Paper solves Minkowski problem for anisotropic p-torsional rigidity.
problem Solving the Minkowski problem for anisotropic p-torsional rigidity.
method Using the anisotropic p-Laplacian equation, presenting sufficient and necessary conditions for existence. result Presented sufficient and necessary conditions for the existence of a solution.
Paper solves Minkowski problem for q-torsional rigidity using curvature flow.
problem Solving Minkowski problem for q-torsional rigidity.
method Method of curvature flows.
result Existence of smooth non-even solutions.
Study scattering rigidity on stationary manifolds using geodesics.
problem Scattering rigidity on standard stationary manifolds.
method Use Hamiltonian reduction to relate to MP-systems. result New rigidity results for stationary manifolds.
The paper proves rigidity results for Serrin-type problems in manifolds.
problem Proving rigidity for Serrin-type problems in Riemannian manifolds.
method Integral identities and Soap Bubble theorem.
result Rigidity results for annular regions in Einstein manifolds.
New method solves generalized Minkowski problem for torsional rigidity.
problem Generalized Minkowski problem for torsional rigidity.
method Flow method
result Existence of solutions for general measures.
Researchers solve boundary and scattering rigidity problems for magnetic systems.
problem Recovering magnetic systems from boundary or scattering data.
method Reduced to magnetic systems and applied results from [DPSU07].
result Recovering MP-system up to a gauge. This paper solves the dual Minkowski problem for q-torsional rigidity.
problem The dual Minkowski problem for q-torsional rigidity.
method Introduced the p-th dual q-torsional measure and solved the p-th dual Minkowski problem for q-torsional rigidity using a Gauss curvature flow.
result Existence of smooth even and non-even solutions to the p-th dual Minkowski problem for q-torsional rigidity.
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.
Study shows critical width for rigidity of equatorial zones on spheres.
problem Mean curvature rigidity of equatorial zones on spheres.
method Used tangency principle and trap-slice lemma for strong rigidity, and constructed nontrivial perturbations using Delaunay surfaces for non-rigidity.
result Critical width exists for rigidity, beyond which zones are non-rigid.
Proves gap rigidity theorem for Hermitian symmetric spaces.
problem Gap rigidity problems in compact Hermitian symmetric spaces.
method Dual analogy to Mok's noncompact case theorem, theorem on higher dimensional submanifolds.
result Proves gap rigidity theorem for diagonal curves in tube type spaces.
Study calculates Mather β-function for ellipses and applies it to rigidity problems.
problem Calculating Mather β-function for ellipses and its application to rigidity.
method Used non-standard generating function of billiard problem to derive Mather β-function for ellipses. Applied to rigidity problems.
result Explicit formula for Mather β-function for ellipses and its application to rigidity.
Study recovers Lorentzian metrics from boundary data, proving local rigidity.
problem Recovering a Lorentzian metric from scattering data on a boundary.
method Analyzes jet and real analyticity of metrics near lightlike points.
result Metric can be recovered up to gauge transformations near lightlike strictly convex points.
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.
Discussing rigidity in codimension 2, extending rigidity concepts.
problem Local isometric rigidity problem in codimension 2.
method Extending rigidity concepts to include genuine and honest rigidity, studying isometric immersions in semi-Euclidean spaces.
result Necessity of natural singularity in inner product for transitivity.
Several rigidity problems in toric topology are addressed in \cite{ma-su08}. In this paper, we survey results on those problems including recent development.
Paper introduces new Lp q-torsional measure and solves Minkowski problem.
problem Solving the Minkowski problem for q-torsional rigidity. method Established Lp variational formula and proved existence of solutions. result Existence of solutions to Lp Minkowski problem for specific measures. 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 on recovering Lorentzian metrics from scattering data.
problem Recovering Lorentzian metrics from scattering data on a boundary.
method Analyzing the role of boundary distance functions and linearizing the light ray transform.
result Scattering rigidity can be reduced to boundary rigidity of magnetic systems.
Study rigidity of geodesic balls on manifolds with boundary.
problem Rigidity of geodesic balls on manifolds with boundary.
method Combining generalized Reilly formula with Steklov-type boundary value problems to derive integral inequalities.
result Characterizations of geodesic balls in space forms.
Study finds bifurcation and local rigidity points for solutions to the Yamabe problem on Aloff-Wallach Spaces.
problem Yamabe problem on Aloff-Wallach Spaces
method Constructing 1-parameter families of solutions and examining changes in the Morse index as the parameter varies.
result Identifies bifurcation and local rigidity points for homogeneous solutions to the Yamabe problem.
We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary r…
The paper explores gaps in curvature-related metrics and rigidity.
problem Understanding gaps in curvature-related metrics and rigidity.
method Analyzes three types of gaps: spectral, metric-rigidity, and topological-rigidity.
result Proposes open problems in the field.
We prove an Obata-type rigidity result for the spherical cap and apply it for an eigenvalue problem with mixed boundary condition.
