arXiv research

In this article, we discuss the local rigidity of Clifford-Klein forms of homogeneous spaces of 1-connected completely solvable Lie groups. In fact, we introduce a splitting of the local rigidity: vertical rigidity and horizontal rigidity. By using this splitting, we refine some existing results about the local rigidit…

Totally geodesic subvarieties in moduli space are locally rigid.

Quantum representations of mapping class groups are locally rigid at prime levels.

Local rigidity results for Bergman and Kähler Carathéodory metrics on domains.

Local rigidity proved for convex hypersurfaces in spaces of constant curvature.

In this paper we extend the local scalar curvature rigidity result in [6] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [10]. We obtain the local scalar curvature rigidi…

Study recovers Lorentzian metrics from boundary data, proving local rigidity.

Let $Γ\stackrel{i}{\hookrightarrow} L$ be a lattice in the real simple Lie group $L$. If $L$ is of rank at least 2 (respectively locally isomorphic to $Sp(n,1)$) any unbounded morphism $ρ: Γ\longrightarrow G$ into a simple real Lie group $G$ essentially extends to a Lie morphism $ρ_L: L \longrightarrow G$ (Margulis's s…

Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $Γ<J$ be a lattice. We prove a very general local rigidity result about actions of $J$ or $Γ$. This shows that almost all so-called "standard actions" are locally rigid. As a special case, we see that any action of $Γ$ by toral aut…

The paper proves rigidity of certain solitons with specific properties.

Area-charge inequalities and local rigidity of free boundary MOTS in charged initial data sets

The paper develops techniques to study dynamical systems with Carnot metrics.

Study finds bifurcation and local rigidity points for solutions to the Yamabe problem on Aloff-Wallach Spaces.

In a recent paper Hodgson and Kerckhoff prove a local rigidity theorem for finite volume, three dimensional hyperbolic cone-manifolds. In this paper we extend this result to geometrically finite cone-manifolds. Our methods also give a new proof of a local version of the classical rigidity theorem for geometrically fini…

The paper proves local rigidity theorems for scalar curvature and related inequalities.

In this paper, we proved a rigidity theorem of the Hodge metric for concave horizontal slices and a local rigidity theorem for the monodromy representation.

Let $Γ$ be a discrete group with property $(T)$ of Kazhdan. We prove that any Riemannian isometric action of $Γ$ on a compact manifold $X$ is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisi…

Paper studies rigidity phenomena for elliptic systems on Riemannian manifolds.

The aim of this note is to give a geometric proof for classical local rigidity of lattices in semisimple Lie groups. We are reproving well known results in a more geometric (and hopefully clearer) way.

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

We study the local Killing Lie algebra of meromorphic almost rigid geometric structures on complex manifolds. This leads to classification results for compact complex manifolds bearing holomorphic rigid geometric structures.

In this article, we survey recent developments in defining the quasi-local mass in general relativity. We discuss various approaches and the properties and applications of the different definitions. Among the expected properties, we focus on the rigidity property: for a surface in the Minkowski spacetime, one expects t…

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-mi…

In this paper, we construct a right-angled 5-polytope P of finite volume such that all the right-angled Coxeter groups with Fuchsian ends obtained from P are locally rigid.

Relations between parameter rigidity of locally free Lie group actions on closed manifolds and the 1st leafwise cohomology of the orbit foliations are discussed. Some computational results of the leafwise cohomology are included.

New rigidity theorem on static manifolds with boundary.

New proof shows all conformal fields are Killing on specific spaces.

Inspired by the work of Chen-Zhang \cite{Chen-Zhang}, we derive an evolution formula for the Wang-Yau quasi-local energy in reference to a static space, introduced by Chen-Wang-Wang-Yau \cite{CWWY}. If the reference static space represents a mass minimizing, static extension of the initial surface $Σ$, we observe that …

Under the assumption that the X-ray transform over symmetric solenoidal 2-tensors is injective, we prove that smooth compact connected manifolds with strictly convex boundary, no conjugate points and a hyperbolic trapped set are locally marked boundary rigid.

In this note, we study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic plane than in complex hyperbolic plane.

We show the rigidity of the hexagonal Delaunay triangulated plane under Luo's PL conformality. As a consequence, we obtain a rigidity theorem for a particular type of locally finite convex ideal hyperbolic polyhedra.

The paper proves rigidity and ε-regularity theorems for Ricci shrinkers.

Localized curvature bounds ensure harmonic maps are constant.

In this work we determine bifurcation instants for 1-parameter families of solutions to the Yamabe problem defined on maximal flag manifolds. We also study the local rigidity points, namely, a isolated solution of the Yamabe problem.

We show the local rigidity of complex hyperbolic lattices in classical Hermitian semisimple Lie groups, $SU(np,p), Sp(2n+2,\mathbb R), SO^*(2n+2), SO(2n,2)$. This reproves or generalizes some results in \cite{GM, KKP, Klingler-inv, Pozzetti}.

The paper proves rigidity of certain Dirac operators using theta functions.

New proof confirms noncompact locally conformally flat manifolds are compact.

Our aim in this paper is to study local rigidity for metrics defined on a compact manifold $M$ with boundary satisfying constant scalar curvature on $M$ and constant mean curvature on $\partial M$. We present some geometrical hypotheses ensuring local rigidity for both, the general Riemannian and the warped metric case…

We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let $\nabla$ be …

Investigate AR-Finsler metrics for local dual flatness and projective flatness.

This article investigates a few questions about orbits of local automorphisms in manifolds endowed with rigid geometric structures. We give sufficient conditions for local homogeneity in a broad class of such structures, namely Cartan geometries, extending a classical result of Singer about locally homogeneous Riemanni…

Let $M$ be a finite volume analytic pseudo-Riemannian manifold that admits an isometric $G$-action with a dense orbit, where $G$ is a connected non-compact simple Lie group. For low-dimensional $M$, i.e. $\dim(M) < 2\dim(G)$, when the normal bundle to the $G$-orbits is non-integrable and for suitable conditions, we pro…

Paper proves rigidity of 3-manifolds with boundary using modified Hawking mass.

Article strengthens initial data rigidity theorem to show unique spacetime extension.

Paper proves rigidity for certain spacelike hypersurfaces in de Sitter space.

In this article, we prove a rigidity theorem for isometric embeddings into the Schwarzschild manifold, by using the variational formula of quasi-local mass.

We prove that, for a hyperbolic two bridge knot, infinitely many Dehn fillings are rigid in $SO_0(4,1)$. Here rigidity means that any discrete and faithful representation in $SO_0(4,1)$ is conjugate to the holonomy representation in $SO_0(3,1)$. We also show local rigidity for almost all Dehn fillings.

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov…