Study CR Yamabe constant, flow, and soliton on CR manifolds.
problem Analyzing CR Yamabe constant, flow, and soliton on CR manifolds.
method Using maximum principles, proving uniqueness for CR Yamabe flow, and studying properties of CR Yamabe soliton.
result Uniqueness theorem for CR Yamabe flow and properties of CR Yamabe soliton.
Study on convergence rate of weighted Yamabe flow.
problem Weighted Yamabe problem on smooth metric measure spaces.
method Weighted Yamabe flow and its convergence rate analysis.
result Study and analysis of convergence rate of the weighted Yamabe flow.
The paper classifies a type of solitons in Euclidean spaces.
problem Classifying generalized Yamabe solitons on hypersurfaces.
method Completely classified solitons arising from the position vector field.
result Classification of generalized Yamabe solitons on hypersurfaces in Euclidean spaces.
Study properties of hyperbolic Yamabe solitons in submanifolds.
problem Characterize and understand hyperbolic Yamabe solitons in submanifolds.
method Define hyperbolic Yamabe flow, prove properties, and classify solitons.
result Prove that hyperbolic Yamabe solitons are pseudosymmetric or metallic shaped.
The Yamabe flow converges to a specific function on compactified manifolds.
problem Analyzing the Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant.
method Studied the Yamabe flow on asymptotically flat manifolds with Y≤0 and showed convergence after rescalings. result The Yamabe flow converges to the unique positive function solving the Yamabe problem on a compactification of the original manifold.
Study on η-Ricci-Yamabe solitons on Riemannian submersions.
problem Characterizing η-Ricci-Yamabe solitons on Riemannian submersions. method Analyzing conditions for η-Ricci-Yamabe solitons on submersions and deriving Laplacian equations. result Classification of fiber and target manifolds as η-Ricci-Yamabe solitons under various conditions. Paper examines properties of specific solutions to Yamabe flow.
problem Characterizing quotient almost Yamabe solitons.
method Investigates quotient almost Yamabe solitons and presents conditions for their rigidity.
result Sufficient conditions for quotient almost Yamabe solitons to be trivial or isometric with a sphere.
The study classifies h-almost Ricci-Yamabe solitons in various paracontact manifolds.
problem Classifying h-almost Ricci-Yamabe solitons in paracontact geometry.
method Characterization and classification of para-Kenmotsu, para-Sasakian, and para-cosymplectic manifolds.
result Characterizations and classifications of various paracontact manifolds.
In this paper we initiate the study of Yamabe and quasi-Yamabe solitons on Euclidean submanifolds whose soliton fields are the tangential components of their position vector fields. Several fundamental results of such solitons were proved. In particular, we classify such Yamabe and quasi-Yamabe solitons on Euclidean hy…
Paper proves uniqueness of Type II Yamabe metrics on manifolds.
problem Uniqueness of Type II Yamabe metrics on compact manifolds.
method Investigates sufficient conditions for metric uniqueness and proves corresponding theorems.
result Establishes sufficient condition for a metric to be the unique Type II Yamabe metric.
Paper finds conditions for non-Einstein relative Yamabe metrics.
problem Finding relative Yamabe metrics with positive scalar curvature.
method Sufficient condition for positive constant scalar curvature metrics on manifolds with boundary.
result Examples of non-Einstein relative Yamabe metrics with positive scalar curvature.
Study CR Yamabe constant and CR structures on manifolds.
problem Understanding CR Yamabe constant and its role in CR geometry.
method Developed integral formulae and constructed families of CR structures.
result Found an infinite family of CR structures with varying CR Yamabe constants.
The paper classifies 3D complete gradient Yamabe solitons.
problem Classifying nontrivial 3D complete gradient Yamabe solitons.
method Analyzing properties of Yamabe solitons to show rotationally symmetry.
result Nontrivial non-flat 3D complete steady gradient Yamabe solitons are rotationally symmetric.
Paper shows k-Yamabe solitons have constant curvature under certain conditions.
problem Understanding the properties of k-Yamabe solitons.
method Analyzing the curvature and gradient conditions for k-Yamabe solitons.
result Compact k-Yamabe solitons have constant σk-curvature under certain conditions. Study convergence of Yamabe flow on singular spaces with positive constant.
problem Analyzing convergence of Yamabe flow on singular spaces.
method Normalized Yamabe flow with positive Yamabe constant on pseudo-manifolds, including stratified spaces.
result Established convergence under low energy condition and investigated alternatives.
