This study finds the shape of centrally symmetric octahedra with specific angles.
problem Finding the shape of centrally symmetric octahedra with prescribed cone-deficits.
method Inspired by Thurston's work, the paper studies the set of shapes of centrally symmetric octahedra with prescribed cone-deficits, showing it forms a real hyperbolic ideal tetrahedron.
result The set of shapes of centrally symmetric octahedra with prescribed cone-deficits forms a real hyperbolic ideal tetrahedron with dihedral angles half of the prescribed cone-deficits.
Polyhedral surfaces can be broken down into parallelograms.
problem Decomposing polyhedral surfaces into simpler shapes.
method Analyzing moduli spaces and using geometric properties.
result Polyhedral surfaces with 8 vertices can be decomposed into at most 20 parallelograms.
Study on the convergence rate of prescribed scalar curvature flow.
problem Prescribing scalar curvature on manifolds.
method Inspired by Yamabe flow convergence rate study, analyze the prescribed scalar curvature flow convergence rate.
result Determine the convergence rate of the prescribed scalar curvature flow.
Proves optimal regularity for sphere minimizers in 3-sphere.
problem Finding optimal regularity for sphere minimizers.
method Proves C1,1 regularity for minimizers of prescribed mean curvature over isotopy classes. result Proves optimal C1,1 regularity for minimizers. Develops variational approach for Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
problem Existence and characterization of Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
method Variational approach, algebraic approximation of singularities, function α_ω, continuity method.
result Many K-stable manifolds admit all possible Kähler-Einstein metrics with prescribed singularities.
Study classifies rotational hypersurfaces with prescribed mean curvature.
problem Classifying rotational hypersurfaces with prescribed mean curvature.
method Phase space analysis to classify hypersurfaces.
result Delacunay-type classification for even prescribed functions.
Proves existence and uniqueness of Killing graphs with prescribed curvature.
problem Existence and uniqueness of Killing graphs with prescribed curvature.
method Proves existence and uniqueness of Killing graphs with prescribed mean curvature considering non-constant functions.
result Existence and uniqueness of Killing graphs with prescribed curvature.
Paper estimates curvature of convex hypersurfaces with prescribed curvature.
problem Estimating curvature of p-convex hypersurfaces with prescribed curvature. method Establishes curvature estimates for p-convex hypersurfaces in Rn+1 with p≥2n. result Proves existence of star-shaped hypersurface of prescribed curvature and interior C2 estimates. Theory proves existence of hypersurfaces with prescribed curvature.
problem Existence of hypersurfaces with prescribed mean curvature in noncompact manifolds.
method Developed min-max theory for noncompact manifolds.
result Proved existence of closed and finite area hypersurfaces.
Introduce a flow for prescribed Hermitian-Yang-Mills tensors
problem Solve the prescribed Hermitian-Yang-Mills tensor equation
method Long-time convergence of the flow
result Uniform C0-estimate of the flow We develop a compactness result near the boundary for families of locally convex immersions. We also develop a mod 2 degree theory for immersion of constant (and prescribed) Gaussian curvature with prescribed boundary. These are then used to solve the Plateau problem for immersions of constant (and prescribed) Gaussian…
Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…
Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.
problem Prescribed curvature problem for convex capillary hypersurfaces.
method Reformulated as Hessian quotient equation with Robin boundary condition.
result Existence of strictly convex capillary hypersurface with prescribed curvature.
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.
Paper solves Dirichlet problem for p-convex hypersurfaces with curvature constraints.
problem Solving the Dirichlet problem for p-convex hypersurfaces with prescribed curvature. method Proved existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition, obtained an interior curvature estimate.
result Existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition.
Study on prescribing Ricci curvature on compact Lie groups.
problem Prescribing Ricci curvature in naturally reductive metrics on compact Lie groups.
method Derive necessary and sufficient conditions for solvability.
result Provide a series of examples.
Study on t-graphs with prescribed mean curvature in Heisenberg groups.
problem Existence and uniqueness of t-graphs with prescribed mean curvature. method Characterization of classical solutions without Dirichlet boundary data, conditions for uniqueness, approximation technique for non-constant mean curvature.
result Conditions for existence and uniqueness of t-graphs in Heisenberg groups. The paper studies prescribed angle surfaces in Riemannian manifolds with torse-forming vector fields.
problem Characterizing surfaces with prescribed angles in Riemannian geometry.
method Introducing and analyzing prescribed angle hypersurfaces associated with torse-forming vector fields.
result Classification of prescribed angle surfaces in 3D Riemannian manifolds.
We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.
The paper constructs surfaces with prescribed mean curvature in a specific space.
problem Finding surfaces with a given mean curvature in a particular geometric space.
method Phase plane analysis to construct entire rotational graphs and catenoid-type surfaces.
result Classification result for surfaces with linearly prescribed mean curvature.
Study of prescribed mean curvature flow on noncompact hypersurfaces in Lorentz manifolds.
problem Short time existence and long time existence of prescribed mean curvature flow on noncompact spacelike hypersurfaces.
method Finding sufficient conditions for short time existence and discussing long time existence and convergence.
result Sufficient conditions for short time existence of prescribed mean curvature flow on noncompact spacelike hypersurfaces.
The paper solves a curvature problem on a ball's surface near constant values.
problem Prescribing almost constant curvatures on a manifold with boundary.
method Perturbative approach and ansatz by Han and Li.
result New existence results for conformal metrics when curvatures are near constants.
