3D spherical caps are rigid under certain perturbations.
problem Rigidity of 3D spherical caps under specific perturbations.
method Gromov's μ-bubble technique
result 3D spherical caps are rigid under perturbations that maintain metric, scalar curvature, and mean curvature.
Paper proves rigidity of spherical ring patterns on surfaces.
problem Proving rigidity of spherical orthogonal ring patterns on closed surfaces.
method Modification of combinatorial total geodesic curvature and variational principles.
result Rigidity of spherical orthogonal ring patterns on closed surfaces proved.
Study inequalities involving torsional rigidity and spherical deficit on Riemannian manifolds.
problem Inequalities involving torsional rigidity and spherical deficit on Riemannian manifolds.
method Establish inequalities and derive an integral identity for a Dirichlet problem.
result Characterize metric balls and measure spherical deficit on Riemannian manifolds.
Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid.
problem Rigidity of minimal Lagrangian diffeomorphisms between spherical surfaces.
method Proving that any minimal Lagrangian diffeomorphism between two closed spherical surfaces with cone singularities is an isometry.
result Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid (i.e., they are isometries).
We prove that every smooth rigid spherical hypersurface in C2 is in fact real-analytic. As an application of this result, it follows that the classification of real-analytic rigid spherical hypersurfaces in C2 found by V. Ezhov and G. Schmalz applies in the smooth case.
The paper classifies 3D spherical Sasakian manifolds using geometric and algebraic methods.
problem Classifying 3D spherical Sasakian manifolds with specific properties.
method Establishing correspondence between different sets of parameters and geometrically describing the moduli space.
result Determination of Sasakian automorphism groups and detection of homogeneous Sasakian manifolds.
The study proves rigidity and non-rigidity of spherical caps in mean curvature.
problem Understanding mean curvature rigidity and non-rigidity on spherical caps.
method Used a Tangency Principle to prove rigidity and constructed counterexamples for non-rigidity.
result Contrast between rigidity and non-rigidity phenomena on spherical caps.
Rigidity shown for spherical product Ricci solitons.
problem Characterizing Ricci solitons on spherical products.
method Ricci flow analysis on S2imesS2 and S2imesN. result Isolated rigidity of S2imesS2 as a shrinking Ricci soliton. In the current paper, first we give the correct version of the formula for mean Berwald curvature of a spherically symmetric Finsler metric given in paper \cite{YCheWSon2015}. Further, we establish differential equations characterizing projectively as well as dually flat spherically symmetric Finsler metrics. Finally, …
We establish Bochner-type formulas for operators related to CR automorphisms and spherical CR structures. From such formulas, we draw conclusions about rigidity by making assumptions on the Tanaka-Webster curvature and torsion.
Rigidity theorem for spherical sectors in Riemannian manifolds.
problem Rigidity of spherical sectors in Riemannian manifolds under overdetermined conditions.
method Analyzing solutions to the inhomogeneous Helmholtz equation with constant Dirichlet and Neumann boundary conditions.
result Spherical sectors are the only solutions under given conditions.
Study proves rigid spectral properties of planets with metric discontinuities.
problem Establishing spectral rigidity for spherically symmetric planets with discontinuities.
method Novel trace formula applied to two wave types in spherically symmetric manifolds with boundary and interior interfaces.
result Spectral rigidity of spherically symmetric planets with discontinuities is proven.
The study proves a rigidity theorem for compact manifolds with boundary.
problem Rigidity of compact manifolds with boundary in low dimensions.
method Dimension reduction argument for mean curvature, extending Schoen-Yau's for scalar curvature.
result Sharp spherical radius rigidity and best NNSC fill-in in terms of mean curvature.
We give a counterexample of Bowers-Stephenson's conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances.
We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles ≤π (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles ≤π, possibly with boundary consisting of totally geodesic hyperbo…
New rigidity found for 3D warped product domains.
problem Finding rigidity conditions for warped product domains.
method Developed scalar curvature rigidity for a general class of domains.
result Identified domains satisfying a boundary condition analogous to logarithmic concavity.
We prove two rigidity results for complete Riemannian three-manifolds of higher rank. Complete three-manifolds have higher spherical rank if an only if they are spherical space forms. Complete finite volume three-manifolds have higher hyperbolic rank if and only if they are hyperbolic space forms.
Unique sphere solution for expanding flows converging to a point.
problem Classifying solutions of expanding curvature flows.
method Reflection argument to prove uniqueness of the homothetic sphere.
result Homothetic sphere is the unique solution converging to a point.
The study characterizes compact submanifolds with pinched Ricci curvature in Euclidean and spherical space forms.
problem Characterizing compact submanifolds with specific Ricci curvature bounds.
method Proving rigidity results for submanifolds with Ricci curvature bounded below by a function of mean curvature.
result Submanifolds are either isometric to the Einstein Clifford torus or have vanishing homology groups up to a certain degree.
Compact proof for Brown-York mass positivity and rigidity in flat and spherical spaces.
problem Positivity of Brown-York mass and rigidity of manifolds with mean-convex boundaries.
method Spinorial proof and optimal lower bound for eigenvalues.
result Optimal lower bound for first non-null eigenvalue of Dirac operator.
We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the…
We study the deformation of spherical conical metrics with at least some of the cone angles larger than 2π. We show in this note via synthetic geometry that for one family of such metrics, there is local rigidity in the choice of cone positions if angles are fixed. This gives an evidence of the analytic obstruction c…
A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obt…
We prove an Obata-type rigidity result for the spherical cap and apply it for an eigenvalue problem with mixed boundary condition.
Simple geodesics on spherical tetrahedra identified for specific angles.
problem Identifying simple closed geodesics on regular tetrahedra in spherical space.
method Analyzing pairs of coprime integers (p,q) to find angles α1 and α2.
result Existence and non-existence of simple closed geodesics for specific angles.
