Survey on rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
problem Rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
method Survey and observation on Cheeger-Yau inequality on RCD spaces.
result Observations on the Cheeger-Yau inequality and its applications.
The extrinsic Bonnet-Myers theorem is proven for positive Ricci curvature manifolds.
problem Understanding the structure of compact Riemannian manifolds with positive Ricci curvature.
method Establishing the extrinsic Bonnet-Myers theorem and showing almost rigidity for hypersurfaces.
result Proven the extrinsic Bonnet-Myers theorem for positive Ricci curvature manifolds and demonstrated almost rigidity for hypersurfaces.
Study almost rigidity of super Ricci flow with non-negative Muller quantity.
problem Almost rigidity properties of super Ricci flow with non-negative Muller quantity.
method Almost splitting and quantitative stratification theorems established by Bamler for Ricci flow.
result Obtained almost constancy for a certain integral quantity concerning scalar curvature at an almost self-similar point.
New volume comparison theorem for gradient Ricci almost solitons.
problem Volume comparison and rigidity of gradient Ricci almost solitons.
method Established a new volume comparison theorem with Bakry-Emery Ricci curvature.
result New volume rigidity result for gradient Ricci almost solitons.
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.
The study provides optimal estimates for surfaces close to constant mean curvature.
problem Optimizing estimates for surfaces near constant mean curvature.
method Bi-Lipschitz and W2,2 parametrization for surfaces with density close to one and small Willmore energy. result Quantitative rigidity for L2-almost CMC surfaces. The paper shows inequality and rigidity for manifolds with integral Ricci curvature.
problem Analyzing structures of manifolds with integral Ricci curvature.
method Using segment inequality and similar methods as in \cite{CC1}, derive almost rigidity structure results.
result Sharp Hölder continuity result holds in the limit space of manifolds with integral Ricci curvature bound.
Study on non-gradient Ricci almost solitons in warped products.
problem Understanding non-gradient Ricci almost solitons.
method Construction method and explicit example in warped products.
result Rigidity result for Gaussian soliton.
We prove the rigidity and vanishing of several indices of "geometrically natural" twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.
The paper proves rigidity results for Einstein manifolds with specific geometric constraints.
problem Understanding the rigidity of Einstein manifolds under bounded covering geometry.
method Analyzing Einstein manifolds with bounded covering geometry to prove rigidity results.
result Compact Einstein manifolds with specific geometric properties are isometric to space forms.
We prove rigidity of any properly immersed noncompact Lagrangian shrinker with single valued Lagrangian angle for Lagrangian mean curvature flows. Our pointwise approach also provides an ele- mentary proof to the known rigidity results for graphical and almost graphical shrinkers of mean curvature flows.
The paper studies the limits and rigidity of almost homogeneous spaces with Ricci curvature bounds.
problem Understanding the limits and rigidity of almost homogeneous spaces with Ricci curvature bounds.
method Analyzes sequences of almost homogeneous RCD(K,N) spaces and their Gromov-Hausdorff limits.
result The Gromov-Hausdorff limit of a sequence of almost homogeneous RCD(K,N) spaces is a nilpotent Lie group with Ric ≥ K.
Almost-isometries are quasi-isometries with multiplicative constant one. Lifting a pair of metrics on a compact space gives quasi-isometric metrics on the universal cover. Under some additional hypotheses on the metrics, we show that there is no almost-isometry between the universal covers. We show that Riemannian mani…
Formula derived for mass of almost Kähler manifolds, extending previous results.
problem Calculating mass in almost Kähler geometry.
method SpinC adaptation of Witten's proof, extending previous complex-geometric methods. result Explicit formula for ADM mass in terms of Hermitian scalar curvature and topological data.
We prove that certain volume preserving actions of Lie groups and their lattices do not preserve rigid geometric structures in the sense of Gromov. The actions considered are the "exotic" examples obtained by Katok and Lewis and the first author, by blowing up closed orbits in the well known actions on homogeneous spac…
Kahler manifolds with specific curvature properties are close to projective spaces.
problem Understanding the shape of Kahler manifolds with maximal volume.
method Combining results on holomorphic rigidity and structure of almost Einstein manifolds.
result Kahler manifolds with lower Ricci bounds and almost maximal volume are close to projective spaces.
