Study prescribed curvature tensor in locally conformally flat manifolds.
problem Solving the Prescribed Curvature Tensor problem in locally conformally flat manifolds.
method Explicit solutions provided for special cases of the tensor R, including complete metrics on Rn.
result Explicit examples of metrics g and conformal metrics g that solve the problem are exhibited.
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.
Researchers solve metric curvature equations on manifolds with boundary.
problem Finding complete conformal metrics with specific curvature functions.
method Revealed algebraic structure of fully nonlinear equations; used topological obstructions.
result Solved a class of fully nonlinear equations for conformal metrics.
Solving Ricci curvature problem on homogeneous spaces.
problem Finding metrics with prescribed Ricci curvature on homogeneous spaces.
method Surveying recent progress in solving the problem for homogeneous spaces.
result Recent progress in solving the problem for homogeneous spaces.
The paper solves a problem in 3D geometry about conformally deforming metrics.
problem Solving the problem of conformally deforming a metric to match a prescribed k-curvature. method Analyzes the k-curvature defined by the k-th elementary symmetric function of the eigenvalues of the Einstein tensor. result Proves the solvability of the problem and compactness of solution sets on manifolds.
We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.
Researchers examine global properties of a scalar curvature functional to solve the prescribed Ricci curvature problem.
problem Solving the prescribed Ricci curvature problem for homogeneous metrics.
method Examining global properties of the scalar curvature functional, focusing on its critical points and maximum.
result Conditions for a global maximum of the scalar curvature functional on a general homogeneous space.
Existence proved for Ricci curvature on sphere product.
problem Existence of metrics with prescribed Ricci curvature on product spheres.
method Proved existence for certain doubly warped product metrics.
result Existence of metrics with prescribed Ricci curvature on Sd1+1imesSd2. We consider the problem of conformally deforming a metric to one with a prescribed symmetric function of the eigenvalues of the Ricci tensor, in the case of negative curvature.
The paper finds solutions for specific curvature conditions on 5D Lie groups.
problem Finding metrics with prescribed Ricci curvature on 5D nilpotent Lie groups.
method Applied Milnor-type theorem technique to prove global existence of (g, c).
result Global existence of (g, c) for prescribed Ricci curvature on 5D nilpotent Lie groups.
New method bounds sectional curvatures and proves Spectral Theorem.
problem Computing and bounding sectional curvatures for algebraic curvature tensors.
method Algebraic and geometric methods to construct hypersurfaces with prescribed sectional curvatures.
result A relatively short proof of the Spectral Theorem for self-adjoint operators.
Equations link metrics with tensors, revealing curvature constraints.
problem Understanding curvature properties of geometric structures.
method Formal analogies to Einstein-Maxwell equations, studying Codazzi and conformal Killing equations.
result Constraints on scalar curvature of metrics in solutions.
Generalizes DeTurck's theorem to non-compact manifolds.
problem Uniquely determine Levi-Civita connection from Ricci curvature.
method Extends DeTurck's theorem to non-compact manifolds.
result Generalization to non-compact manifolds with finite total scalar curvature.
Paper classifies Schouten-like metrics on 5D nilpotent Lie groups.
problem Finding Riemannian metrics with prescribed Ricci curvature.
method Introduced Schouten-like metrics and classified them on 5D nilpotent Lie groups.
result Comprehensive classification of 5D nilpotent Lie groups' Schouten-like metrics.
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 consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold (Mn,g) endowed with the complementary orthogonal distributions D1 and D2. We provide conditions for symmetric (0,2)-tensors T of a simple form (defined on M) to admit metrics g~, conformal to …
Conditions for torsion-free connections with specific curvature maps are derived.
problem Finding conditions for torsion-free connections with prescribed curvature.
method Using a power series approach to derive necessary and sufficient conditions for a curvature map to arise from a torsion-free connection.
result A unique torsion-free connection is derived from a given curvature map.
