Paper finds a graph Steklov eigenvalue estimate with rigidity results.
problem Estimating Steklov eigenvalues on graphs.
method Lichnerowicz-type estimate for the first Steklov eigenvalues.
result Rigidity results for the Steklov eigenvalues on graphs.
Sharp estimate shown to be rigid on curved surfaces.
problem Sharp Bezout estimate on nonnegatively curved Riemann surfaces.
method General three circle theorem applied.
result Rigidity of the sharp Bezout estimate.
The study examines rigidity and stability of gradient estimates on surfaces and manifolds.
problem Rigidity and stability of gradient estimates for positive harmonic functions and solutions to heat equations.
method Sharp gradient estimates for positive harmonic functions and solutions to heat equations on surfaces and manifolds with nonnegative curvature.
result Obtained rigidity and stability results for gradient estimates.
Alternative approach to rigidity of high-dimensional isometric immersions.
problem Rigidity of high-dimensional isometric immersions between compact manifolds.
method Quantitative rigidity estimates, reducing to Euclidean setting and applying Friesecke-James-Müller rigidity estimate.
result Quantitative results showing close proximity to isometric immersions for small stretching and bending energy.
New rigidity estimate derived via harmonic map flow.
problem Rigidity of maps from S2 to S2. method Harmonic map flow approach.
result Rigidity estimate derived.
Paper proves rigidity estimates for hyperbolic shells and applies them to \(Γ\)-limit theory.
problem Rigidity of hyperbolic shells and their \(Γ\)-limit behavior.
method Nonlinear rigidity estimates for \(H^1\) deformations and hyperbolic shells with clamped lateral boundary.
result Derives the optimal exponent \(h^{-4/3}\) for hyperbolic shells.
Paper extends Steklov eigenvalue estimate to weighted graphs.
problem Steklov eigenvalue estimation on weighted graphs.
method Extended Perrin's estimate to general weighted graphs.
result Characterized rigidity of the extended estimate.
The paper proves estimates and theorems for Kähler manifolds.
problem Curvature conditions on Kähler manifolds.
method Volume comparison and rigidity theorems.
result Conjugate radius estimates for Kähler manifolds.
Proves rigidity for eigenvalue estimate on three-manifolds.
problem Eigenvalue estimate for Kohn Laplacian on three-manifolds.
method Rigidity proof for Lichnerowicz-type estimate.
result Rigidity for eigenvalue estimate on specific three-manifolds.
Study proves rigidity theorems for ancient solutions to mean curvature flow with convex image.
problem Rigidity of ancient solutions to mean curvature flow with convex Gauss image.
method Refined curvature estimates.
result Better rigidity theorems for ancient solutions in higher codimension.
Proves better rigidity theorems for special solitons.
problem Understanding rigidity properties of specific solitons.
method Refined point-wise estimates for mean curvature.
result Stronger rigidity results for Lagrangian and symplectic translating solitons.
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. Sharp curvature estimates for mean curvature flow in spheres.
problem Understanding the behavior of surfaces evolving under mean curvature flow in spheres.
method Proving asymptotically sharp curvature pinching estimates and using them to derive derivative and convexity estimates.
result Partial classification of singularity models and new rigidity results for ancient solutions.
Rigidity for 4D Willmore submanifolds with boundary.
problem Understanding critical points of Willmore energy with boundary conditions.
method Proving a 4-Willmore equation and establishing curvature estimates.
result Four dimensional Willmore submanifolds with totally geodesic boundary are umbilic.
Eigenfunction gradients on curved spaces imply rigid structure.
problem Eigenfunction gradient estimates on curved manifolds.
method Sharp Li-Yau type gradient estimates for Neumann or Dirichlet eigenfunctions.
result Compact manifolds with specific curvature properties are rigidly structured.
Paper proves new rigidity results for biconservative hypersurfaces.
problem Rigidity of non-negatively curved compact biconservative hypersurfaces.
method Alternative proofs and new estimates of the squared norm of the shape operator.
result New rigidity results for biconservative hypersurfaces in space forms.
