The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
This paper proves a curvature entropy inequality for non-symmetric convex bodies.
problem Proving a curvature entropy inequality for non-symmetric convex bodies.
method Demonstrated the log-Minkowski inequality of curvature entropy for general convex bodies in 2D.
result Equivalence of cone-volume measure uniqueness, log-Minkowski volume inequality, and curvature entropy inequality for general convex bodies in 2D.
New flow method solves Christoffel-Minkowski problem.
problem Solving Christoffel-Minkowski problem.
method Entropy preserving curvature flow with global term.
result Entropy preserving flow solves Christoffel-Minkowski problem.
Study on curvature flow in Minkowski space for cocompact hypersurfaces.
problem Investigating curvature flow in Minkowski space for cocompact hypersurfaces.
method Investigation of the cocompact inverse \(σ_k\) curvature flow in Minkowski space.
result Longtime existence and convergence of the curvature flow established.
Unified estimates for mean curvature in Lorentz-Minkowski space.
problem Estimating mean curvature for space-like and time-like graphs.
method Using gradient bounds to derive Heinz-type estimates.
result Unified vanishing theorem for mean curvature of constant mean curvature graphs.
New proof of Minkowski identities for hypersurfaces in curved spaces.
problem Proving Minkowski identities for hypersurfaces in constant curvature manifolds.
method Using a differential system and position vector field.
result New proof of Minkowski identities for hypersurfaces.
Study classifies non-removable singularities in Minkowski 3-space equations.
problem Classifying non-removable singularities in Minkowski 3-space equations.
method Classification through real analytic solutions of the prescribed mean curvature equation.
result Classification of non-removable isolated singularities.
Study smooth hypersurfaces with prescribed curvature in Minkowski space.
problem Existence of smooth spacelike hypersurfaces with prescribed curvature.
method Proving existence based on C2 estimates. result Existence of smooth spacelike hypersurfaces with prescribed curvature.
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.
New method solves a generalized Minkowski problem using a curvature flow.
problem Generalized Minkowski problem for smooth measures.
method Flow involving Gauss curvature and support function.
result Existence of solutions for the dual Orlicz-Minkowski problem.
Paper solves Minkowski problem for p-harmonic measures.
problem Solving the Minkowski problem for p-harmonic measures on convex domains.
method Using the Gauss curvature flow method.
result Existence of smooth solution to the Minkowski problem for p-harmonic measures.
Study Minkowski formula for conformal Killing-Yano 2-forms in constant curvature spacetimes.
problem Derive Minkowski formula for conformal Killing-Yano 2-forms.
method Analyze spacetime Alexandrov theorem with a free boundary.
result Established spacetime Alexandrov theorem with a free boundary.
The paper studies surfaces in Minkowski space with constant curvature.
problem Properly embedded spacelike surfaces in Minkowski 3-space with constant Gaussian curvature.
method Classification of surfaces with bounded prescribed Gaussian curvature, proving uniqueness of foliations.
result Every regular domain not a wedge is uniquely foliated by properly embedded convex surfaces of constant Gaussian curvature.
Study translating solitons in Minkowski space with prescribed Gauss image.
problem Second boundary value problem for mean curvature flow in Minkowski space.
method Construct translating solitons with prescribed Gauss image.
result Constructing translating solitons with prescribed Gauss image in Minkowski space.
Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.
problem Solving dual Orlicz Minkowski problems for anisotropic flows.
method Anisotropic inverse Gauss curvature flows and stationary solutions.
result New existence results for dual Orlicz Minkowski problems for smooth measures.
Paper solves curvature equations in Minkowski space for non-convex domains.
problem Solving curvature equations in non-convex domains of Minkowski space.
method Existence theorem proved via \emph{a priori} estimates and Serrin-type condition.
result Existence of solutions for curvature equations in non-convex domains.
Study of curvature flow in Minkowski space converging to a hyperboloid.
problem Understanding curvature flows in Minkowski space.
method Analyzing fully nonlinear curvature flows of noncompact spacelike hypersurfaces in Minkowski space.
result The flow converges to the future timelike hyperboloid after rescaling.
