Paper estimates diameter for Minkowski problem solutions.
problem Estimating diameter of solutions to Minkowski problem.
method Uniform diameter estimate for Lp dual Minkowski problem. result Uniform diameter estimate for planar Lp dual Minkowski problem. 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.
Solves capillary Lp-Christoffel-Minkowski problem in half-space.
problem Capillary surfaces in half-space geometry.
method Non-collapsing estimate for height and capillary support function.
result Extends Christoffel-Minkowski existence result.
In this article, we estimate the quasi-local energy with reference to the Minkowski spacetime [16,17], the anti-de Sitter spacetime [4], or the Schwarzschild spacetime [3]. In each case, the reference spacetime admits a conformal Killing-Yano 2-form which facilitates the application of the Minkowski formula in [15] to …
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.
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. The paper examines stability of Minkowski inequalities in warped product spaces.
problem Stability of Minkowski-type inequalities for hypersurfaces in warped product spaces.
method Established a stability estimate for the norm of the traceless second fundamental form.
result Proved stability of Minkowski inequalities in specific warped product examples.
New proof of Minkowski spacetime stability in exterior regions.
problem Stability of Minkowski spacetime in exterior regions.
method Unified treatment of decay of initial data, use of rp-weighted estimates. result Reduced number of derivatives and simplified last slice treatment.
Extends global stability of Minkowski spacetime to minimal decay assumptions.
problem Global stability of Minkowski spacetime under minimal decay assumptions.
method Uses rp-weighted estimates instead of vectorfield method. result Proves global stability of Minkowski spacetime under minimal decay assumptions.
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
Geodesic completeness and optimal Sobolev index proven for Minkowski spacetimes.
problem Geodesic completeness and optimal Sobolev index for Minkowski spacetimes.
method Null non-trapping condition and real principal type estimate.
result Optimal Sobolev index proven for asymptotically Minkowski spacetimes.
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.
The paper establishes Pogorelov type estimates for solutions to Hessian quotient equations in hyperbolic space.
problem Estimating solutions to Hessian quotient equations in Lorentz-Minkowski space.
method Using a priori estimates, the paper establishes Pogorelov type estimates for k-convex solutions.
result Pogorelov type estimates for k-convex solutions to Hessian quotient equations in hyperbolic space.
The paper estimates curvature for specific hypersurfaces in a special space.
problem Estimating curvature for spacelike graphic hypersurfaces in Lorentz-Minkowski space.
method Analyzing the geometry of spacelike admissible graphic hypersurfaces.
result Existence of specific hypersurfaces with prescribed curvature and boundary conditions.
New proof of injectivity for broken non-abelian X-ray transform in Minkowski space.
problem Injectivity of broken non-abelian X-ray transform in Minkowski space.
method Stability estimate considering gauge, light-sink connections, Bayesian inversion.
result Consistent recovery of light-sink connections from noisy data.
We extend Eardley and Moncrief's L∞ estimates for the conformally invariant Yang-Mills-Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localise their estimates, and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates to…
Proves stability of Minkowski space-time for Einstein-Yang-Mills equations.
problem Stability of Minkowski space-time for perturbations governed by Einstein-Yang-Mills equations.
method Proves exterior energy estimates for tensorial non-linear wave equations in Minkowski space-time.
result Proves exterior stability of Minkowski space-time for Einstein-Yang-Mills equations.
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.
Proves estimate similar to De Lellis-Müller on Minkowski lightcone.
problem Estimating spacelike cross sections of the Minkowski lightcone.
method Geometric scaling invariant estimate, using singularity models and almost-Schur lemma.
result Spacelike cross sections are W2,2-close to a round surface. We prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space. The proof is based on barrier constructions and local a priori estimates.
Paper proves uniqueness of solutions to a geometric inequality problem.
problem Uniqueness of solutions to the isotropic Lp Minkowski problem. method Analysis of the Hilbert-Brunn-Minkowski operator LK to derive stability estimates. result Uniqueness of S2-isotropic solutions to the isotropic Lp Minkowski problem in Rn for specific ranges of p. 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.
New Sobolev inequality found for mean convex spacelike submanifolds in Minkowski space.
problem Finding a Sobolev inequality for mean convex spacelike submanifolds in Minkowski space.
method Applying the ABP estimate method to spacelike submanifolds in Rn,1. result Obtained a Sobolev inequality without a mean curvature term for mean convex hypersurfaces.
Paper proves smoothness of solutions to a complex geometric problem.
problem Smoothness of solutions to the degenerate Lp Dual Minkowski problem. method Inspired by Guan and Li's approach for the Aleksandrov problem, the authors derive C1,1 estimates. result Proves solutions are C1,1 regular. Paper solves a geometric problem involving mixtures of area and curvature measures.
problem Investigates a geometric problem involving mixtures of area and curvature measures.
method Establishes a gradient estimate to prove the existence of a solution.
result Proves the existence of an even, smooth, strictly convex solution for 1<p<q≤k+1. Paper solves dual Minkowski problem in 2D plane for specific curvature cases.
problem Finding the number of solutions to the dual Minkowski problem in 2D with constant curvature.
method Combining theoretical analysis and numerical estimation of an integral with parameters.
result Found the number of solutions for the constant dual curvature case when 0<q≤4. Study complete space-like stationary surfaces with graphical Gauss image, estimating exceptional values and classifying degenerate surfaces.
problem Estimating exceptional values and classifying degenerate surfaces in Minkowski spacetime.
method Generalizing Fujimoto's theorem, estimating upper bounds, introducing conjugate similarity, and establishing structure theorems.
result Sharp contrast to Bernstein type results for minimal surfaces, estimating upper bounds of exceptional values.
The present note is a result of an on-going investigation into the logarithmic Brunn-Minkowski inequality. We obtain lower estimates on the volume product for convex bodies in Rn not necessarily symmetric with respect to the origin from a modified logarithmic Brunn-Minkowski inequality.
Unified study of Brunn-Minkowski conjectures for log-concave measures.
problem Understanding the role of symmetry in inequalities of Brunn-Minkowski type.
method Unified framework, new results for conjectures, improved estimates for Lebesgue and Gaussian measures.
result Unified framework and new results for Brunn-Minkowski conjectures.
We consider a non-trapping n-dimensional Lorentzian manifold endowed with an end structure modeled on the radial compactification of Minkowski space. We find a full asymptotic expansion for tempered forward solutions of the wave equation in all asymptotic regimes. The rates of decay seen in the asymptotic expansion a…
Minkowski space is shown to be globally stable as a solution to the Einstein--Vlasov system in the case when all particles have zero mass. The proof proceeds by showing that the matter must be supported in the "wave zone", and then proving a small data semi-global existence result for the characteristic initial value p…
We prove a generalization of Hsiung-Minkowski formulas for closed submanifolds in semi-Riemannian manifolds with constant curvature. As a corollary, we obtain volume and area upper bounds for k-convex hypersurfaces in terms of a weighted total k-th mean curvature of the hypersurface. We also obtain some Alexandrov-type…
Perturbs area-minimizing hypersurfaces to reduce singular set's dimension.
problem Reduces the dimension of the singular set of area-minimizing hypersurfaces.
method Perturbs a smooth hypersurface to minimize the Minkowski dimension of the singular set.
result The singular set of the perturbed minimizing current has Minkowski dimension less than n-9.
Sharp inequality proved in 3D hyperbolic spaces using flow methods.
problem Proving a sharp Minkowski-type inequality in Cartan-Hadamard 3-spaces.
method Using harmonic mean curvature flow to prove the inequality.
result Sharpened estimates for total mean curvature in hyperbolic 3-space.
Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
Stability of Minkowski space-time in Einstein-Yang-Mills system proven.
problem Stability of Minkowski space-time in Einstein-Yang-Mills system.
method Null frame decomposition, wave coordinates, dispersive estimates.
result Solutions converge to zero Yang-Mills curvature and Minkowski space-time.
We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of…
We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacel…
Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.
problem Deriving upper bounds for the Alexandrov-Fenchel deficit.
method Using weighted Minkowski integral formulas and an integral formula for the deficit in Jensen's inequality.
result Quantitative estimates under weaker convexity assumptions, including a distance term.
In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional Lorentz-Minkowski space L3 and get the uniqueness of cr…
Böröczky, Lutwak, Yang and Zhang recently proved the log-Brunn-Minkowski inequality which is stronger than the classical Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper establishes the log-Brunn-Minkowski, log-Minkowski, Lp-Minkowski and Lp-Brunn-Minkowski inequalities f…
The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the q-th dual curvature measure of an origin-symmetric convex body in Rn. A full solution to this is given when 1<q<n. The necessary and suffic…
Solves Christoffel-Minkowski problem for axially symmetric bodies.
problem Necessary and sufficient conditions for mixed area measures of axially symmetric convex bodies.
method Introduced a new method to transform mixed area measures and mixed volumes of axially symmetric bodies, refining Firey's classification and improving estimates.
result Complete solution to the mixed Christoffel-Minkowski problem for axially symmetric bodies without regularity assumptions.
The paper applies a capillary John ellipsoid theorem to solve capillary curvature problems.
problem Solving capillary curvature problems in Euclidean half-spaces.
method Applying a capillary John ellipsoid theorem to derive non-collapsing estimates and gradient estimates.
result Established existence of solutions to capillary curvature problems in certain ranges of p and q. 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.
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. Paper solves a new Minkowski problem for a specific type of rigidity.
problem Solving a new Minkowski problem for a specific type of rigidity.
method Developed a nonlinear partial differential equation and used a curvature flow method.
result Existence of smooth non-even solutions to the p-th dual Minkowski problem for p < n-2.
Researchers prove uniqueness and continuity of solution to L_p dual Minkowski problem.
problem Proving uniqueness and continuity of solution to L_p dual Minkowski problem.
method Established new Minkowski-type inequalities related to optimization problem.
result Uniqueness and continuity of solution for general convex bodies when q<p.