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.
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.
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.
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.
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.
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.
Unique AdS spacetime found with prescribed metric on a convex surface.
problem Finding an AdS spacetime with a specific metric on a convex surface.
method Constructing a quasifuchsian AdS spacetime with a past-convex Cauchy surface.
result Existence and uniqueness of a quasifuchsian AdS spacetime with the specified properties.
We solve the modified Kazdan-Warner problem of finding metrics with prescribed scalar curvature and unit total volume.
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.
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.
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.
We prove the existence of metrics with prescribed Q-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a pre…
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.
Study fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.
problem Finding conformal Hermitian metrics with prescribed curvature functions.
method Blow-up argument and partial uniform ellipticity.
result Our assumptions are almost sharp, with some geometric function theory obstructions.
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.
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.
Paper introduces a complete metric topology for low energy spaces.
problem Defining a topology for low energy spaces with prescribed singularity.
method Introduces a completely metrizable topology stronger than capacity convergence.
result Low energy spaces have a natural completely metrizable topology.
Reproves results on spherical metrics using parabolic bundles.
problem Existence and uniqueness of conformal spherical metrics with prescribed angles.
method Kobayashi-Hitchin correspondence for parabolic bundles.
result Reproves Troyanov and Luo-Tian's results.
The paper finds metrics for manifolds with prescribed volumes and eigenvalues.
problem Finding Riemannian metrics with specific volumes and eigenvalues.
method Prescribes Dirichlet eigenvalues for a compact manifold with a non-empty boundary.
result Existence of metrics with prescribed volume and eigenvalues.
Conditions for scalar curvature on compact manifolds under conformal deformation.
problem Finding conditions for scalar curvature functions on compact manifolds.
method Analyzing sufficient and necessary conditions for scalar curvature problems within conformal classes.
result Conditions for scalar curvature functions on various compact manifolds.
We study the problem of prescribing the Ricci curvature in the class of naturally reductive metrics on a compact Lie group. We derive necessary as well as sufficient conditions for the solvability of the equations and provide a series of examples.
Survey on metrics with conic singularities on Riemann surfaces.
problem Conformal metrics with conic singularities on Riemann surfaces.
method Study of metrics with constant positive curvature and conic singularities.
result Simplified proofs for interesting cases, including those by C. L. Chai, C. S Lin, and C. L. Wang.
The prescribed Ricci curvature problem consists in finding a Riemannian metric g on a manifold M such that the Ricci curvature of g equals a given (0,2)-tensor field T. We survey the recent progress on this problem in the case where M is a homogeneous space.
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.
We study metrics on the shape space of curves that induce a prescribed splitting of the tangent bundle. More specifically, we consider reparametrization invariant metrics G on the space Imm(S1,R2) of parametrized regular curves. For many metrics the tangent space $T_c\operatorname{Imm}(S^1,…
The paper solves curvature problems on infinite hyperbolic surfaces.
problem Prescribed curvature flow on hyperbolic surfaces with infinite topological type.
method Introduced a prescribed curvature flow adapted to infinite cellular decompositions.
result Established well-posedness of the flow and proved convergence under certain conditions.
In 1993, Bartnik introduced a quasi-spherical construction of metrics of prescribed scalar curvature on 3-manifolds. Under quasi-spherical ansatz, the problem is converted into the initial value problem for a semi-linear parabolic equation of the lapse function. The original ansatz of Bartnik started with a background …
Investigates solving curvature equations on special Lie groups.
problem Solving curvature equations on non-compact simple Lie groups.
method Analyzes left-invariant naturally reductive metrics and conditions for solvability.
result Obtains conditions for the solvability of curvature equations.
Study finds conditions for metrics on curved spaces.
problem Finding metrics with specific curvature properties.
method Analyzes a conformal class with prescribed scalar and boundary mean curvatures.
result Establishes a necessary and sufficient condition involving a conformal invariant.
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…
Solves a problem in Riemannian geometry for scalar-flat metrics with boundary conditions.
problem Finding a conformal metric with zero scalar curvature and prescribed boundary mean curvature.
method Construction of local test functions to resolve open cases and establish new solvability conditions.
result Established new solvability conditions for the problem.
Study on surfaces with conical singularities and geodesic boundaries, deriving existence results.
problem Existence of conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundaries.
method Variational argument to derive existence results for surfaces with at least two boundary components.
result First result in this setting for surfaces with conical singularities of both positive and negative orders.
In this paper,we obtain two results on closed Reimainnian manifold M×[0,T].When T is small enough,to any prescribed scalar curvature, the existence and uniqueness of metrics are obtained on the volume element preserving deformation.When T is large and the given scalar curvature is small enough,the same resu…
The paper studies metrics that match prescribed geodesics and introduces a variational problem.
problem Finding Riemannian metrics whose geodesics match given paths.
method Introduces a functional E on Riemannian metrics and computes its variational equations.
result Existence of conformally critical metrics in certain cases.
Constructs ancient solutions to mean curvature flow with prescribed singular sets.
problem Creating mean-convex ancient solutions with specific singular sets.
method Constructs solutions with a prescribed singular set Kimes{0} using mean curvature flow in a Riemannian metric. result Constructs ancient solutions with a first-time singular set exactly Kimes{0}. 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. By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with n cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension n−3. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a n…
This paper considers the existence of conformally compact Einstein metrics on 4-manifolds. A reasonably complete understanding is obtained for the existence of such metrics with prescribed conformal infinity, when the conformal infinity is of positive scalar curvature. We find in particular that general solvability in …
Study on curvature flow in 4D ball, proving existence and convergence.
problem Existence of metrics with prescribed T-curvature on the 4D unit ball. method Using T-curvature flow and Morse-theoretic approach, combining Ache-Chang's inequality. result Existence results and exponential convergence to extremal metric.
In this note we show that every compact spin manifold of dimension ≥3 can be given a Riemannian metric for which a finite part of the spectrum of the Dirac operator consists of arbitrarily prescribed eigenvalues with multiplicity 1.
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 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.
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 …
Proves existence of static vacuum metrics with specific boundary data.
problem Existence of static vacuum metrics with prescribed boundary data.
method Proves existence and local uniqueness of static vacuum metrics close to the Euclidean metric.
result Existence of static vacuum metrics with prescribed Bartnik boundary data.
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.
The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.
problem Finding decorated piecewise hyperbolic metrics with prescribed combinatorial curvature.
method Introduced combinatorial α-Ricci flow with surgery to handle potential singularities and prove longtime existence and convergence.
result Existence of decorated piecewise hyperbolic metrics with prescribed combinatorial α-curvature.
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.
Making use of integral representations, we develop a unified approach to establish blow up profiles, compactness and existence of positive solutions of the conformally invariant equations Pσ(v)=Kvn−2σn+2σ on the standard unit sphere Sn for all σ∈(0,n/2), where Pσ is the intertwining …