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.
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 Ollivier Ricci flow with prescribed curvature on infinite graphs.
problem Ricci flow with prescribed curvature on infinite graphs.
method Existence and uniqueness of the solution to the Ricci flow.
result Convergence of the Ricci flow for graphs with girth at least 6.
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.
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. 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.
Study solves Ricci curvature problem for specific noncompact spaces.
problem Solving the Prescribed Ricci Curvature problem for noncompact spaces with two isotropy summands.
method Classified and solved for all simply connected, noncompact G/H with semi-simple G and connected H having two irreducible summands. result Provided solutions to the Prescribed Ricci Curvature problem for all such spaces.
Study Palais-Smale sequences for Ricci curvature on homogeneous spaces.
problem Characterize Palais-Smale sequences for prescribed Ricci curvature.
method Complete description of divergent sequences on compact homogeneous spaces.
result Existence of saddle points on generalized Wallach and flag manifolds.
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 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.
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.
Researchers study metrics with maximal Ricci curvature on homogeneous spaces.
problem Maximizing Ricci curvature in G-invariant metrics on homogeneous spaces.
method Used a formula for the Lichnerowicz Laplacian in terms of the moment map for the variety of algebras.
result Such metrics are generic in the compact case.
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.
The paper studies Ricci flow on graphs with prescribed curvature.
problem Characterizing weight evolution on graphs with prescribed curvature.
method Ricci flow with Lin-Lu-Yau curvature prescription.
result Ricci flow converges to weights of prescribed curvature under certain conditions.
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.
The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.
problem Creating minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature.
method Develops a min-max theory and shows deformability of surfaces in a generic metric.
result Establishes a theorem for producing minimal surfaces with prescribed genus.
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.
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.
Study optimal partition problem for Q-curvature equations on Einstein manifolds.
problem Optimal partition problem for prescribed Q-curvature equation.
method Cohomogeneity one actions, higher order conformal operators, weakly coupled elliptic systems.
result Existence and multiplicity of least energy symmetric and sign-changing solutions.
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 …
Consider a compact Lie group G and a closed Lie subgroup H<G. Let M be the set of G-invariant Riemannian metrics on the homogeneous space M=G/H. By studying variational properties of the scalar curvature functional on M, we obtain an existence theorem for solutions to the prescribed Ricci …
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.
Proves existence of smooth metrics with specific curvature properties.
problem Existence of smooth metrics with prescribed negative Ricci curvature.
method Formulated and proved for general domains in Euclidean space.
result Existence of smooth complete conformal metrics with prescribed negative Ricci curvature.
Study of Ricci flow convergence on surfaces with boundary.
problem Convergence of singular solutions to Ricci flow on compact surfaces with boundary.
method Subsequential convergence analysis of Ricci flow with prescribed geodesic curvature.
result Convergence does not depend on the sign of geodesic curvature of the boundary in the case of rotational symmetry.
Maps from metrics to Ricci curvature are locally invertible near Einstein manifolds.
problem Understanding the invertibility of maps from metrics to Ricci curvature near Einstein manifolds.
method Analyzing the invertibility of maps involving Ricci curvature, conformal classes, and mean curvature.
result The map is locally invertible near an Einstein manifold with boundary.
Extends Eells-Sampson theorem for manifolds with positive sectional curvature bounds.
problem Rigidity of harmonic maps between manifolds with curvature constraints.
method Proves an extension of Eells-Sampson theorem under positive sectional curvature upper bounds.
result Recover Hamilton's rigidity result for positive Ricci curvature.
We study the Ricci iteration for homogeneous metrics on spheres and complex projective spaces. Such metrics can be described in terms of modifying the canonical metric on the fibers of a Hopf fibration. When the fibers of the Hopf fibration are circles or spheres of dimension 2 or 7, we observe that the Ricci iteration…
The paper proves the existence of embedded hypersurfaces of constant mean curvature in manifolds with positive Ricci curvature.
problem Proving the existence of embedded hypersurfaces of constant mean curvature in manifolds with positive Ricci curvature.
method Using the Allen--Cahn min-max scheme with a non-zero constant prescribing function.
result The existence of embedded, closed λ-CMC hypersurfaces with Morse index 1 for any prescribed non-zero constant λ.
