The paper examines complete Yamabe solitons with finite total scalar curvature.
problem Characterizing complete Yamabe solitons with specific curvature properties.
method Analyzing steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature.
result Steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive scalar curvature have zero scalar curvature.
Study non-positive scalar curvature on hyperbolic manifolds, finding necessary and sufficient conditions.
problem Finding metrics with non-positive scalar curvature on asymptotically hyperbolic manifolds.
method Analyzing the prescribed scalar curvature problem for non-positive scalar curvature on asymptotically hyperbolic manifolds.
result Obtained a necessary and sufficient condition for the existence of solutions.
Conformal invariants of manifolds of non-positive scalar curvature are studied in association with growth in volume and fundamental group.
The paper proves stability for Einstein metrics with special twisted spinors.
problem Stability of Einstein metrics with specific spinor conditions.
method Proves linear semi-stability for a class of Einstein metrics with non-positive scalar curvature.
result Linear semi-stability for Einstein metrics carrying a parallel twisted spinr spinor. In this paper, we consider the scalar curvature of Yamabe solitons. In particular we show that, with natural conditions and non positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that the quadratic decay at infinity of the Ricci curvature of …
The paper classifies biharmonic hypersurfaces with constant scalar curvature.
problem Classifying biharmonic hypersurfaces with constant scalar curvature.
method Analyzing biharmonic hypersurfaces in space forms and spheres.
result Supports conjectures on biharmonic submanifolds and hypersurfaces.
The study characterizes quasi Yamabe solitons with potential vector fields.
problem Characterizing quasi Yamabe solitons with specific properties.
method Analyzing potential vector fields and their norms in quasi Yamabe solitons.
result If the potential vector field has a finite global norm in a complete non-trivial, non-compact quasi Yamabe soliton with finite volume, the scalar curvature becomes constant and the soliton reduces to a Yamabe soliton.
In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi-Civita connections and explicit formulas for computing sectional curvatures of these metrics and show that all these spaces have …
The paper solves a problem related to curvature in complex geometry.
problem Resolving the prescribed Chern scalar curvature problem.
method Divided into three cases based on the sign of the Gauduchon degree, analyzed separately.
result Proves that certain functions are Chern scalar curvatures of conformal metrics.
The paper explores properties of Finsler manifolds with specific curvature conditions.
problem Investigating Finsler metrics with various curvature conditions.
method Analyzing non-Riemannian (α,β)-metrics and compact Finsler manifolds with specific curvature properties. result Compact Finsler manifolds with relatively non-negative stretch curvature are Landsberg metrics.
Global existence of Yamabe flow on non-compact manifolds with unbounded initial curvature.
problem Global existence of Yamabe flow on non-compact manifolds with unbounded initial curvature.
method Assumption of conformally equivalent initial metric to a complete background metric with bounded scalar curvature and positive Yamabe invariant.
result Global existence of Yamabe flow without requiring initial curvature bounds.
We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines…
The paper explores conditions for constructing gradient Ricci solitons on warped product spaces.
problem Conditions for constructing gradient Ricci solitons on warped product spaces.
method Analyzes conditions for expanding or steady gradient Ricci solitons on warped product spaces.
result Necessary and sufficient conditions for constructing gradient Ricci solitons on warped product spaces.
We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces (Theorem 3). Thus we generalise [25,Theorem 3] by Nunes. This splitting result follows from an area comparison theorem for hypersurfaces with non-positi…
Solves geodesic equations on special Kähler manifolds, proving global regularity.
problem Geodesic equations on ALE Kähler manifolds.
method Solving geodesic equations under ALE conditions, proving regularity.
result Global C1,1 regularity of geodesic solutions. Minimal Lagrangians in certain curved spaces are stable under specific flows.
problem Stability of minimal Lagrangians in Kähler-Einstein manifolds of non-positive curvature.
method Proved stability under Lagrangian mean curvature flow.
result Equivalence between linear and dynamical stability for C1-close Lagrangians. In this short paper, we prove a Hitchin-Thorpe type inequality for closed 4-manifolds with non-positive Yamabe invariant, and admitting long time solutions of the normalized Ricci flow equation with bounded scalar curvature.