Let M be a domain enclosed between two principal orbits on a cohomogeneity one manifold M1. Suppose T and R are symmetric invariant (0,2)-tensor fields on M and ∂M, respectively. The paper studies the prescribed Ricci curvature equation Ric(G)=T for a Riemannian metric G on M subject…
Study eigenvalues of curvature operators to annihilate cobordism invariants.
problem Annihilating rational cobordism invariants on spin manifolds.
method Linear inequalities on curvature operator eigenvalues.
result Curvature conditions stabilize to annihilate invariants.
Affine spheres can be realized with a specific Blaschke metric condition.
problem Realizing affine spheres with a prescribed Blaschke metric.
method Proving the necessary and sufficient condition for local realizability.
result The equality Δln∣κ−λ∣=6κ is a condition for local realizability of the metric as an affine sphere's Blaschke metric. Suppose M is a manifold with boundary. Choose a point o∈∂M. We investigate the prescribed Ricci curvature equation $\Ric(G)=T$ in a neighborhood of o under natural boundary conditions. The unknown G here is a Riemannian metric. The letter T in the right-hand side denotes a (0,2)-tensor. Our main the…
Solves modified Schouten tensor problems in conformal metric classes.
problem Prescribed problems for modified Schouten tensors in conformal classes of metrics.
method Uniform ellipticity confirmation under topological and functional constraints.
result Extends results from previous work on smooth complete metrics.
Researchers find conditions for a specific curvature on Lie groups.
problem Conditions for a specific Ricci curvature on three-dimensional unimodular Lie groups.
method Provided necessary and sufficient conditions for the existence of a pair (g, c) with left-invariant Riemannian metric g and positive constant c such that Ric(g) = cT.
result Existence and uniqueness of solutions for the prescribed Ricci curvature problem on Lie groups.
On a manifold with boundary, we deform the metric conformally. This induces a deformation of the Schouten tensor. We fix the metric at the boundary and realize a prescribed value for the product of the eigenvalues of the Schouten tensor in the interior, provided that there exists a subsolution.
In this paper we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is non-negative definite.
Paper proves existence of curves with specific geometric properties.
problem Existence of isometric immersions with prescribed second fundamental form.
method Introducing developments of curves with symmetric tensors and geometric construction.
result Existence of isometric immersions with prescribed second fundamental form.
Prescribing, by conformal transformation, the kth-elementary symmetric polynomial of the Schouten tensor P to be constant is a generalisation of the Yamabe problem. On compact Riemannian n-manifolds we show that, for k between and including 3 and n, this prescription equation is an Euler-Lagrange equation of some act…
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. Study on Yamabe flow for negative scalar curvature.
problem Prescribed scalar curvature problem on compact manifolds.
method Yamabe flow with conditions on scalar curvature function.
result Long time existence and convergence of the flow.
Study on prescribing scalar curvature on spheres using flow method.
problem Prescribing scalar curvature on n-sphere with sign-changing function.
method Scalar curvature flow approach.
result Flow converges to a metric with prescribed sign-changing function.
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.
Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in Hn+1, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with t…
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.
Let G be a compact connected Lie group and H a closed subgroup of G. Suppose the homogeneous space G/H is effective and has dimension 3 or higher. Consider a G-invariant, symmetric, positive-semidefinite, nonzero (0,2)-tensor field T on G/H. Assume that H is a maximal connected Lie subgroup of G. We p…
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.
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.
Paper solves tensor problem for holomorphic vector bundles.
problem Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors.
method New comparison theorem for Hermitian-Yang-Mills tensors.
result Existence of unique smooth Hermitian metric for any positive-definite tensor.
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 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.
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 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. 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. In this paper, we employ a nonlocal Q-curvature flow inspired by Gursky-Malchiodi's work \cite{gur_mal} to solve the prescribed Q-curvature problem on a class of closed manifolds: For n≥5, let (Mn,g0) be a smooth closed manifold, which is not conformally diffeomorphic to the standard sphere, satisfying e…
Extends initial data to asymptotically flat solutions in general relativity.
problem Solving constraint equations for asymptotically flat solutions in general relativity.
method New method for solving prescribed divergence equation and geometric variant of conformal method.
result Global solution extending initial data to asymptotically flat solutions.