Study anisotropic flows solving Orlicz-Minkowski problems, proving existence and new results.
problem Anisotropic non-homogeneous Gauss curvature flows and Orlicz-Minkowski problems.
method Long-time existence and behavior analysis, parabolic approximation method, curvature flow.
result Existence and new results for Orlicz-Minkowski problems, including Lp versions. Solves Lp-Gaussian chord Minkowski problem using Gauss curvature flow.
problem Solving the Lp-Gaussian chord Minkowski problem. method Using Gauss curvature flow to obtain smooth even solutions.
result Obtains smooth even solutions to the Lp-Gaussian chord Minkowski problem. Sharp Minkowski inequality for convex surfaces in curved spaces.
problem Establishing a precise lower bound for total mean curvature of convex surfaces.
method Harmonic mean curvature flow applied to Cartan-Hadamard manifolds.
result Improved Minkowski inequality for convex surfaces in nonpositively curved 3-spaces.
Researchers solved the even Lp-Minkowski problem under curvature pinching.
problem Solving the even Lp-Minkowski problem under curvature pinching. method Anisotropic Riemannian metric comparison and anisotropic curvature analysis.
result The even Lp-Minkowski inequality and uniqueness are proven for all p≥pγ. Unique CMC foliation in Minkowski space solved.
problem Classifying entire spacelike CMC hypersurfaces in Minkowski space.
method Proved uniqueness of spacelike CMC foliation for regular domains.
result Uniquely foliated by spacelike CMC hypersurfaces for any regular domain.
The paper proves new Minkowski inequalities for flows in warped spaces.
problem Proving new Minkowski inequalities for flows in warped spaces.
method Locally constrained inverse curvature flows in Riemannian warped spaces.
result New Minkowski inequalities are derived for flows in warped spaces.
Paper studies mean curvature flow in Minkowski spaces, proving existence and uniqueness.
problem Existence and uniqueness of mean curvature flow in Minkowski spaces.
method Introduced mean curvature flow on Finsler manifolds, proved existence and uniqueness for Minkowski spaces.
result Proved existence and uniqueness of mean curvature flow in Minkowski spaces.
The study examines spacelike hypersurfaces in Minkowski space with constant σn−1 curvature.
problem Characterizing spacelike hypersurfaces with constant σn−1 curvature in Minkowski space. method Analyzing hypersurfaces with bounded principal curvatures and proving properties of their convexity.
result Hypersurfaces with constant σn−1 curvature in Minkowski space are either convex or can be split into a product form. Unified Minkowski problem discussed for (p,q)-mixed quermassintegrals.
problem Unified Minkowski problem for (p,q)-mixed quermassintegrals.
method Introducing (p,q)-mixed quermassintegrals and (p,q)-dual mixed curvature measure to study the Minkowski problem.
result Derivation of important properties and geometric inequalities for (p,q)-mixed quermassintegrals.
Investigates differences in solving mean curvature problems in Euclidean and Lorentz-Minkowski spaces.
problem Solving the Dirichlet problem for mean curvature equations in different spacetimes.
method Compares techniques and conditions for solvability in Euclidean and Lorentz-Minkowski spaces.
result Lorentz-Minkowski spacelike condition allows dropping mean convexity hypothesis.
The paper calculates curvature limits and proves Gauss-Bonnet theorems in affine and Minkowski groups.
problem Computing curvature limits in affine and Minkowski groups.
method Analyzing Euclidean C2-smooth surfaces and curves in affine and Minkowski groups. result Gauss-Bonnet theorems in affine and Minkowski groups are proven.
Paper solves Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.
problem Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.
method Proved existence of solutions using a new full rank theorem.
result Existence of smooth, origin-symmetric, strictly horospherically convex solutions.
Study proves only origin-centered spheres solve certain curvature problems.
problem Proving uniqueness of solutions to curvature problems.
method Using the Heintze-Karcher inequality, the study proves the uniqueness of smooth, strictly convex solutions to a class of Minkowski type problems.
result Only origin-centered spheres solve isotropic and Lp-Gaussian-Minkowski problems. Geometric description of Riemann zero mean curvature surfaces in Lorentz-Minkowski space.
problem Characterizing surfaces with zero mean curvature in Lorentz-Minkowski space.
method Geometric description of surfaces in spacelike and timelike planes.
result New features of zero mean curvature surfaces in Lorentz-Minkowski space.
New general volume concept solves Minkowski problem for star bodies.
problem Minkowski problem for star bodies
method General volume concept, new curvature measure, variational formulas, Minkowski-type inequality
result Solution to the Minkowski problem for the new general dual Orlicz curvature measure
Sharp curvature estimates lead to optimal C1,1 regularity for Lp Minkowski problems.
problem Optimal regularity for solutions to Lp Minkowski problems. method Anisotropic Gauss curvature flows and curvature estimates.
result Sharp C1,1 regularity for solutions to Lp Minkowski problems. Study of a flow related to the Orlicz-Minkowski problem for convex hypersurfaces.
problem Orlicz-Minkowski problem involving Gauss curvature and support function.
method Generalized Gauss curvature flow for convex hypersurfaces in Euclidean n-space.
result Long-time existence and convergence of the flow, leading to existence results for the Orlicz-Minkowski problem.
Study bifurcations of curves on surfaces in Minkowski 3-space.
problem Understanding the behavior of curves on surfaces in Minkowski 3-space.
method Analyzing the degeneracy of induced pseudo metric, discriminant of principal curvatures, parabolic curve, and mean curvature vanishing points.
result Bifurcations of robust features on surfaces in Minkowski 3-space.
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.
Paper solves a generalized chord Minkowski problem using Gauss curvature flows.
problem Generalized chord Minkowski problem using Orlicz functions.
method Gauss curvature flows to achieve existence of smooth solutions.
result Existence of smooth solutions to the orlicz chord Minkowski problem.
Study of σk flow in Minkowski space for noncompact hypersurfaces.
problem Understanding the behavior of σk curvature flow in Minkowski space. method Proving flow existence and convergence to self-expander after rescaling.
result The flow converges to a self-expander after rescaling.
Study solves a generalized Christoffel-Minkowski problem using curvature flow.
problem Generalization of the Lp-Christoffel-Minkowski problem. method Anisotropic curvature flow to derive long-time existence and smooth solutions.
result Existence of smooth solutions for c=1 under certain initial data. The study uses isothermic coordinates to analyze space-like surfaces with constant curvature.
problem Global properties of space-like surfaces with constant mean curvature in Lorentz-Minkowski space.
method Isothermic coordinate systems
result Global properties of space-like surfaces with constant mean curvature explored.
The study identifies cylinders in certain submanifolds of Minkowski space.
problem Characterizing submanifolds with specific geometric properties in Minkowski space.
method Analyzing Finsler submanifolds with nonnegative Ricci curvature in Minkowski space, proving cylinder properties under certain conditions.
result Submanifolds containing a line or with positive relative nullity index are cylinders under specific conditions.
Timelike curvature and Brunn-Minkowski inequality linked in non-smooth spacetimes.
problem Equivalence between timelike Ricci curvature and Brunn-Minkowski inequality in synthetic Lorentzian spaces.
method Introducing strong q-timelike Brunn-Minkowski condition and proving equivalence to curvature conditions. result Timelike curvature dimension condition equivalent to timelike Brunn-Minkowski inequality in specific settings.
New optimal surfaces found in Heisenberg group defy Euclidean sphere optimality.
problem Optimizing mean curvature in Heisenberg group sub-Riemannian setting.
method Developed variational theory, established first and second variation formulas, introduced new critical surfaces.
result Identified and characterized a new family of rotationally invariant critical surfaces, the Pansu-Minkowski spheres.
Paper proves a Minkowski-like inequality in de Sitter space.
problem Proving a Minkowski inequality in Lorentzian de Sitter space.
method Locally constrained inverse mean curvature flow.
result Properly defined Minkowski inequality in de Sitter space.
New examples of mixed-type zero-curvature graphs found.
problem Finding new examples of zero-curvature graphs in Lorentz-Minkowski space.
method Using Konderak's representation formula to construct entire zero-curvature graphs over specific planes.
result Existence of new types of entire zero-curvature graphs in mixed-type in Lorentz-Minkowski space.
The study generalizes Minkowski inequalities for curves on surfaces.
problem Generalizing Minkowski inequalities for curves on Riemannian surfaces.
method Introducing a generalized Minkowski average using parametrized curves and proving the inequality for constant-speed curves.
result A family of constant-speed curves on a Riemannian surface satisfies the Brunn-Minkowski inequality with respect to the Riemannian area form if and only if the geodesic curvature is determined by a function κ satisfying a specific inequality.
Solves capillary curvature problems for specific p values.
problem Capillary curvature problems for −n<p<1 and θ∈(0,2π). method Iterative scheme based on capillary Minkowski problem and capillary curvature image operators.
result Fixed points of capillary curvature image operators correspond to solutions of capillary Lp-Minkowski problem. Discover new identities linking hypersurface mean curvatures.
problem Understanding mean curvatures of hypersurfaces in Riemannian manifolds.
method Developed three most general Minkowski or Hsiung-Minkowski identities.
result Classical Minkowski identity is natural to all Riemannian manifolds.