This is a continuation of a previous paper of same title. The degeneration, i.e. curvature blow-up, of sequences of metrics appoaching the Sigma constant, assumed non-positive, is analysed. The degeneration is related to the sphere decomposition of the 3-manifold M, in case M is sigma-tame.
New approach linking CR Yamabe invariant to Sasaki structures.
problem Existence of constant transversal scalar curvature Sasaki structures.
method Drawing on CR Yamabe problem ideas, establishing link between invariant, Sasaki structures, and K-stability.
result CR Yamabe invariant value determines K-semistability of Sasaki manifolds.
We initiate the study of an analogue of the Yamabe problem for complex manifolds. More precisely, fixed a conformal Hermitian structure on a compact complex manifold, we are concerned in the existence of metrics with constant Chern scalar curvature. In this note, we set the problem and we provide a positive answer when…
New tensor integration theorem with reflecting boundary.
problem Integrating tensor fields over broken rays with reflections.
method Two proofs in non-positive curvature geometry with convex obstacles.
result Symmetrized covariant derivatives integrate to zero under given conditions.
In this article, we prove new rigidity results for compact Riemannian spin manifolds with boundary whose scalar curvature is bounded from below by a non-positive constant. In particular, we obtain generalizations of a result of Hang-Wang \cite{hangwang1} based on a conjecture of Schroeder and Strake \cite{schroeder}.
The paper classifies special Riemannian manifolds with cyclic parallel Ricci tensor.
problem Classifying Riemannian manifolds with specific properties.
method Analyzing solutions to a specific partial differential equation under cyclic parallel Ricci tensor conditions.
result Classification of manifolds with positive static triples, critical metrics, and total scalar curvature.
The paper proves properties of complex surfaces and their curvature.
problem Understanding curvature properties on compact complex surfaces.
method Establishing Chern number identities and applying to curvature conditions.
result Compact complex surfaces with specific curvature conditions are Kähler surfaces.
The study classifies gradient Ricci solitons with specific vector fields.
problem Characterizing gradient Ricci solitons with closed conformal vector fields.
method Analyzing properties of gradient Ricci solitons with constant scalar curvature and closed conformal vector fields.
result Gradient Ricci solitons with these properties are isometric to specific spaces.
A complex disproves a curvature property.
problem Proving non-equivalence of curvature properties.
method Analyzing a specific 2-complex presentation.
result The contracting non-positive immersions property is not equivalent to positive maximal irreducible curvature.
Study non-positive curvature Weyl connections on homogeneous spaces.
problem Characterize Weyl connections with non-positive sectional curvatures.
method Analyze canonical families and use properties of unimodular Lie groups.
result Non-positive Weyl connections on homogeneous spaces are locally product type.
In this paper, firstly we study some left invariant Riemannian metrics on para-hypercomplex 4-dimensional Lie groups. In each Lie group, the Levi-Civita connection and sectional curvature have been given explicitly. We also show these spaces have constant negative scalar curvatures. Then by using left invariant Riemann…
Study shows non-positivity of Einstein-Hilbert action for certain metrics.
problem Analyzing the non-positivity of the Einstein-Hilbert action for specific metrics.
method Using spectral triples and modular operator computations.
result Recovery of earlier results on noncommutative tori and new Gauss-Bonnet theorem.
The paper proves rigidity results for non-positively curved homogeneous Finsler metrics.
problem Rigidity of non-positively curved homogeneous Finsler metrics.
method Analyzes and proves rigidity results for specific Finsler metrics with non-positive flag curvature.
result Homogeneous Finsler spaces with non-positive flag curvature and isotropic S-curvature are either Riemannian or locally Minkowskian.
We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kahler-Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. …
Splitting theorem for non-positively curved Lorentzian spaces.
problem Understanding curvature in Lorentzian spaces.
method Proving a splitting theorem with global non-positive timelike curvature and extending first variation formula.
result Splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature.
Survey on groups acting on non-positive curvature spaces.
problem Understanding groups acting on spaces with non-positive curvature.
method Presentation of examples and properties, linking algebraic/analytic and geometric properties.
result Discussion of links between group properties and space geometry.
Rigidity results for Hawking mass in curved spaces with bounds on Bartnik capacity.
problem Rigidity of surfaces in curved spaces with mass bounds.
method Analyzing Hawking mass and applying rigidity results to specific geometric settings.
result Explicit lower bounds on Hawking and Bartnik masses in non-flat spaces.