The present paper gives an example of a rigid spherical cone-manifold and that of a flexible one which are both Seifert fibred.
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…
We provide an explicit description of all rigid hypersurfaces that are equivalent to a Heisenberg sphere. These hypersurfaces are determined by 4 real parameters. The defining equations of the rigid spheres can also be viewed as the complete solution of a non-linear PDE that expresses the vanishing Cartan curvature con…
New theorem on spheres with punctures using infinity metric.
problem Rigidity of metrics on spheres with punctures.
method Proof of Llarull's theorem for L∞ metrics on spheres with finitely many points removed. result The rigidity theorem holds for L∞ metrics on spheres with finitely many points removed. Study spherical metrics on flat torus with cone singularities.
problem Existence and uniqueness of spherical metrics with specific cone angles.
method Analysis of Green function critical points and torus geometry.
result Existence of unique spherical metrics determined by torus geometry.
Polyhedra's structure is uniquely defined by edge lengths and dihedral angles, even nonconvex.
problem Determining the structure of polyhedra based on edge lengths and dihedral angles.
method Proved rigidity under specific conditions in Euclidean, hyperbolic, and spherical geometries.
result Polyhedra's structure is uniquely defined by edge lengths and dihedral angles, even nonconvex.
The paper proves stability of the positive mass theorem in spherical symmetry.
problem The stability of the positive mass theorem in spherical symmetry when mass is small.
method Formulated a conjecture and proved it under spherical symmetry assumption.
result A sequence of asymptotically flat initial data converging to Minkowski space.
The paper proves rigidity theorems for area widths of Riemannian manifolds.
problem Characterizing metrics on Riemannian manifolds using their area widths.
method Analyzing the volume spectrum and spherical area widths.
result Rigidity theorems for specific metrics on projective spaces.
Study on membranes under confinement, proving existence and regularity of minimizers.
problem Existence and regularity of minimizers for constrained Helfrich energy.
method Elliptic system analysis, careful study of measure-valued Lagrange multiplier.
result Optimal regularity for solutions throughout branch points, rigid behavior for unit ball minimizers.
The paper explores rigid geometric structures near surfaces with equality in area-charge inequalities.
problem Geometric constraints near surfaces with equality in area-charge inequalities.
method Investigation of equality in area-charge inequalities for spherical minimal surfaces and MOTS within the Einstein-Maxwell equations framework.
result Equality in area-charge inequalities imposes rigid geometric structures, including normal electric and magnetic fields and isometric Riemannian products.
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…
Paper solves degenerated circle pattern metric problem in spherical geometry.
problem Existence and rigidity of (degenerated) circle pattern metrics with prescribed total geodesic curvatures.
method Defined prescribed combinatorial Ricci flows and studied their convergence.
result First degenerated result for total geodesic curvatures in spherical background geometry.
Study the Hessian geometry of an ideal gas in a centrifuge.
problem Understanding the Hessian geometry of an ideal gas in a centrifuge.
method Investigate the Hessian geometry associated with an ideal gas in a spherical centrifuge, using the action of the Euclidean rotation group.
result The Hessian geometry of a spherical rigid body is isometric to a hyperbolic space in the high angular velocity limit.
Paper proves a rigidity result for static perfect fluids.
problem Proving a rigidity result for static perfect fluids.
method Robinson's divergence formula and boundary conditions.
result Rigidity result for static perfect fluids.
Rigidity results are obtained for Riemannian d-manifolds with sec⩾1 and spherical rank at least d−2>0. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini--Study metric. This conjecture is verified in all odd dimensions, for …
Injectivity of geodesic ray transform on specific Finsler manifolds proven.
problem Injectivity of geodesic ray transform on spherically symmetric reversible Finsler manifolds.
method Reduction to invertibility of generalized Abel transforms using angular Fourier series and Taylor expansions of geodesics.
result Injectivity of geodesic ray transform proven on specified Finsler manifolds.
The paper examines rigidity of metric constructions in Wasserstein spaces.
problem Isometric rigidity of metric constructions in Wasserstein spaces.
method Analyzes spaces like Hilbert, rays, half-cylinders, and spherical suspensions.
result Different spaces exhibit varying levels of isometric rigidity in Wasserstein spaces.
Unique steady and expanding solitons with spherical links identified.
problem Characterizing steady and expanding Ricci solitons with specific asymptotic symmetries.
method Symmetry principle applied to asymptotically cylindrical and conical GRSs, proving uniqueness for Bryant solitons.
result Bryant steady and expanding solitons are the unique asymptotically cylindrical and conical GRSs with spherical links under certain conditions.
Proves metric spaces with Euclidean heat kernel are isometric to Euclidean space.
problem Characterizing metric measure spaces with specific heat kernels.
method Analyzes Dirichlet forms and heat kernels to prove rigidity.
result Metric measure spaces with Euclidean heat kernel are isometric to Euclidean space.
A Riemannian manifold M has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of M lie in the interval [−1,−41], and M is closed, we show that M is a locally symmetric space of rank one. This…
It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional at compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.
Survey of rigidity and gap phenomena in sphere-ball submanifolds.
problem Rigidity and gap phenomena in submanifolds of sphere and ball.
method Comparison of techniques, pinching and gap theorems, Morse index and topology.
result Free boundary condition in ball forces stronger rigidity than in sphere.
Embeds Riemannian manifolds with Anosov flows, linking classical and new theorems.
problem Embedding Riemannian manifolds with Anosov flows.
method Isometric embedding into a closed Riemannian manifold with Anosov geodesic flow.
result Direct link between classical and new theorems.