In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. To such a solution, we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data an…
Proves existence of graph on torus with prescribed curvature.
problem Existence of graphs with prescribed mean curvature on torus.
method Proves existence using graph theory and curvature conditions.
result Existence of a graph on the n-dimensional torus with prescribed curvature.
In this work, we study the Yamabe flow corresponding to the prescribed scalar curvature problem on compact Riemannian manifolds with negative scalar curvature. The long time existence and convergence of the flow are proved under appropriate conditions on the prescribed scalar curvature function.
Study local curvature estimates and existence of conformal metrics on noncompact manifolds.
problem Deriving local C0-estimates and existence of conformal metrics with prescribed curvature. method Utilizing Aviles-McOwen's result and its nonlinear extension, combined with asymptotic conditions.
result Proved existence of complete conformal metrics with prescribed curvature functions.
The paper defines and proves the existence of curves in Riemannian manifolds with prescribed angles to torse-forming vector fields.
problem Existence of curves with prescribed angles to torse-forming vector fields in Riemannian manifolds.
method Introducing the notion of a prescribed angle curve and proving its existence for torse-forming vector fields.
result Existence of prescribed angle curves in Riemannian manifolds associated with torse-forming vector fields.
We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions. Our duality extends the Calabi correspondence between minimal graphs in the Euclid…
Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. In surprise, that methods has also brought us to promote Ivochkina's C2 estimates for prescribed curvature equation in \cite{I1, I}.
Classification of surfaces with prescribed mean curvature in Heisenberg space and SL2(R).
problem Surfaces with prescribed mean curvature in Heisenberg space and SL2(R).
method Classification result for rotational surfaces with prescribed mean curvature.
result Existence of embedded tori as counterexamples to the Alexandrov problem.
Study finds loops with specific curvature exist using Hardy's inequality.
problem Existence of closed planar loops with prescribed curvature.
method Variational approach, Hardy's inequality and associated functional space.
result Existence of loops with specific curvature proven.
We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…
Study solves complex Hessian equations with prescribed singularities on compact Kähler manifolds.
problem Solving complex Hessian equations with specific singularity types on compact Kähler manifolds.
method Analyzes the total mass of complex Hessian measures and solves equations with prescribed singularities.
result Proves non-decreasing total mass of complex Hessian measures and solves complex Hessian equations.
Study of metrics with prescribed curvature and geodesic curvature on a disc.
problem Existence and behavior of conformal metrics with prescribed curvature and boundary geodesic curvature.
method Variational characterization and gradient flow approach.
result Existence of solutions or blow-up to a spherical cap, leading to existence results via shadow flow.
The paper solves curvature measure problem in hyperbolic space.
problem Prescribed curvature measure problem in hyperbolic space.
method Establishing C^2 regularity estimates for solutions to fully nonlinear PDE.
result Existence of star-shaped k-convex bodies with prescribed curvature measures.
Proves Chen-Lin conjecture for sphere scalar curvature problem.
problem Existence of solutions for prescribed scalar curvature on spheres.
method Criterion proving existence based on Chen-Lin conjecture.
result Proves conjecture for standard n-dimensional sphere.
Study eigenvalues for special curvature equations on star-shaped surfaces.
problem Eigenvalue problem for prescribed curvature equations on star-shaped, k-convex hypersurfaces.
method Established existence of a unique eigenvalue and hypersurface through uniform estimates in p for Lp-type equations.
result Existence of a unique eigenvalue and associated hypersurface under certain conditions.
Study prescribed scalar curvature on orbifolds with isolated singularities.
problem Prescribed scalar curvature problem on orbifolds with singularities.
method Compactness theorem in dimension 4, existence theorem in dimensions n ≥ 4, use of Leray-Schauder degree.
result Existence of solutions in dimensions n ≥ 4, compactness in dimension 4.
We give a new existence proof for closed hypersurfaces of prescribed mean curvature in Lorentzian manifolds.
Proves existence of metrics with prescribed Q-curvature in Riemannian manifolds.
problem Existence of metrics with prescribed Q-curvature under natural assumptions.
method Proves existence of metrics with prescribed Q-curvature in Riemannian manifolds.
result Existence of metrics with prescribed Q-curvature in Riemannian manifolds.
In this paper we find strictly locally convex hypersurfaces in Rn+1 with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an applicatio…
We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a …
We study the prescribed mean curvature equation with a prescribed boundary contact angle condition in M×R where Mn is a Riemannian submanifold in Rn+1. The main purpose is to establish a priori gradient estimates for solutions, from which the long time existence of the solution are derived.
Constructs hyperspheres with prescribed mean curvature in Euclidean space.
problem Creating hyperspheres with a specific curvature in Euclidean space.
method Constructs families of smooth functions to fill Euclidean space with hyperspheres of prescribed mean curvature.
result Euclidean space can be filled with hyperspheres of prescribed mean curvature.
The existence of closed hypersurfaces of prescribed curvature in globally hyperbolic Lorentzian manifolds is proved provided there are barriers.
Paper explores prescribing Chern scalar curvatures on noncompact Hermitian manifolds.
problem Prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds.
method Generalizes Aviles-McOwen's existence results to higher-dimensional Hermitian manifolds.
result Existence results for Chern scalar curvatures on Hermitian manifolds.
It is proved the existence and uniqueness of Killing graphs with prescribed mean curvature in a large class of Riemannian manifolds.
We give a proof of the Kazdan-Warner conjecture concerning the prescribed scalar curvature problem in the null case.