The study extends and generalizes a result about quasi Einstein manifolds, proving conditions for Killing vector fields.
problem Characterizing conditions for Killing vector fields in quasi Einstein manifolds.
method Extending and generalizing Cochran's result, proving conditions for Killing vector fields under specific integrals and conformal conditions.
result Conditions for Killing vector fields in quasi Einstein manifolds, including integral identities and global isometry to spheres.
Injectivity theorems for geodesic ray transform on curved 2D manifolds.
problem Understanding geodesic ray transform on curved 2D manifolds.
method Injectivity theorems for geodesic ray transform on Cartan-Hadamard manifolds with non-positive curvature.
result Proved two injectivity theorems for geodesic ray transform on 2D Cartan-Hadamard manifolds.
Non-positively curved surfaces can be embedded in flat spacetime.
problem Embedding surfaces with non-positive curvature in flat spacetime.
method Polyhedral approximation to prove isometric embedding.
result Metric of non-positive curvature can be embedded as a convex spacelike Cauchy surface.
The paper characterizes Einstein 4-manifolds with semi-definite curvature and derives inequalities.
problem Characterizing Einstein 4-manifolds with semi-definite sectional curvature.
method Using pointwise inequalities involving scalar curvature and Weyl curvatures.
result Closed 4-dimensional Einstein metrics saturating the pointwise inequality are completely characterized.
4-manifolds with non-positive curvature are essentially Euclidean.
problem Understanding the structure of 4-manifolds with specific curvature properties.
method Proving homeomorphism to Euclidean space using globally non-positive curvature.
result CAT(0) 4-manifolds are homeomorphic to Euclidean space.
Paper examines non-positive curvature properties of Hilbert metric in convex domains.
problem Investigating curvature properties of Hilbert metric in convex domains.
method Surveying and proving relationships among concepts, showing conditions for rigidity.
result If Hilbert metric is Berwald, domain is an ellipsoid and metric is Riemannian.
In this paper we study the set of balanced metrics (in Donaldson's terminology) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this set is finite when M admits…
Sharp stability estimate for tensor tomography in non-positive curvature.
problem Stability estimate for tensor tomography on manifolds with non-positive curvature.
method Pestov identity with localized frequency boundary term.
result Stability estimate of the form L2↦HT1/2. The study proves local and global Holder continuity of quasi-n-harmonic mappings into spaces with non-positive curvature.
problem Holder continuity of quasi-n-harmonic mappings into metric spaces with non-positive curvature.
method Local and global Holder continuity proved using Euclidean domains and bounded Lipschitz domains.
result Local and global Holder continuity of quasi-n-harmonic mappings.
The article proves isometry theorems for specific types of manifolds.
problem Investigating properties of Cartan-Hadamard manifolds and related solitons.
method Analyzing steady, gradient shrinking, and expanding Ricci solitons.
result Specific manifolds are isometric to Euclidean space under certain conditions.
Study finds criteria for surfaces with specific curvature properties.
problem Understanding Kählerian or projective structures on surfaces with non-positive curvature.
method Established a criterion for compact Hermitian surfaces with non-positive second Chern-Ricci curvature.
result Found conditions for Kählerian or projective structures on surfaces with non-positive curvature.
The study of harmonic maps into non-positively curved spaces proves a Bochner formula.
problem Understanding harmonic maps in non-positively curved spaces.
method Expanding domain variation and Bochner formulas in terms of curvature.
result Harmonic maps from spaces of non-negative Ricci curvature into non-positively curved spaces have subharmonic energy density.
The paper studies curvatures in metric Jordan algebras, proving unique connections and curvature formulas.
problem Investigating curvatures in metric Jordan algebras.
method Defined the Jordan-Levi-Civita connection, introduced curvature tensors, and proved curvature formulas.
result Every formally real Jordan algebra admits a metric of non-positive Jordan curvature and a Jordan-Einstein metric of negative Jordan scalar curvature.
Using the Perron method, we prove the existence of hypersurfaces of prescribed special Lagrangian curvature with prescribed boundary inside complete Riemannian manifolds of non-positive curvature.