We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf fo…
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48 results for “Yamabe type PDE”
The toric CR Yamabe problemmath.DG
Study of CR Yamabe problem on toric contact manifolds.
problem CR Yamabe problem on toric contact manifolds.
method Equivalence to boundary value problem for elliptic PDE.
result Many contact toric manifolds admit CR structures with distinct CR Yamabe invariants.
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 solves two cases of the asymptotically flat scalar-flat Yamabe problem with boundary.
problem Finding asymptotically flat scalar-flat metrics on manifolds with boundary.
method Solving elliptic PDEs with sub- and supersolutions.
result Existence of conformally equivalent asymptotically flat scalar-flat metrics.
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.
Solves Yamabe problem for Sobolev-class asymptotically hyperbolic manifolds.
problem Solving the Yamabe problem for specific types of asymptotically hyperbolic manifolds.
method Introduces new function spaces and uses Fredholm theorems for elliptic operators.
result Solves the Yamabe problem for asymptotically hyperbolic manifolds with Sobolev-class metrics.
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, -curvature problems, and isoperimetric-ratio type problems.
result Product metrics are Yamabe metrics for sufficiently small flat tori.
Paper solves Gauduchon scalar curvature problem on almost Hermitian manifolds.
problem Prescribed Gauduchon scalar curvature problem on almost Hermitian manifolds.
method Reduced to solving a semi-linear partial differential equation with exponential nonlinearity using super and sub-solution method.
result Existence of solution depends on the sign of a constant associated to Gauduchon degree.
In this paper we study elliptic PDEs on compact Gromov-Hausdorff limit spaces of Riemannian manifolds with lower Ricci curvature bounds. In particular we establish continuities of geometric quantities, which include solutions of Poisson's equations, eigenvalues of Schrodinger operators, generalized Yamabe constants and…
Study on Yamabe flow convergence and singularities.
problem Understanding convergence and singularities in Yamabe flow solutions.
method Analysis of complete non-compact conformally flat solutions to the Yamabe flow.
result Existence of Type II singularities that develop either at a finite time or as .
Multiplicity of Solutions and Degenerate Solutions of Yamabe-Type Equations on Projective Spacesmath.DG
Study on solutions of Yamabe-type equations on projective spaces.
problem Existence and multiplicity of solutions of Yamabe-type equations on projective spaces.
method Investigation of solutions invariant under cohomogeneity one actions of U(n) and Sp(n).
result Existence of degenerate solutions on projective spaces.
Solves a complex mathematical problem on curved surfaces.
problem Boundary Yamabe problem with constant scalar and mean curvature.
method Iterative schemes and perturbation methods to solve nonlinear elliptic PDEs.
result Existence of a real, positive, smooth solution in compact manifolds.
Properties of hypersurface singular sets of solutions to the -Yamabe equation in the negative conemath.AP
Study properties of solutions with singularities in the negative cone.
problem Properties of solutions with singularities in the negative cone.
method Proved PDE for trace and normal derivatives, showed hypersurface is minimal for k=2.
result Hypersurface is minimal for k=2 and satisfies certain PDE.
New ancient compact solutions found for Yamabe flow.
problem Finding compact solutions to the Yamabe flow.
method Constructed rotationally symmetric ancient compact solutions.
result Found type I ancient compact solutions converging to self-similar solutions.
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.
The study of the -th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called curvature, has produced many fruitful results in conformal geometry in recent years. In these studies, the deforming conformal factor is considered to be a solution of a fully nonlinea…
Study shows how solutions to Yamabe flow can develop Type II singularities.
problem Existence and detailed analysis of Type II singularities in Yamabe flow.
method Detailed asymptotic analysis and blow-up rate calculation.
result Yamabe flow solutions can converge to a steady soliton after blow-up.
Study on solutions of Yamabe-type equations on sphere products, proving nodal solutions.
problem Yamabe-type equations on sphere products.
method Cohomogeneity one diagonal action of O(n+1) to find solutions.
result Existence of nodal solutions on sphere products.
Generalizes Aubin's result for Yamabe-type problem on smooth metric measure spaces.
problem Solving Yamabe-type problem on smooth metric measure spaces.
method Generalization of Aubin's result for nonlocally conformally flat manifolds with dimension ≥ 6 and parameter m close to nonnegative integers.
result Generalization of Aubin's result for Yamabe-type problem.
A gluing construction of singular solutions for a fully non-linear equation in conformal geometrymath.DG
Paper studies solutions to a specific equation in conformal geometry with singular sets.
problem Singular solutions to a fully non-linear equation in conformal geometry.
method Uses a classical gluing method adapted to the fully non-linear setting.
result Shows the classical gluing method can be applied to the --Yamabe equation.
This paper classifies Kähler manifolds with specific Einstein-type properties.
problem Classifying gradient Einstein-type Kähler manifolds with .
method Unified framework of Einstein-type manifolds, focusing on classification with .
result Complete classification of non-trivial, complete gradient Einstein-type Kähler manifolds with .
New iterative schemes solve Yamabe-type equations on closed manifolds.
problem Solving Yamabe-type equations on closed manifolds.
method Double iterative scheme and local variational method.
result Reproof of classical Yamabe problem as a consequence of Yamabe-type equations.
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.
Survey of non-uniqueness in anisotropic Calderón problem with disjoint boundary data.
problem Non-uniqueness in anisotropic Calderón problem with disjoint boundary measurements.
method Analysis of conformal metrics and nonlinear elliptic PDEs.
result Existence of infinite conformal metrics with identical boundary measurements on disjoint sets.