The goal of this paper is twofold. First we prove a rigidity estimate, which generalises the theorem on geometric rigidity of Friesecke, James and Müller to 1-forms with non-vanishing exterior derivative. Second we use this estimate to prove a kind of spontaneous breaking of rotational symmetry for some models of cryst…
This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
A rigidity theorem for smooth Legendrian self-shrinkers is proven.
problem Understanding the structure of Legendrian self-shrinkers.
method Estimating weighted volume to prove optimal volume growth.
result Rigidity theorem for entire smooth Legendrian self-shrinkers.
Abstract: Survey on quadratic Hessian equations, their properties, and open problems.
problem Understanding quadratic Hessian equations and their solutions.
method Survey and review of existing research.
result Survey of entire solutions, viscosity solutions, and Hessian estimates.
Paper solves Minkowski problem for k-torsional rigidity.
problem Solving Minkowski problem for k-torsional rigidity.
method Constructing Hadamard variational formula, proposing k-torsional measure, using curvature flow method.
result Existence of smooth non-even solutions to the Minkowski problem.
Paper generalizes inscribed radius estimate to hyperbolic Einstein manifolds.
problem Bounding inscribed radius in asymptotically hyperbolic Einstein manifolds.
method Generalized inscribed radius estimate to AH Einstein manifolds, combining recent work.
result Rigidity result achieved for upper bound of relative volume.
Proves rigidity of ancient solutions in mean curvature flow.
problem Rigidity of ancient solutions in mean curvature flow.
method Point-wise estimate for second fundamental form.
result Rigidity theorem of ancient solutions in codimension one.
Researchers prove rigidity of 2D manifolds from boundary geodesic lengths.
problem Reconstructing a Riemann surface from boundary geodesic lengths.
method Re-casting lens data as generalized Riemannian circles and solving a system of equations.
result Essentially optimal results on boundary and lens rigidity for 2D manifolds.
Study stability of rigid motions and Möbius transformations on spheres, proving new rigidity estimates.
problem Stability of rigid motions and Möbius transformations on spheres.
method Investigates both linear and nonlinear stability aspects of rigid motions and Möbius transformations of S^(n-1) into R^n.
result Optimal rigidity estimates for isometric and conformal maps from S^(n-1) to R^n, including new Korn-type inequalities.
The paper shows how heat flows and Wasserstein distances relate to space rigidity.
problem Understanding rigidity in Wasserstein contraction along heat flows.
method Establishing equivalence between rigidity and Bakry-Émery gradient estimates, applying results from Ambrosio-Brué-Semola and Han.
result Spaces with specific curvature bounds exhibit rigidity in Wasserstein contraction.
This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains.
Sharp focal radius estimate for hypersurfaces in manifolds with positive curvature.
problem Estimating the focal radius of hypersurfaces in manifolds with positive curvature.
method Proved a sharp Clifford-threshold focal-radius estimate and rigidity under specific curvature conditions.
result Any closed two-sided immersion satisfies a focal radius estimate of π/4, with equality case rigid.
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.
A new method for non-rigid point set registration reduces computational complexity.
problem Efficiently registering non-rigid point sets with large numbers of points.
method Structured Analytic Coherent Point Drift (Analytic-CPD) reformulates CPD for structured analytic mappings.
result Analytic-CPD reduces computational complexity by controlling the deformation model's dimensionality.
In this paper we show that in some cases the E.Hopf rigidity phenomenon admits quantitative interpretation. More precisely we estimate from above the measure of the set M swept by minimal orbits. These estimates are sharp, i.e. if M occupies the whole phase space we recover the E.Hopf rigidity. …
The paper explores rigidity theorems for spectral curvature bounds in 3-manifolds.
problem Classical rigidity results in scalar curvature geometry are extended to the spectral setting.
method Warped μ-bubble method is systematically employed to classify stable weighted minimal hypersurfaces and establish band width estimates. result Classification theorems and band width estimates for spectral Ricci and scalar curvatures are proven.
In this paper, we firstly establish a new volume growth estimate for spacelike entire graphs in the pseudo-Euclidean space Rnm+n. Then by using this volume growth estimate and the Co-Area formula, we prove various rigidity results for spacelike entire self-shrinking graphs.