The study compares and finds Yamabe constants on warped products.
problem Comparing Yamabe constants on warped products.
method Using fiberwise spherical symmetrization.
result Existence of radially-symmetric Yamabe minimizers on product manifolds.
The paper studies Yamabe metrics and stability in Riemannian manifolds.
problem Existence of complete Yamabe metrics with zero scalar curvature.
method Yamabe flow and local L1-stability analysis. result Local L1-stability of the Yamabe flow on manifolds with non-negative Ricci curvature. Quantitative stability for nearly minimizing Yamabe metrics.
problem Understanding the stability of nearly minimizing metrics in Riemannian geometry.
method Proving quantitative closeness of nearly minimizing metrics to minimizing metrics in a specific sense.
result The distance between nearly minimizing metrics and minimizing metrics is controlled quadratically by the Yamabe energy deficit.
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.
Characterizes ∗-k-Ricci-Yamabe solitons on Kenmotsu manifolds.
problem Understanding ∗-k-Ricci-Yamabe solitons on Kenmotsu manifolds. method Analyzes the geometry of ∗-k-Ricci-Yamabe solitons and gradient solitons on Kenmotsu manifolds. result Characterizes the nature of ∗-k-Ricci-Yamabe solitons and gradient solitons. We consider the equivariant Yamabe problem, i.e. the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their …
The (α,β)-Ricci-Yamabe flow exists on closed manifolds.
problem Existence of solutions to the (α,β)-Ricci-Yamabe flow. method Showed short time existence and established long time existence theorems.
result Existence of smooth solutions to the (α,β)-Ricci-Yamabe flow on closed manifolds. Study on Yamabe solitons with applications and structure elucidation.
problem Understanding the structure of Yamabe solitons and their applications.
method Investigation of complete gradient conformal solitons under specific conditions.
result Affirmative partial answer to Yamabe soliton conjecture.
Study properties of 3D trans-Sasakian manifolds with η-Yamabe solitons.
problem Properties of 3D trans-Sasakian manifolds with η-Yamabe solitons.
method Investigated curvature conditions and constructed a manifold.
result Construct a 3D trans-Sasakian manifold satisfying η-Yamabe soliton.
In this paper, we completely classify almost Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field. Some results of almost Yamabe solitons with a concurrent vector field and almost Yamabe solitons on submanifolds in Riemannian manifolds equipped with a concurrent vector field are al…
Proves product metrics are Yamabe metrics under small flat torus conditions.
problem Yamabe metrics on product spaces with small flat tori.
method Extends earlier results to Type~I and Type~II Yamabe constants, Q-curvature problems, and isoperimetric-ratio type problems. result Product metrics are Yamabe metrics for sufficiently small flat tori.
The paper extends a connected-sum inequality to calculate λ-Yamabe invariants of certain manifolds.
problem Calculating λ-Yamabe invariants for specific compact manifolds with boundary.
method Generalizing Kobayashi's connected-sum inequality and applying it to specific manifolds.
result The paper proves that certain manifolds have the same λ-Yamabe invariants as the hemi-sphere.
We provide the classification of locally conformally flat gradient Yamabe solitons with positive sectional curvature. We first show that locally conformally flat gradient Yamabe solitons with positive sectional curvature have to be rotationally symmetric and then give the classification and asymptotic behavior of all r…
Study shows long-term flow on special manifolds with positive Yamabe constant.
problem Analyzing long-time behavior of Yamabe flow on singular spaces.
method Formulated axioms for long-time existence, used parabolic Moser iteration for bounds.
result Established long-time existence of normalized Yamabe flow on specified manifolds.
Paper examines stability of minimizing metrics on manifolds with boundary.
problem Stability of minimizing Yamabe metrics on compact manifolds with boundary.
method Investigates stability in the sense introduced by Escobar, showing closeness of nearly minimizing metrics.
result Quantitative closeness of nearly minimizing metrics to minimizing Yamabe metrics within their conformal class.
We prove several facts about the Yamabe constant of Riemannian metrics on general noncompact manifolds and about S. Kim's closely related "Yamabe constant at infinity". In particular we show that the Yamabe constant depends continuously on the Riemannian metric with respect to the fine C^2-topology, and that the Yamabe…
This paper classifies solitons under specific tensor conditions.
problem Classifying solitons under vanishing conditions on the Weyl, Cotton, and Cao-Chen tensors.
method Analyzing complete conformal gradient solitons and using tensor conditions.
result Classification of complete nontrivial locally conformally flat conformal gradient solitons.
Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.
problem Relating Yamabe invariants on asymptotically Poincare-Einstein manifolds.
method Established a sharp inequality using lower Ricci curvature bounds.
result Sharp inequality relating type II Yamabe invariant of the interior to the Yamabe invariant of the conformal infinity.
The paper studies special solitons on Riemannian manifolds with specific vector fields.
problem Characterizing conformal and ∗-Yamabe solitons with torse forming potential vector fields. method Analyzing solitons under different connections (Riemannian, semi-symmetric, projective semi-symmetric) and developing examples.
result Characterizations and properties of conformal and ∗-Yamabe solitons with torse forming vector fields. In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several s…
We define a relative Yamabe invariant of a smooth manifold with given conformal class on its boundary. In the case of empty boundary the invariant coincides with the classic Yamabe invariant. We develop approximation technique which leads to gluing theorems of two manifolds along their boundaries for the relative Yamab…
The study explores δ-almost gradient Yamabe solitons on pseudo-Riemannian manifolds.
problem Characterizing δ-almost gradient Yamabe solitons on pseudo-Riemannian manifolds.
method Analyzing δ-almost Yamabe solitons within the framework of para-contact metric manifolds, proving properties and conditions for solitons.
result Characterization of δ-almost gradient Yamabe solitons on K-paracontact metric manifolds.
The Yamabe invariant is linked to static potentials and eigenvalues.
problem The relationship between Yamabe invariant and static potentials/eigenvalues.
method Analyzes the Yamabe invariant in the context of static potentials and eigenvalues of the Laplacian.
result The Yamabe invariant is closely tied to static potentials and the first eigenvalue of the Laplacian.
We study a particular class of open manifolds. In the category of Riemannian manifolds these are complete manifolds with cylindrical ends. We give a natural setting for the conformal geometry on such manifolds including an appropriate notion of the cylindrical Yamabe constant/invariant. This leads to a corresponding ve…
Study combinatorial Yamabe flow on infinite triangulated surfaces.
problem Solve discrete Yamabe problem on noncompact surfaces.
method Introduced and analyzed combinatorial Yamabe flow with short-time and long-time existence proofs.
result Established short-time and long-time existence of the flow, and proved convergence in hexagonal triangulations.
In this paper, we investigate the behavior of ADM mass and Einstein-Hilbert functional under the Yamabe flow. Through studying the Yamabe flow by weighted spaces, we show that ADM mass and Einstein-Hilbert functional are well-defined and monotone non-increasing under the Yamabe flow on n-dimensional, n≥3, asymp…
We prove that a nontrivial complete generalized quasi Yamabe gradient soliton (M; g) must be a quasi Yamabe gradient soliton on each connected component of M and that a nontrivial complete locally conformally at generalized quasi Yamabe gradient soliton has a special warped product structure.
The paper classifies expanding gradient Yamabe solitons based on scalar curvature.
problem Classifying expanding gradient Yamabe solitons based on scalar curvature.
method Rigorous analysis of scalar curvature in both cases: greater than and less than the soliton constant.
result Complete classification of nontrivial complete expanding gradient Yamabe solitons.
In this paper, we introduce the concept of quasi Yamabe gradient solitons, which generalizes the concept of Yamabe gradient solitons. By using some ideas in [7,8], we prove that n-dimensional (n≥3) complete quasi Yamabe gradient solitons with vanishing Weyl curvature tensor and positive sectional curvature must …
New proof of Yamabe invariant for RP^3 using harmonic functions.
problem Yamabe invariant of RP3 method Using harmonic functions
result New proof of Yamabe invariant for RP3 The fractional Yamabe problem, proposed by González-Qing (2013, Anal. PDE) is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the classical Yamabe problem and the boundary Yamabe problem to the realm of nonlocal co…
The paper examines conditions for a vector field to be Killing in almost Yamabe solitons.
problem Conditions for a vector field to be Killing in almost Yamabe solitons.
method Investigation of almost Yamabe solitons on compact and non-compact manifolds.
result Sufficient conditions for the defining conformal vector field to be Killing.
Paper shows constant σk-curvature for quasi k-Yamabe solitons.
problem Understanding constant curvature in quasi k-Yamabe solitons.
method Analyzes conditions for solitons to be gradient and constant curvature.
result Compact quasi k-Yamabe solitons have constant σk-curvature.