-manifold with a lower Ricci curvature bound and a lower bound on the volume of balls of radius one, a solution to the Ricci flow exists for short time. Actually our proof also yields a (non-canonical) way to flow and regularize some interior region of a non-complete initial data satisfying the aformentioned bounds.
On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (Riemannain real vector bundle) with an arbitrary metric connection over a compact Hermitian manifold, we can derive various vanishing theorems for Hermitian manifolds and complex vector bundles by the second Ricci curvature tensors. We will also introduce a natural geometric flow on Hermitian manifolds by using the second Ricci curvature tensor.
We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the uniformity of all estimates throughout the proof. The $C^{k,α}$-smoothness result is optimal with respect to the spectral gap condition involved. The core of the persistence proof is based on the Perron method. In the process we derive new results on noncompact submanifolds in bounded geometry: a uniform tubular neighborhood theorem and uniform smooth approximatio…
We establish the theory of Berezin-Toeplitz quantization on symplectic manifolds of bounded geometry. The quantum space of this quantization is the spectral subspace of the renormalized Bochner Laplacian associated with some interval near zero. We show that this quantization has the correct semiclassical limit.
We study the geometry of PQKT-connections. We find conditions to the existence of a PQKT-connection and prove that if it exists it is unique. We show that PQKT geometry persist in a conformal class of metrics.
In this paper the notion of the intrinsic geometry of an almost contact metric manifold is introduced. Description of some classes of spaces with almost contact metric structures in terms of the intrinsic geometry is given. A new type of almost contact metric spaces, more precisely, Hermitian almost contact metric spaces, is introduced.
We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In particular, for any noncompact covering of a compact manifold, there is a metric on the base so that the lifted metric has an arbitrarily large finite number of gaps in its essential spectrum. Also, for any complete noncompact manifold with bounded curvature and positive injectivity radius we construct a metric uniformly equivalent to the given one (also of bounded c…
Let $ρ_n(V)$ be the number of complete hyperbolic manifolds of dimension n with volume less than $V$. Burger, Gelander, Lubotzky, and Moses showed that when n>3 there exist a,b>0 depending on the dimension such that aV log(V) < log(ρ_n(V)) < bV log(V), for V >> 0. In this note, we use their methods to bound the number of hyperbolic manifolds with diameter less than d and show that the number grows double-exponentially. Additionally, this bound holds in dimension 3.
In this paper we establish a Gaffney type inequality, in $W^{\ell,p}$-Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind non-compact manifolds). Here $p\in]1,\infty[$ and $\ell=1,2$ depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin's complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be pro…
We will construct surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we will prove that a complete non-compact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary, if the manifold M is not less curved than a non-compact model surface of revolution, and if the total curvature of the model surface is finite and less than $2π$. Hence, in the first result mentioned above, we may treat a much wide…
The spectral geometry of hyperbolic and spherical manifolds: analogies and open problems
本文综述双曲流形的谱几何已知结果，探讨其在球面流形的类比成立程度，并列出开放问题。
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