The paper proves a quantitative rigidity result for spaces with specific curvature bounds.
problem Understanding the rigidity of spaces with almost maximal volume entropy.
method Analyzing Riemannian manifolds and RCD-spaces with specific curvature conditions. result Spaces with almost maximal volume entropy are closely related to hyperbolic space forms.
Study clarifies almost Ricci-Bourguignon solitons and their properties.
problem Understanding the properties of almost Ricci-Bourguignon solitons.
method Revisit and compare with known results of Barros and Ribeiro.
result Identify conditions for compact almost RB-solitons to be trivial or have special properties.
The main results of this paper consists of two parts. Firstly, we obtain an almost rigidity theorem which says that on a RCD(0, N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Seco…
Paper examines properties of specific solutions to Yamabe flow.
problem Characterizing quotient almost Yamabe solitons.
method Investigates quotient almost Yamabe solitons and presents conditions for their rigidity.
result Sufficient conditions for quotient almost Yamabe solitons to be trivial or isometric with a sphere.
Survey of recent results in weak almost contact structures.
problem New geometric structures replacing complex structure in contact manifolds.
method Survey of recent findings in weak almost contact manifolds.
result Recent results on geodesic and Killing fields, rigidity and splitting theorems, etc.
We show that an almost Hermitian manifold (M,g) of real dimension 4n which is strongly asymptotic to CH2n and satisfies a certain scalar curvature bound must be isometric to the complex hyperbolic space. Assuming Kähler instead of almost Hermitian this gives the already known rigidity result by H. Bou…
The paper proves surfaces close to spheres under specific conditions.
problem Proving rigidity of almost constant mean curvature spheres.
method Linearized analysis around the sphere, Willmore bound, and small defect.
result Almost-CMC surfaces are close to the round sphere with linear control.
We consider complete noncompact Riemannian manifolds with quadratically decaying lower Ricci curvature bounds and minimal volume growth. We first prove a rigidity result showing that ends with strongly minimal volume growth are isometric to warped product manifolds. Next we consider the almost rigid case in which manif…
Sharp bounds on Alexandrov spaces' boundaries with rigidity analysis.
problem Volume bounds on Alexandrov spaces' boundaries.
method Sharp volume bounds and rigidity analysis of Alexandrov spaces.
result New sharp volume bounds and classification of rigidity cases.
We study the equations governing rigid N=1 supersymmetry in five dimensions. If the supersymmetry spinor satisfies a reality condition, these are foliations admitting families of almost complex structures on the leaves. In other words, all these manifolds have families of almost Cauchy-Riemann (CR) structures. After de…
Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle
problem Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle
method Classification using rigidity of structures
result Complete classification in the homogeneous setting
Proves almost profinite rigidity for certain free-by-cyclic groups.
problem Profinite rigidity of free-by-cyclic groups.
method Analyzes ranks of fibres, characteristic polynomials, and stretch factors of monodromies.
result Generic free-by-cyclic groups are almost profinitely rigid.
We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyper…
Researchers extend monotonicity formulas for harmonic functions in RCD(0,N) spaces.
problem Generalizing monotonicity formulas for harmonic functions in mRCD(0,N) spaces. method New estimates for harmonic functions and a functional version of the outer volume cone theorem.
result Proven rigidity and almost rigidity statements for harmonic functions in mRCD(0,N) spaces. We prove that, for a hyperbolic two bridge knot, infinitely many Dehn fillings are rigid in SO0(4,1). Here rigidity means that any discrete and faithful representation in SO0(4,1) is conjugate to the holonomy representation in SO0(3,1). We also show local rigidity for almost all Dehn fillings.
This paper proves a rigidity result for annuli in RCD(K,N)-spaces.
problem The rigidity of annuli in RCD(K,N)-spaces. method The approach uses second order differentiation and a method similar to Cheeger-Colding's.
result Annuli in RCD(K,N)-spaces with certain curvature conditions are measured Gromov-Hausdorff close to a warped product. Paper shows regions close to negatively curved metrics are minimal fillings and rigid.
problem Boundary rigidity and minimality of metrics near negatively curved ones.
method Generalizes previous work on filling volume minimality and boundary rigidity for almost hyperbolic metrics.
result Regions with metrics close to a negatively curved symmetric metric are strict minimal fillings and boundary rigid.
Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
problem Characterizing and proving rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
method Analysis of Riemannian manifolds and metric measure spaces with synthetic lower Ricci curvature bounds, using concentration compactness and Polya-Szego inequalities.
result Closed Riemannian manifolds with optimal Sobolev constant are isometric to the sphere, and almost equality implies close measure Gromov-Hausdorff convergence to a spherical suspension.
The paper proves rigidity estimates for varifolds almost minimizing the Willmore energy.
problem Optimal rigidity estimates for varifolds almost minimizing the Willmore energy.
method Analyzes integral 2-varifolds with generalized mean curvature in Rn. result Varifolds are close to the standard embedding of the round sphere in a quantitative way.
This paper generalizes a rigidity result of complex hyperbolic spaces by M. Herzlich. We prove that an almost Hermitian spin manifold (M,g) of real dimension 4n+2 which is strongly asymptotic to $\hyp{\C}^{2n+1}$ and satisfies a certain scalar curvature bound must be isometric to the complex hyperbolic space. The f…
The study shows almost maximal volume entropy rigidity for certain manifolds with integral Ricci curvature.
problem Volume entropy rigidity for manifolds with lower integral Ricci curvature bound.
method Analyzing manifolds with specific integral Ricci curvature bounds, diameter, and volume entropy.
result The universal cover of the manifold is close to a hyperbolic space form under certain conditions.
This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also provide a more invariant view on the approach used in the above mentioned paper.
The study examines gradient almost Yamabe solitons in warped product manifolds and their geometric properties.
problem Investigating the geometry of gradient almost Yamabe solitons in warped product manifolds.
method Presenting geometric rigidity results, investigating existence conditions, and classifying specific solitons.
result Classification of rotational gradient almost Yamabe solitons in RimesfRn. New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.
problem Generalizing Sasakian and cosymplectic structures to new metric structures.
method Introducing weak structures and proving rigidity of Sasakian structures.
result Any weak Sasakian structure is homothetically equivalent to a Sasakian structure.
We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with R2. As an application, we show that positively curved metrics on S3 and RP3 with almost maximal width must be nearly round.
Sharp gradient estimate for Green functions on non-parabolic RCD(0,N) spaces.
problem Analyzing the gradient of Green functions on non-parabolic RCD(0,N) spaces.
method Defining a smoothed distance function and proving a sharp upper bound for its gradient.
result The gradient of the smoothed distance function has a sharp upper bound and is sharp at points where the space is isomorphic to an RCD(N-2, N-1) space.
We extend our family rigidity and vanishing theorems in [{\bf LiuMaZ}] to the Spin^c case. In particular, we prove a K-theory version of the main results of [{\bf H}], [{\bf Liu1}, Theorem B] for a family of almost complex manifolds.
This paper studies gradient almost Ricci-harmonic soliton with respect to a fixed metric. We rely on analytic techniques to estabilish some basic elliptic and integral equations for the structure of almost Ricci-harmonic soliton which generalizes that of Ricci-hamonic solitons on one hand and that of almost Ricci solit…
We prove that if two non-trapping obstacles in Rn satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.
Study rigidity of self-maps and classify manifolds homotopy equivalent to Stiefel manifolds.
problem Rigidity of self-maps and classification of manifolds homotopy equivalent to Stiefel manifolds.
method Finding explicit inverses in the structure set via normal invariants of specific tangential homotopy equivalences.
result Classification of manifolds tangentially homotopy equivalent to Vn,2imesSk up to almost diffeomorphism. 3-manifolds with toral boundary are uniquely determined by their profinite completions.
problem Determining the homeomorphism type of 3-manifolds based on their profinite completions.
method JSJ-decomposition and profinite completions of fundamental groups.
result Profinitely almost rigid 3-manifolds have finitely many homeomorphism types.
In this paper we give pinching theorems for the first nonzero eigenvalue of the Laplacian on the compact hypersurfaces of ambient spaces with bounded sectional curvature. As application we deduce rigidity results for stable constant mean curvature hypersurfaces M of these spaces N. Indeed, we prove that if M is i…