Proves rigidity for maps between manifolds using degree theory and current developments.
problem Lipschitz-volume rigidity for maps between metric manifolds and Riemannian manifolds.
method Degree theory and recent developments of Lipschitz-volume rigidity for integral currents.
result Proves a Lipschitz-volume rigidity result for 1-Lipschitz maps.
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.
Paper studies rigidity phenomena for elliptic systems on Riemannian manifolds.
problem Local radial rigidity of elliptic systems on Riemannian manifolds.
method Reduction to singular ordinary differential equations of Euler type.
result Local uniqueness and existence results for solutions with prescribed initial jets.
We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1-rigidity of a submanifold under admissible deformations is introduced. It measures how a deformation deviates from a one parameter family of motions u…
Study on Serrin's problem in convex cones with rigidity results and geometric inequalities.
problem Serrin's overdetermined problem in convex cones of Riemannian manifolds.
method Rigidity results, soap bubble theorem, Heintze-Karcher inequality, drift Laplacian analysis.
result Characterization of intersections of geodesic balls with cones in Riemannian manifolds.
Study rigidity in Riemannian manifolds using Pohozoaev and P-function approaches.
problem Rigidity in Serrin's overdetermined problems in Riemannian manifolds.
method Prove a Pohozoaev-type identity, use conformal vector field, and apply P-function approach.
result Show Serrin's type rigidity result in Riemannian manifolds.
The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.
problem Rigidity of inequalities on weighted Riemannian manifolds.
method Theorems of rigidity on curvature and measure for the Borell-Brascamp-Lieb inequality, generalizing a theorem by Balogh and Kristály.
result A generalization of the curvature rigidity theorem to the weighted setting.
The hemisphere rigidity theorem connects to the Gelfand problem, providing a precise value for the extremal parameter.
problem Finding the extremal parameter for a specific nonlinear equation on a hemisphere.
method Interpreting the hemisphere rigidity theorem within the context of the Gelfand problem and applying it to a fourth-order Gelfand problem.
result A precise value for the extremal parameter is derived for the Gelfand problem under certain conditions.
Compact method proves Brown-York mass positivity and connects to major conjectures.
problem Proving positivity of Brown-York's mass and its connections to conjectures.
method Compact approach to proving mass positivity and exploring connections.
result Proved the positivity of Brown-York's mass and its relation to conjectures.
Extends symmetry and rigidity to surfaces with soap film-like singularities.
problem Symmetry and rigidity of minimal surfaces with singularities.
method Method of moving planes applied to surfaces with Plateau-like singularities.
result Extends classical results to surfaces with singularities.
In this paper we consider the lens rigidity problem with partial data for conformal metrics in the presence of a magnetic field on a compact manifold of dimension ≥3 with boundary. We show that one can uniquely determine the conformal factor and the magnetic field near a strictly convex (with respect to the magne…
Surveying recent progress on hyperbolic 3-manifold rigidity.
problem Profinite rigidity of hyperbolic 3-manifolds.
method Review of profinite completion and rigidity of groups, evidence from other types of 3-manifolds, and existing ideas.
result Positive evidence for profinite rigidity in hyperbolic 3-manifolds.
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.
In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown …
By using certain idea developed in minimal submanifold theory we study rigidity problem for self-shrinkers in the present paper. We prove rigidity results for squared norm of the second fundamental form of self-shrinkers, either under point-wise conditions or under integral conditions.
Paper proves rigidity theorems for geodesically reversible Finsler metrics.
problem Understanding geodesically reversible Finsler metrics in closed manifolds.
method Applied theory of volumes and areas on Finsler spaces to establish rigidity theorems.
result Partial explanation of the scarcity of geodesically reversible Finsler metrics in closed manifolds.
Solves the gauge problem for Ricci flow cylinders, proving strong rigidity.
problem Recognizing metrics in different coordinates and diffeomorphisms.
method Solves a nonlinear system of PDEs to produce a diffeomorphism fixing a gauge.
result Strong rigidity of cylinders in Ricci flow, proving all tangent flows are cylinders.
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.
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.
The paper proposes conjectures about moduli space rigidity.
problem Rigidity of moduli spaces in algebraic geometry.
method Group-theoretic, topological, and holomorphic approaches.
result Some conjectures have been proved.
The paper explores rigidity of hypersurfaces with constant curvature in Euclidean spaces.
problem Rigidity of hypersurfaces with constant mean and scalar curvature.
method Characterizations and rigidity results under various conditions of Gaussian-Kronecker and r-th mean curvatures. result Rigidity theorems for hypersurfaces in dimensions 4, 5, and 6, and general dimensions under pinching conditions.
Study shows curvature rigidity of specific metric types.
problem Curvature rigidity of specific metric types.
method Spin geometry based arguments.
result Scalar curvature rigidity of specific metric types.
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 ∂M. We present some geometrical hypotheses ensuring local rigidity for both, the general Riemannian and the warped metric case…
We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.
New method proves length spectrum rigidity in various geometric settings.
problem Length spectrum rigidity in geometric settings.
method Combination of dynamical systems and geometric group theory.
result Provides concise proofs and extends classical results.