arXiv research

The paper explores conditions for topological rigidity in quotients of the Davis complex.

Study on topological rigidity of ALE vector bundles with specific conditions.

The paper proposes conjectures about moduli space rigidity.

The paper shows that certain 3-manifolds are essentially Euclidean space.

Non-rigidity of hyperbolic manifold under scalar curvature constraints.

Strong rigidity proven for non-compact surfaces.

We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of top…

The paper explores gaps in curvature-related metrics and rigidity.

Several rigidity problems in toric topology are addressed in \cite{ma-su08}. In this paper, we survey results on those problems including recent development.

Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.

Paper proves rigidity of de-Sitter tori with conical singularities.

New rigidity results for manifolds with maximal symmetry rank and positive intermediate Ricci curvature.

Round cylinders are rigid in Ricci shrinkers close to the standard product.

We prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a group action. We also…

Profinite rigidity proven for many hyperbolic manifolds.

Simplified proof of a famous geometry result for students.

In \cite{BK02}, M. Bonk and B. Kleiner proved a rigidity theorem for expanding quasi-Möbius group actions on Ahlfors $n$-regular metric spaces with topological dimension $n$. This led naturally to a rigidity result for quasi-convex geometric actions on CAT$(-1)$-spaces that can be seen as a metric analog to the "entrop…

We prove that any rigid representation of $π_1Σ_g$ in $\mathrm{Homeo}_+(S^1)$ with Euler number at least $g$ is necessarily semi-conjugate to a discrete, faithful representation into $\mathrm{PSL}(2,\mathbb{R})$. Combined with earlier work of Matsumoto, this precisely characterizes Fuchsian actions by a topological rig…

We introduce the notion of a topological geodesic in a 3-manifold. Under suitable hypotheses on the fundamental group, for instance word-hyperbolicity, topological geodesics are shown to have the useful properties of, and play the same role in several applications as, geodesics in negatively curved spaces. This permits…

The study proves topological rigidity for certain geometric shapes using Poincaré inequalities.

Extends rigidity results to non-homogeneous manifolds.

Study uses equivariant topology to measure distances between G metric spaces.

New rigidity results for complex and quaternionic moment-angle manifolds.

Topologically and geometrically engaging actions have proved to be useful to obtain rigidity results for semisimple Lie group actions. We show that the action of a simple noncompact Lie group on a compact manifold preserving a unimodular rigid geometric structure of algebraic type (e.g. a connection together with a vol…

In this paper we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of f-minimal hypersurfaces.

Study shows curvature constraints force submanifolds to have specific topology or geometry.

New characterizations for manifolds with boundary rigidity results.

We give a survey on recent development of the Novikov conjecture and its applications to topological rigidity and non-rigidity. .

We show a rigidity theorem for the Seiberg-Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and tota…

Sharp pinching conditions restrict the geometry and topology of submanifolds.

Finite rigid sets found in surface curve complexes.

The paper studies hypersurfaces in 5D space forms with topological and rigidity results.

Entropy rigidity for Finsler flows but collapse for Reeb flows.

In [5], a rigidity result was obtained for outermost marginally outer trapped surfaces (MOTSs) that do not admit metrics of positive scalar curvature. This allowed one to treat the "borderline case" in the author's work with R. Schoen concerning the topology of higher dimensional black holes [8]. The proof of this rigi…

Study rigidity of real moment-angle manifolds using cubical geometry.

In this paper, we prove a version of the classical Cartan-Hadamard theorem for negatively curved manifolds, of dimension $n\neq 5$, with non-empty totally geodesic boundary. More precisely, if $M_1^n,M_2^n$ are any two such manifolds, we show that (1) $\partial ^\infty \tilde M_1^n$ is homeomorphic to $\partial ^\infty…

In this note we present various extensions of Obata's rigidity theorem concerning the Hessian of a function on a Riemannian manifold. They include general rigidity theorems for the generalized Obata equation, and hyperbolic and Euclidean analogs of Obata's theorem. Besides analyzing the full rigidity case we also chara…

The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f: N --> M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, …

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and app…

Study pinched submanifolds in space forms, proving rigidity results.

For an orientable surface $S$ of finite topological type with genus $g \geq 3$, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph of $S$. The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set, and in fact a …

Let $S$ be a connected orientable surface of finite topological type. We prove that there is an exhaustion of the curve complex $\mathcal{C}(S)$ by a sequence of finite rigid sets.

Study topological 4-manifolds with specific fundamental groups.

Projective rigidity of circle packings on complex surfaces proved.

We introduce a category of rigid geometries on singular spaces which are leaf spaces of foliations and are considered as leaf manifolds. We single out a special category $\mathfrak F_0$ of leaf manifolds containing the orbifold category as a full subcategory. Objects of $\mathfrak F_0$ may have non-Hausdorff topology u…

We prove a topological rigidity result for simple, thick, hyperbolic P-manifolds of dimension 2: isomorphism of the fundamental groups implies homeomorphism of the P-manifolds. An immediate application is a diagram rigidity theorem for certain amalgamations of free groups: the direct limits of two such diagrams are iso…

Following a survey of the abstract boundary definition of Scott and Szekeres, a rigidity result is proved for the smooth case, showing that the topological structure of the regular part of this boundary in invariantly defined.

In this note we prove the Borel Conjecture for closed, irreducible and sufficiently collapsed three-dimensional Alexandrov spaces. We also pose several questions related to characterization of fundamental groups of three-dimensional Alexandrov spaces, finite groups acting on them and rigidity results.