Ricci solitons as critical points of quadratic curvature functionals
problem Einstein metrics and Ricci solitons as critical points of quadratic Riemannian functionals
method Study of Ricci solitons as critical points of a special quadratic curvature functional
result Ricci solitons are non-Einstein critical points of these functionals
Given a definable function f, enough differentiable, we study the continuity of the total curvature function t --> K(t), total curvature of the level {f=t}, and the total absolute curvature function t-->|K| (t), total absolute curvature of the level {f=t}. We show they admits at most finitely many discontinuities.
New findings show functional inequalities fail on Finsler manifolds with positive S-curvature.
problem Failure of functional inequalities on Finsler manifolds with positive S-curvature.
method Analysis of Finsler metric measure manifolds with reversibility, flag curvature, and S-curvature.
result Functional inequalities fail on Finsler manifolds with positive S-curvature.
The study links Ricci curvature and convexity in complex tori.
problem Characterizing Ricci curvature signs in toric manifolds.
method Characterization through convexity of volume functional.
result Relationships between Ricci curvature, volume, submanifolds, and pluri-subharmonic functions.
Paper shows CR Q-curvature orthogonal to CR pluriharmonic functions.
problem Understanding CR Q-curvature and pluriharmonic functions on CR manifolds. method Obtained a cohomological expression for the integral of CR Q-curvature and pluriharmonic functions. result CR Q-curvature is orthogonal to CR pluriharmonic functions. We obtained that any 2-form and any smooth function on 2-manifolds with boundary can be realized as the curvature form and the gaussian curvature function of some Riemmanian metric, respectively.
Small Weyl infimum on 4-manifolds with positive scalar curvature.
problem Understanding scalar curvature on 4-manifolds.
method Analyzing the Weyl functional and comparing scalar and self-dual Weyl curvatures.
result The infimum of the Weyl functional is small on many 4-manifolds with positive scalar curvature.
Researchers found all special metrics in 4D for certain curvature functionals.
problem Identifying special metrics in 4D for quadratic curvature functionals.
method Determined all homogeneous metrics that are critical for quadratic curvature functionals.
result All homogeneous metrics in 4D for some quadratic curvature functionals have been identified.
We present a novel and comprehensive approach to the study of the parametric Plateau problem for locally strictly convex (LSC) hypersurfaces of prescribed curvature for general convex curvature functions inside general Riemannian manifolds. We prove existence of solutions to the Plateau problem with outer barrier for L…
We consider the evolution of starshaped hypersurfaces in the Euclidean space by general curvature functions. Under appropriate conditions on the curvature function, we prove the global existence and convergence of the flow to a hypersurface of prescribed curvature.
First example of open manifold with positive Ricci curvature and non-proper Busemann function.
problem Counterexample to Busemann function properness in open manifolds with nonnegative Ricci curvature.
method Provided an open manifold with positive Ricci curvature and non-proper Busemann function.
result First example of open manifold with positive Ricci curvature and non-proper Busemann function.
The paper extends a Liouville theorem to biharmonic functions on manifolds with nonnegative Ricci curvature.
problem Proving that biharmonic functions with certain growth conditions are constant or harmonic.
method Using a new local L2 estimate for the Laplacian of biharmonic functions combined with a mean value inequality. result Any biharmonic function of subquadratic growth on a manifold with nonnegative Ricci curvature must be harmonic, and any of sublinear growth must be constant.
Study examines maximal domains of radial harmonic functions across different curvature types.
problem Understanding maximal domains of radial harmonic functions in various curvature settings.
method Analysis of harmonic spaces with positive, zero, and negative curvature.
result Characterization of maximal domains for radial harmonic functions in different curvature contexts.
The paper reformulates Bakry-Émery curvature on graphs using eigenvalues.
problem Analyzing curvature on weighted graphs.
method Reformulating curvature as the smallest eigenvalue of a rank one perturbation of the curvature matrix.
result The curvature function is analytic, strictly monotone increasing, and concave until a threshold, after which it is constant.
