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 paper examines Euclidean domains with nearly maximal Yamabe quotients.
problem Understanding domains with nearly maximal Yamabe quotients in Euclidean space.
method Analyzes the properties of domains in R3 with nearly maximal Yamabe quotients, proving conditions for equality and comparing to quasi-conformal maps. result Domains with nearly maximal Yamabe quotients are diffeomorphic to balls and are close to a ball in a metric space sense.
Positive mass theorem and Yamabe equation on CR manifolds
problem Positive mass theorem and Yamabe equation on CR manifolds
method Positive mass theorem and Yamabe equation on CR manifolds
result Positive mass theorem in 3-dimensional CR geometry
Proves existence of Yamabe metrics on conical manifolds with conical points and links.
problem Existence of Yamabe metrics on singular manifolds with conical points and links.
method Derives a counterpart of Aubin's result, uses conical links and Fourier analysis, adds lower-order correction to standard bubbles.
result Derives asymptotic expansions on the Yamabe quotient for generic type metrics.
The study proves CR structures on specific three-manifolds are equivalent to standard structures.
problem Proving CR structures on three-manifolds are equivalent to standard structures.
method Analyzing Yamabe constant and total Q′-curvature to deduce CR equivalence. result Closed CR three-manifolds with certain curvature properties are equivalent to standard structures.
Study solves Yamabe problems on metric measure spaces with or without boundary.
problem Yamabe-type problems on compact metric measure spaces with or without boundary.
method Analyzes uniqueness, characterization, and existence of minimizers.
result Characterizes weighted Yamabe solitons and existence of positive minimizers.
Given a compact Riemannian manifold, with positive Yamabe quotient, not conformally diffeomorphic to the standard sphere, we prove a priori estimates for solutions to the Yamabe problem. We restrict ourselves to the dimensions less than or equal to 7, where the Positive Mass Theorem is known to be true. We also show th…
We let (M^m, g) be a closed smooth Riemannian manifold (m >1) with positive scalar curvature S_g, and prove that the Yamabe constant of (M \times R^n,g+g_E) is achieved by a metric in the conformal class of (g+g_E), where g_E is the Euclidean metric. We also show that the Yamabe quotient of (M \times R^n,g+g_E) is impr…
The paper solves a Yamabe problem involving the quotient of Q-curvature and scalar curvature.
problem Solving a Yamabe problem for the quotient of Q-curvature and scalar curvature.
method Introduced a new Sobolev inequality and a new Yamabe constant to prove the existence of solutions.
result Existence of solutions to the Yamabe problem for the quotient of Q-curvature and scalar curvature.
Global convergence proved for Gursky-Malchiodi Q-curvature flow in dimensions n≥5.
problem Resolving the constant Q-curvature problem in dimensions n≥5. method Established a non-local version of the Łojasiewicz-Simon inequality for the Paneitz-Sobolev quotient, constructed test bubbles, and derived a stability inequality for the Paneitz-Sobolev quotient.
result Global convergence of the flow for arbitrary initial energy under the same positivity assumptions.
We obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of S1 inside Sm, m≥5, that are conformal to the round (incomplete) metric and "periodic" in the sense of…
We prove that a n-dimensional, 4≤n≤6, compact gradient shrinking Ricci soliton satisfying a Ln/2-pinching condition is isometric to a quotient of the round Sn. The proof relies mainly on sharp algebraic curvature estimates, the Yamabe-Sobolev inequality and an improved rigidity result f…
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 consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension n≥3. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric g, there is a conformally equivalent asymptotically flat scal…
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. 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…
Study on discrete Gaussian curvature for polyhedral surfaces.
problem Discretization of Gaussian curvature for polyhedral surfaces.
method Generalization of discrete conformal equivalence to define discrete Gaussian curvature and classify polyhedral surfaces.
result Existence of polyhedral surfaces with constant discrete Gaussian curvature in every discrete conformal class.
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.