The paper studies rigidity and stability of minimal submanifolds in hyperbolic space.
problem Conditions for a minimal submanifold to be totally geodesic.
method Analyzes the length of the second fundamental form and eigenvalues of the super stability operator.
result Sharp upper bounds for the first eigenvalue of the super stability operator for surfaces in hyperbolic 4-space.
Closed hyperbolic manifolds and manifolds with nonpositive sectional curvature are geometrically rigid under certain curvature conditions.
problem Geometric rigidity under scalar curvature lower bound
method Prove rigidity in the equality case of the sharp bottom spectrum estimate
result Closed manifolds with specific curvature conditions must be hyperbolic
Study rigidity on CR Yamabe equation on Sasakian manifolds.
problem Proving rigidity on CR Yamabe equation on Sasakian manifolds.
method Using Jerison-Lee's differential identity and integral estimates.
result Prove that the manifold is CR isometric to Heisenberg group \(\mathbb{H}^n\).
New stability estimate for metric rigidity in hyperbolic dynamics.
problem Metric rigidity in hyperbolic dynamics.
method Radial source estimates in Hölder-Zygmund spaces for uniformly hyperbolic dynamics.
result Metrics with same marked length spectrum are isometric in C3+ε-close metrics in any dimension ≥2. Paper proves rigidity of initial data sets with boundary and capillary MOTS.
problem Rigidity of initial data sets with boundary and capillary MOTS.
method Estimates area of MOTS, proves rigidity for 3D, extends to high dimensions using Yamabe constant.
result Rigidity results for initial data sets with boundary and capillary MOTS.
The paper compares Steklov and Laplacian eigenvalues on graphs.
problem Understanding the relationship between Steklov and Laplacian eigenvalues on graphs.
method Analyzing eigenvalues and discussing rigidity.
result Obtained Lichnerowicz-type estimates and combinatorial estimates for Steklov eigenvalues.
Paper estimates curvature of minimal surfaces in a specific geometric space.
problem Estimating curvature of minimal hypersurfaces in Heisenberg groups.
method Extending Simons formula and Kato inequality to sub-Riemannian setting, applying to stable hypersurfaces.
result Integral curvature estimates for stable hypersurfaces in Heisenberg groups.
Optimizes dimension estimate for holomorphic functions on Kähler manifolds.
problem Determining the optimal dimension for holomorphic functions with polynomial growth.
method Analyzes Kähler manifolds with non-negative holomorphic bisectional curvature.
result Identifies the specific gap and optimal dimension for maximal volume growth.
The study proves triviality and rigidity results for Ricci solitons and estimates their conjugate radius.
problem Understanding the properties and behavior of Ricci solitons.
method Analytical proofs and estimates for various types of Ricci solitons.
result Upper bounds and estimates for conjugate radius of Ricci solitons.
Survey on rigidity results for graphs with prescribed mean curvature.
problem Rigidity of graphs with prescribed mean curvature.
method Analysis of mean curvature operator, maximum principles, gradient estimates.
result Detailed geometric applications, including Bernstein theorem and splitting theorem.
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.
Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.
problem Investigating rigidity phenomena for weighted Ricci curvature bounds.
method Derived comparison geometric estimates and generalized for non-symmetric Laplacian.
result Obtained rigidity results for Laplacian comparison theorem, diameter comparisons, and volume comparisons.
We give rigidity results for the discrete Bonnet-Myers diameter bound and the Lichnerowicz eigenvalue estimate. Both inequalities are sharp if and only if the underlying graph is a hypercube. The proofs use well-known semigroup methods as well as new direct methods which translate curvature to combinatorial properties.…
Study shows how compact shapes can be rigidly mapped into complete manifolds.
problem Rigidity of isometric immersions in complete manifolds.
method Local quantitative rigidity estimates, reduced to Euclidean setting.
result Subsequence of immersions converges to an isometric immersion.
The paper proves new rigidity results for critical metrics of quadratic curvature functionals.
problem Proving rigidity of critical metrics for specific quadratic curvature functionals.
method Rigidity results for conformal vector fields, ODE argument, and new pointwise and integral estimates.
result Critical metrics are rigid under specific conditions.