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.
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.
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
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.
The Yamabe problem in compact closed Riemannian manifolds is concerned with finding a metric with constant scalar curvature in the conformal class of a given metric. This problem was solved by the combined work of Yamabe, Trudinger, Aubin, and Schoen. In particular, Aubin solved the case when the Riemannian manifold is…
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.
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…
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.
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 ν-entropy. This problem reduces in all dimensions on Euclidean space to the characterization of the minimizers of the family …
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.
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.
Smooth solutions found for a specific type of Yamabe problem.
problem Regularity of viscosity solutions to the σk-Yamabe problem in the negative cone. method Analysis of Lipschitz viscosity solutions with specific assumptions.
result Existence and smoothness of solutions away from a negligible set.
Study on complex manifolds introduces a new deformation of the Yamabe problem.
problem Yamabe-type problems on compact Hermitian manifolds.
method Introducing a one-parameter Hermitian deformation of the Yamabe problem, defined by adding natural torsion terms to the Riemannian scalar curvature.
result Analysis of criteria for the existence of solutions and discussion of examples.
New solutions found for Yamabe problem on spheres with foliations.
problem Yamabe problem on spheres with singular Riemannian foliations.
method Variational methods, symmetries from foliations, Sobolev embedding theorem, Principle of Symmetric Criticality.
result Existence of sign-changing and positive solutions with specific symmetries.
This paper studies non-compactness in spinorial Yamabe-type problems on manifolds.
problem Non-compactness in spinorial Yamabe-type problems on manifolds.
method Analysis of two specific models on the manifold \(S^m\).
result The solution set is not compact for certain perturbations of the background metric.
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.
We construct Delaunay-type solutions for the fractional Yamabe problem with an isolated singularity $(-Δ)^γw = c_{n, γ} w^{\frac{n+2γ}{n-2γ}}, w>0 \ \mbox{in} \ \mathbb{R}^n \backslash \{0\}$ We follow a variational approach, in which the key is the computation of the fractional Laplacian in polar coordinates.
Solutions grow for a special type of math problem on curved spaces.
problem Yamabe problem on manifolds with umbilic boundary
method Building blowing-up solutions for a supercritical perturbation
result Existence of solutions for n>7 and non-vanishing Weyl tensor
Study of conformal logarithmic Laplacian on sphere, connecting Yamabe problems and Sobolev spaces.
problem Yamabe-type problems and Sobolev spaces on the sphere.
method Detailed spectral analysis, conformal invariance, and Hilbert space introduction.
result Established precise connection between sphere and \(\mathbb{R}^N\) logarithmic Laplacian.
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 Neumann problem on CR manifold boundary.
problem Neumann problem on CR manifold boundary.
method Analyzes CR Yamabe operator and contact forms.
result Solves Neumann problem and finds contact form.
Defines contact structures on Heisenberg groups for geometric interpretation.
problem Finding a geometric interpretation for the Yamabe equation on Heisenberg groups.
method Defines contact structures of Heisenberg type, introduces a natural connection, and computes conformal scalar curvature.
result Establishes equivalence between contact Riemannian manifolds and contact structures of Heisenberg type.
We consider a spinorial Yamabe-type problem on open manifolds of bounded geometry. The aim is to study the existence of solutions to the associated Euler-Lagrange-equation. We show that under suitable assumptions such a solution exists. As an application, we prove that existence of a solution implies the conformal Hija…
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.
Study on spinor field equation on spheres, focusing on blow-up analysis.
problem Spinorial Yamabe problem on spheres.
method Variational methods, blow-up analysis.
result Blow-up profile for the spinorial Yamabe type equation on Sm. 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 existence and uniqueness of solutions for Yamabe problem on non-compact manifolds with negative curvature.
problem Existence and uniqueness of solutions for the Yamabe problem on non-compact manifolds of negative curvature type.
method Used partial C2 decay of the metric and local volume ratio condition to establish existence and uniqueness results. result Established existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type.
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.
We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some gene…
This paper classifies Kähler manifolds with specific Einstein-type properties.
problem Classifying gradient Einstein-type Kähler manifolds with α=0. method Unified framework of Einstein-type manifolds, focusing on classification with α=0. result Complete classification of non-trivial, complete gradient Einstein-type Kähler manifolds with α=0. 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.
The paper constructs many ancient solutions to the Yamabe flow on spheres.
problem Ancient solutions to the Yamabe flow on spheres.
method Non-radial inner--outer gluing scheme, conformal invariance, weighted Hölder estimates.
result Uncountably many non-rotationally symmetric ancient solutions.
We study the convergence of complete non-compact conformally flat solutions to the Yamabe flow to Yamabe steady solitons. We also prove the existence of Type II singularities which develop at either a finite time T or as t→+∞.
In the first part of this thesis, we study the Yamabe problem with singularities, that we can announce as follow: Given a compact Riemannian manifold (M,g), find a constant scalar curvature metric, conformal to g, when g has not necessarily the usual regularity (it can be C1). To solve this problem, we start t…
In this paper we study the local behaviour of admissible metrics in the k-Yamabe problem on compact Riemannian manifolds (M,g0) of dimension n≥3. For n/2<k<n, we prove a sharp Harnack inequality for admissible metrics when (M,g0) is not conformally equivalent to the unit sphere Sn and that the set of …
Let (M,g) be a compact oriented Riemannian manifold with an incomplete edge singularity. This article shows that it is possible to evolve g by the Yamabe flow within a class of singular edge metrics. As the main analytic step we establish parabolic Schauder-type estimates for the heat operator on certain Hölder spaces …
We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as t→−∞, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.
We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as t→−∞, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.
Paper extends Kodaira dimension's role in Yamabe invariant for most complex surfaces.
problem Determining the sign of Yamabe invariant for compact complex surfaces.
method Analyzing Kodaira dimension and using simplified proof techniques.
result Pattern of Yamabe invariant sign depends on Kodaira dimension for most surfaces.
We prove that the problem of constructing biharmonic conformal maps on a 4-dimensional Einstein manifold reduces to a Yamabe-type equation. This allows us to construct an infinite family of examples on the Euclidean 4-sphere. In addition, we characterize all solutions on Euclidean 4-space and show that there exists a…
A new flow method solves the weighted Yamabe problem with boundary.
problem Solving the weighted Yamabe problem on metric measure spaces with boundary.
method Introduced a Yamabe-type flow with a specific geometric setup.
result Long-time existence and convergence of the flow proved.
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 n=2 and solutions with pointwise decay assumption in n≥3. 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 paper solves spinorial Yamabe-type problems on spheres, with applications in geometry.
problem Existence of solutions to a conformally invariant Dirac equation on spin manifolds.
method Perturbation techniques to establish the existence of solutions.
result Established existence of solutions for the conformally invariant Dirac equation on Sm. 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.
Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.
problem Yamabe problem and Calabi-Yau with torsion metrics for Bismut connection.
method Analysis of Bismut scalar and Ricci curvatures, construction of examples.
result Existence of metrics with constant Bismut scalar curvature.
The paper proves stability for scalar-flat metrics on manifolds with boundary.
problem Stability of scalar-flat metrics on manifolds with boundary.
method Reduced problem to boundary functional and used deficit control.
result Deficit controls distance to minimizing set on manifolds with boundary.
Study on rigidity of special Riemannian manifolds.
problem Rigidity properties of generalized m-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.