Paper finds critical metrics with pinched curvature are geodesic balls.
problem Identifying critical metrics with specific curvature constraints.
method Proved isometry to geodesic balls in S^n and provided conditions for the gradient of the potential function.
result Critical metrics with pinched curvature are isometric to geodesic balls in S^n.
We study the Bakry-Émery curvature function KG,x:(0,∞]→R of a vertex x in a locally finite graph G systematically. Here KG,x(N) is defined as the optimal curvature lower bound K in the Bakry-Émery curvature-dimension inequality $CD(\mathcal{K},\ma…
Finite graphs with specific curvature have limited harmonic functions and ends.
problem Graphs with nonnegative curvature outside a finite subset.
method Introducing discrete Gromov-Hausdorff convergence to study bounded harmonic functions.
result The space of bounded harmonic functions is finite dimensional, and the number of non-parabolic ends is finite.
Study shows incompatibility of certain scalar curvatures on manifolds.
problem Incompatibility of scalar curvatures on manifolds.
method Analyzing closed manifolds with positive Yamabe invariant and positive Morse functions.
result Existence of energetically bounded solutions for related candidate functions.
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. Study proves finiteness for distance functions on curved surfaces with controlled curvature.
problem Understanding distance functions on curved surfaces with Hölder continuous curvature.
method Proves a finiteness principle using Whitney extension theory for geodesics and points on Riemannian surfaces with Hölder continuous curvature.
result Establishes a finiteness principle for isometric embedding of metric spaces into Riemannian surfaces with controlled curvature.
Study finds solutions to flows by negative curvature powers.
problem Curvature flows with negative powers.
method Closed self-similar solutions in warped product manifolds, proving non-strict convexity.
result Proves self-similar solutions are slices of warped product manifolds.
Directly applies Kazdan--Warner results to prescribe scalar curvature on bundles.
problem Prescribing scalar curvature functions on bundles.
method Direct application of Kazdan--Warner results and variational methods.
result Determines which functions are realizable as scalar curvature functions on bundles.
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.
Paper proves a Liouville theorem for solitons with constant curvature.
problem Understanding harmonic functions on specific geometric structures.
method Proved a Liouville theorem without gradient estimates.
result Finite dimensionality of harmonic functions with polynomial growth.
Research characterizes critical points of scalar curvature functionals.
problem Characterizing critical points of scalar curvature functionals.
method Translation and analysis of a previous Russian paper.
result Provides insights into critical points of scalar curvature functionals.
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 …
We obtain a priori estimates for solutions to the prescribed scalar curvature equation on S3. The usual non-degeneracy assumption on the curvature function is replaced by a new condition, which is necessary and sufficient for the existence of a priori estimates, when the curvature function is a positive Morse functi…
Study distance functions on manifolds linking geometry to topology.
problem Understanding the relationship between curvature and topology on manifolds.
method Analyzing distance functions and their connection to manifold geometry.
result Alternative proofs of theorems linking curvature and topology.
We study rigidity results for the Einstein metrics as the critical points of a family of known quadratic curvature functionals involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor, characterized by some pointwise inequalities involving the Weyl curvature and the traceless Ricci curvat…
Derive Dirichlet scalar curvature energy functional variation formula
problem Dirichlet scalar curvature energy functional
method First variation formula
result Introduce Dirichlet-Einstein metrics
The paper studies a flow of convex hypersurfaces expanding by their support and curvature functions.
problem Analyzing the behavior of expanding hypersurfaces in Euclidean space.
method Introduced a curvature flow with specific speed function and proved the existence and convergence of the flow under certain conditions.
result The flow converges to a round sphere centered at the origin for all time under specific conditions.
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.
The paper studies a flow of convex hypersurfaces using anisotropic curvature functions.
problem Deforming convex hypersurfaces in Euclidean space.
method Fully nonlinear curvature flow involving k-th elementary symmetric function and support function.
result Long-time existence and convergence of the flow under certain assumptions.