Survey on rigidity results for graphs with prescribed mean curvature.
problem Rigidity of graphs with prescribed mean curvature.
method Analysis of mean curvature operator, maximum principles, gradient estimates.
result Detailed geometric applications, including Bernstein theorem and splitting theorem.
In this note we show that any real exact G-invariant (1,1)-form is the Ricci form of a Kaehler metric on the complexification of an irreducible compact symmetric space G/K.
In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold M and a symmetric 2-tensor r, construct a metric on M whose Ricci tensor equals r. In particular, DeTurck and Koiso proved the following celebrated result: the Ricci curvature uniquely determines the Levi-Civita…
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 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.
The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
We show the existence of a deformation process of hypersurfaces from a product space M1×R into another product space M2×R such that the relation of the principal curvatures of the deformed hypersurfaces can be controlled in terms of the sectional curvatures or Ricci curvatures of M1 and M2. In t…
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…
The paper explores Ricci forms on noncompact complex manifolds, including Kähler-Einstein and canonical metrics.
problem Existence of specific metrics on noncompact complex manifolds with prescribed Ricci curvature.
method Analyzes geometric problems on noncompact complex manifolds using Ricci curvature.
result Improves the main theorem in Cheng-Yau [4] and constructs Hesse-Einstein metrics.
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…
The article shows how to create metrics with positive Ricci curvature on twisted suspensions.
problem Creating metrics with positive Ricci curvature on complex manifolds.
method Using twisted suspensions and Riemannian metrics.
result Maximal symmetry rank of positive Ricci curvature manifolds is (n-2) in all dimensions n≥4.
The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.
problem Discrete conformal structures on polyhedral surfaces and their rigidity.
method Parameterized combinatorial curvature, combinatorial α-Ricci flow, and flow extension through singularities.
result Existence and convergence of combinatorial α-Ricci flow for solving the Yamabe problem.
We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial …
The study finds infinite nodal solutions for equations on positive Ricci curvature manifolds.
problem Existence of nodal solutions for equations on manifolds with positive Ricci curvature.
method Analyzes cohomogeneity one Riemannian manifolds with positive Ricci curvature and proves the existence of infinite nodal solutions for specific equations.
result Proves the existence of infinite nodal solutions for equations of the form −Δgu+λu=λuq on positive Ricci curvature manifolds. Some observations about the local and global generality of gradient Kahler Ricci solitons are made, including the existence of a canonically associated holomorphic volume form and vector field, the local generality of solutions with a prescribed holomorphic volume form and vector field, and the existence of Poincare co…
The paper proves the existence of H-spheres with arbitrary codimensions in certain Riemannian manifolds.
problem Existence of H-spheres with arbitrary codimensions in closed Riemannian manifolds.
method Min-max theory and Morse index analysis.
result Existence of branched immersed H-spheres with controlled Morse index and arbitrary codimensions.
Study infinite combinatorial Ricci flow on spherical surfaces.
problem Investigate infinite combinatorial Ricci flow with spherical background.
method Establish existence and convergence of solution for infinite cellular decompositions.
result Existence and convergence of solution for infinite combinatorial Ricci flow in spherical geometry.
Paper proves translating solutions for a specific flow in a product manifold.
problem Existence of translating solutions for nonparametric mean curvature flow with Neumann boundary data.
method Proves existence using product manifold MnimesR with specific conditions. result Existence of translating solutions for the flow in the product manifold.
Consider a compact Lie group G and a closed subgroup H<G. Suppose M is the set of G-invariant Riemannian metrics on the homogeneous space M=G/H. We obtain a sufficient condition for the existence of g∈M and c>0 such that the Ricci curvature of g equals cT for a given $T\in\mathcal …
Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at confor…