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.
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. 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.
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.
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.
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.
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.
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.
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. 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.
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.
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 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. 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.
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 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.
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 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.
We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following earlier results, we o…
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…
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 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…
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.
Paper investigates prescribing Chern scalar curvatures on specific manifolds.
problem Prescribing Chern scalar curvatures on noncompact Hermitian manifolds with nonpositive curvatures.
method Establishes existence results and sufficient conditions for negative curvature metrics.
result Obtains sufficient conditions for the existence of a constant negative Chern scalar curvature metric.
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.
Smooth curves with specific curvature can be closely approximated.
problem Approximating smooth curves with prescribed curvature.
method Application of h-principle to C1-dense approximation of curves. result Existence of C∞ knots with prescribed curvature. Solves curvature problems on manifolds with negative curvature.
problem Prescribed curvature problems on closed manifolds with negative curvature.
method Investigates fully nonlinear prescribed curvature problems for modified Schouten tensor on closed Riemannian manifolds with negative curvature.
result Proves solvability of curvature problems under certain conditions.
In this paper, we consider a class of prescribed Weingarten curvature equations. Under some sufficient condition, we obtain an existence result by the standard degree theory based on the a prior estimates for the solutions to the prescribed Weingarten curvature equations.
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.
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}.
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.
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.
We use a phase space analysis to give some classification results for rotational hypersurfaces in Rn+1 whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in Sn, we show that a Delaunay-type classification hold…
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…
Study provides obstructions for Q-curvature on complete metrics in n-space.
problem Obstructing the Q-curvature prescription for complete conformal metrics.
method Analysis of decay rates and application of Bonnet-Mayer theorem.
result Found obstructions related to decay rates and Q-curvature properties.
Mean curvature flow converges to a translating soliton with prescribed contact angle.
problem Mean curvature flow with contact angle constraints in non-Euclidean settings.
method Existence proof using translating solitons and bounds on convexity and Ricci curvature.
result Graphical solutions converge to a translating soliton as time goes to infinity.
Proves existence and uniqueness of metrics with negative curvature and singularities on compact surfaces.
problem Existence and uniqueness of conformal metrics with negative curvature and singularities.
method Proves existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
result Existence and uniqueness of conformal metrics with negative curvature and singularities on compact surfaces.
In this paper, we consider the problem of prescribing scalar curvature on n-sphere. Assume that the candidate curvature function f, which is allowed to change sign, satisfies some kind of Morse index or symmetry condition. By studying the well-known scalar curvature flow, we are able to prove that the flow converges …
Study rotational surfaces with prescribed Gauss curvature in 3D space.
problem Classify and analyze rotational surfaces with prescribed Gauss curvature.
method Phase plane analysis and mild assumptions on the prescribed function.
result Existence of singular radial solutions intersecting orthogonally the axis of rotation.
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.
Study proves existence and nonexistence for annular surfaces with specific curvature and boundary.
problem Existence and nonexistence of annular surfaces with prescribed mean curvature.
method Proves existence and nonexistence results for normal graphs of unduloids or nodoids.
result Existence and nonexistence results for annular type surfaces with prescribed mean curvature.
The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.
problem Prescribing discrete Gaussian curvature on polyhedral surfaces.
method Discrete conformal theory and variational principles with constraints.
result Proves Kazdan-Warner type theorems for polyhedral surfaces.
Let M be a quasi-Fuchsian three-manifold that contains a closed incompressible surface with principal curvatures within the range of the unit interval, for a prescribed function H (with mild conditions) on M, we construct a closed incompressible surface with mean curvature H . A direct application is the existe…
Compact capillary hypersurfaces with specific curvatures are proven.
problem Proving compactness of capillary hypersurfaces with prescribed mean curvature.
method Extending curvature varifolds to oriented integral varifolds and applying curvature varifolds with capillary boundary.
result Compactness result for capillary hypersurfaces with mean curvature prescribed by ambient functions.
Paper finds smooth convex solutions to curvature problem.
problem Finding smooth, convex solutions to curvature problems.
method Established existence of solutions through mathematical analysis.
result Smooth, origin-symmetric, strictly convex solutions found.
We give a new existence proof for closed hypersurfaces of prescribed mean curvature in Lorentzian manifolds.