Survey on rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
problem Rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
method Survey and observation on Cheeger-Yau inequality on RCD spaces.
result Observations on the Cheeger-Yau inequality and its applications.
In this paper, we study Riemannian functionals defined by L2-norms of Ricci curvature, scalar curvature, Weyl curvature, and Riemannian curvature. We try to understand stability of their critical points that are products of Einstein metrics. In particular, we prove that the product of a spherical space form and a co…
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.
Paper proves Harnack inequality for f-mean curvature flow.
problem Proving Harnack inequality for f-mean curvature flow. method Gradient flow of the weighed area functional with measure density function e−f. result Proves Li-Yau-Hamilton type Harnack estimate.
New rigidity results for critical metrics with curvature pinching.
problem Understanding critical metrics with curvature pinching conditions.
method Proving rigidity for metrics defined on closed smooth manifolds that are critical for a quadratic functional.
result Bach-flat metrics with constant scalar curvature satisfying Sec > 1/48 R are Einstein and isometric to specific spaces.
The paper proves a hierarchy of Liouville theorems for polyharmonic functions on manifolds with nonnegative Ricci curvature.
problem Establishing Liouville theorems for polyharmonic functions on manifolds with nonnegative Ricci curvature.
method A new L2 estimate for the Laplacian of a polyharmonic function, obtained by induction through a cutoff construction combined with a hole-filling argument. result All polyharmonic functions of sublinear growth on manifolds of nonnegative Ricci curvature are constant.
We study closed n-dimensional manifolds of which the metrics are critical for quadratic curvature functionals involving the Ricci curvature, the scalar curvature and the Riemannian curvature tensor on the space of Riemannian metrics with unit volume. Under some additional integral conditions, we classify such manifol…
New rigidity results for critical metrics of a quadratic curvature functional.
problem Proving uniqueness of critical metrics for a specific curvature functional.
method Analyzing complete, possibly non-compact, critical metrics of the quadratic curvature functional.
result Critical metrics with finite energy are scalar flat (global minima) for dimensions n≥10.
The paper classifies hypersurfaces in H2imesH2 with constant curvature.
problem Classifying hypersurfaces in H2imesH2 with constant sectional curvature. method Analyzing the geometry of H2imesH2 and constructing specific examples. result Examples of hypersurfaces in H2imesH2 with non-constant product angle function. Discuss folklore statements about manifolds with curvature bounds.
problem Distance functions in manifolds with curvature bounds.
method Regularity, subsets of positive reach, and cut locus.
result Folklore statements about manifolds with curvature bounds are discussed.
We give existence results for solutions of the prescribed scalar curvature equation on S3, when the curvature function is a positive Morse function and satisfies an index-count condition.
Algorithm approximates functions into manifolds with curvature bounds.
problem Approximating functions into manifolds with lower curvature bounds.
method Algorithm using manifold exponential and logarithm, with error bounds based on sectional curvature.
result Error bounds for nonnegative sectional curvature are similar to linear space approximations.
Analyzes singularities of convex hypersurfaces in hyperbolic space.
problem Understanding singularities of convex hypersurfaces with constant curvature.
method Analyzes the structure of singular sets using convex curvature functions.
result Describes the structure of singular sets in hyperbolic space.
Study on 3-manifolds with nonnegative scalar curvature and positive harmonic functions.
problem Characterizing 3-manifolds with nonnegative scalar curvature.
method Exhaustions by level sets of harmonic functions and refined average gradient estimates.
result Contractible 3-manifolds are diffeomorphic to R^3, and handlebodies have genus at most 1.
The study examines continuous mean curvature functions on manifolds without conjugate points.
problem Understanding properties of manifolds with specific curvature functions.
method Analyzing simply connected Riemannian manifolds with continuous horospherical mean curvature functions.
result Compact rank one manifolds without conjugate points are locally symmetric spaces of negative curvature.