The paper finds sign-changing solutions for a specific type of elliptic equation.
problem Existence of sign-changing solutions for a Yamabe type equation.
method Investigates a critical elliptic equation with a Yamabe type operator on a compact manifold with boundary.
result Existence of sign-changing solutions assured under certain geometric conditions.
Proves biharmonic maps on 4D Einstein manifolds relate to Yamabe-type equations.
problem Constructing biharmonic conformal maps on 4D Einstein manifolds.
method Reduces biharmonic problem to Yamabe-type equations.
result Characterizes all solutions and constructs infinite family of examples.
We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as , to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.
A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem
problem Defining and analyzing a mass-type invariant for smooth metric measure spaces
method Defining a mass-type quantity and showing its geometric invariance properties
result The mass-type quantity has a close relation with the fractional Yamabe problem and the relevant Green's function
New solutions found for fractional Yamabe problem with singularities.
problem Fractional Yamabe problem with isolated singularities.
method Variational approach, focusing on fractional Laplacian computation in polar coordinates.
result Delaunay-type solutions constructed for the fractional Yamabe problem.
The paper proves existence theorems for fractional Yamabe problem on conformal infinity.
problem Existence of solutions to fractional Yamabe problem on conformal infinity.
method Various geometric assumptions on the asymptotically hyperbolic manifold and its conformal infinity.
result Simpler geometric restrictions for existence results compared to previous works.
New Liouville-type results for CR Yamabe equation in Heisenberg group.
problem Characterizing solutions to CR Yamabe equation in Heisenberg group.
method Integral estimates combined with divergence formula.
result Liouville-type results for bounded solutions in and solutions with pointwise decay assumption in .
On a Riemannian compact manifold, we give existence and multiplicity results for solutions of elliptic PDE by introducing isometry invariances. When the groups we used have finite orbits, we get multiplicity results for equations with the classical critical Sobolev exponent, for instance the Yamabe equation. When there…
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.
Study finds solutions for complex problems on non-compact manifolds.
problem Solving fully nonlinear Yamabe-type problems on non-compact manifolds.
method Existence results for a class of problems, considering both positive and negative cases.
result Explicit examples of manifolds satisfying the hypotheses of the theorems.
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.
Paper studies fractional CR Yamabe equation on sphere, proving multiplicity of solutions.
problem Fractional CR Yamabe equation on sphere solutions.
method Analyzed Palais-Smale sequences to characterize bubbling phenomena and prove multiplicity of solutions.
result Existence of infinitely many solutions to the fractional CR Yamabe equation.
Study on rigidity of special Riemannian manifolds.
problem Rigidity properties of generalized -quasi-Einstein manifolds of Yamabe-type.
method Investigation of rigidity properties for the potential vector field in compact and non-compact settings.
result The potential vector field either vanishes identically or becomes a non-trivial Killing vector field under certain assumptions.
We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.
Study rigidity on CR Yamabe equation on Sasakian manifolds.
problem Proving rigidity on CR Yamabe equation on Sasakian manifolds.
method Using Jerison-Lee's differential identity and integral estimates.
result Prove that the manifold is CR isometric to Heisenberg group \(\mathbb{H}^n\).
Inequality found for a specific equation on 5D manifolds.
problem Finding an inequality for a Yamabe type equation on 5D manifolds.
method Analyzing a Yamabe type equation in 5D manifolds.
result An inequality of type sup x inf found for the Yamabe type equation in 5D.
The CR Yamabe flow converges exponentially to a contact form with flat curvature.
problem Analyzing the CR Yamabe flow with zero invariant.
method Used CR Poincaré inequality and Gagliardo-Nirenberg type interpolation inequality.
result The flow converges exponentially to a contact form with flat pseudo-Hermitian scalar curvature.
Study on -Ricci-Yamabe solitons on Riemannian submersions.
problem Characterizing -Ricci-Yamabe solitons on Riemannian submersions.
method Analyzing conditions for -Ricci-Yamabe solitons on submersions and deriving Laplacian equations.
result Classification of fiber and target manifolds as -Ricci-Yamabe solitons under various conditions.
Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type m…
New PDEs of mixed type emerge in fluid mechanics and geometry.
problem Analysis of nonlinear PDEs of mixed type.
method Through historical problems and recent trends.
result Many PDEs are of mixed type, requiring new analysis.
We describe and partially solve a natural Yamabe-type problem on smooth metric measure spaces which interpolates between the Yamabe problem and the problem of finding minimizers for Perelman's -entropy. This problem reduces in all dimensions on Euclidean space to the characterization of the minimizers of the family …
Study degenerate solutions on product of spheres using bifurcation theory.
problem Existence of degenerate solutions on product of spheres.
method Bifurcation theory, isoparametric functions, Gegenbauer polynomials.
result Existence of degenerate solutions that depend on both factors.
On some conformally invariant fully nonlinear equations, Part II: Liouville, Harnack and Yamabemath.AP
The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding Liouville type problem.
The paper classifies specific types of 3D solitons with unique properties.
problem Classifying 3D complete gradient Yamabe solitons with specific tensors.
method Analyzing properties of solitons with divergence-free Cotton tensor and nonpositive Ricci curvature.
result Classification of 3D complete gradient Yamabe solitons with divergence-free Cotton tensor.