arXiv research

Study on solutions of Yamabe-type equations on projective spaces.

Constructs singular Yamabe solutions via equivariant reduction.

We study asymptotic behaviors of positive solutions to the Yamabe equation and the $σ$k-Yamabe equation near isolated singular points and establish expansions up to arbitrary orders. Such results generalize an earlier pioneering work by Caffarelli, Gidas, and Spruck, and a work by Korevaar, Mazzeo, Pacard, and Schoen, …

Positive mass theorem and Yamabe equation on CR manifolds

The paper proves solutions for Yamabe equations on manifolds with boundary.

In this paper, we consider the Yamabe equation on a complete noncompact Riemannian manifold and find some geometric conditions on the manifold such that the Yamabe problem admits a bounded positive solution.

Study on optimal partitions and nodal solutions for the Yamabe equation.

Study on solutions near isolated singularities in 6D Yamabe equation.

$σ_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admis…

The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.

Study bounds derivatives of solutions to a specific equation on domains.

New iterative schemes solve Yamabe-type equations on closed manifolds.

Study rigidity on CR Yamabe equation on Sasakian manifolds.

Study on solutions to spinorial Yamabe equation on manifolds with boundary.

Study on $η$-Ricci-Yamabe solitons on Riemannian submersions.

We study the following $1$-Yamabe equation on a connected finite graph $Δ_1u+g\mathrm{Sgn}(u)=h|u|^{α-1}\mathrm{Sgn}(u),$ where $Δ_1$ is the discrete $1$-Laplacian, $α>1$ and $g, h>0$ are known. We show that the above $1$-Yamabe equation always has a nontrivial solution $u\geq0$, $u\neq0$.

The paper studies geometric structures in perfect fluid spacetimes with specific metrics.

The paper studies global Yamabe flow on AF manifolds, preserving ADM mass.

The paper finds sign-changing solutions for a specific type of elliptic equation.

We prove existence results for nodal solutions of the Yamabe equation that are constant along the level sets of an isoparametric function.

Study on complex manifolds introduces a new deformation of the Yamabe problem.

We prove the existence of a solution of the Yamabe equation on complete manifolds with finite volume and positive Yamabe invariant. In order to circumvent the standard methods on closed manifolds which heavily rely on global (compact) Sobolev embeddings we approximate the solution by eigenfunctions of certain conformal…

We review recent compactness and non-compactness results for the Yamabe equation. We also discuss the asymptotic behavior of the parabolic Yamabe flow.

Study on positive solutions of Yamabe-type equation on spheres.

New Liouville-type results for CR Yamabe equation in Heisenberg group.

Inequality found for a specific equation on 5D manifolds.

The paper studies Yamabe metrics and stability in Riemannian manifolds.

Study on blow-up behavior of sign-changing solutions for Yamabe equation.

A complete solution to the quaternionic contact Yamabe equation on the qc sphere of dimension $4n+3$ as well as on the quaternionic Heisenberg group is given. A uniqueness theorem for the qc Yamabe problem in a compact locally 3-Sasakian manifold is shown.

Study on spinor field equation on spheres, focusing on blow-up analysis.

We consider Yamabe-type equations on the Riemannian product of constant curvature metrics on $\textbf{S}^n \times\textbf{ S}^n$, and study solutions which are invariant by the cohomogeneity one diagonal action of $O(n+1)$. We obtain multiplicity results for both positive and nodal solutions. In particular we prove the …

We give a survey of various compactness and non-compactness results for the Yamabe equation. We also discuss a conjecture of Hamilton concerning the asymptotic behavior of the parabolic Yamabe flow.

We study positive solutions of the Yamabe equation with isolated singularity and prove the existence of solutions with prescribed asymptotic expansions near singular points and an arbitrarily high order of approximation.

Researchers found explicit solutions to a complex equation in advanced geometry.

The paper finds bounds for a spinorial equation and applies it to a Bär-Hijazi-Lott invariant.

In this note we prove an existence result for the Einstein conformal constraint equations for metrics with vanishing Yamabe invariant assuming that the TT-tensor is small in $L^2$.

We show that solutions of the Yamabe equation on certain n-dimensional non-compact Riemannian manifolds which are bounded and L^p for p=2n/(n-2) are also L^2. This L^p-L^2-implication provides explicit constants in the surgery-monotonicity formula for the smooth Yamabe invariant in a previous article of the authors. As…

We study the Yamabe problem on open manifolds of bounded geometry and show that under suitable assumptions there exist Yamabe metrics, i.e. conformal metrics of constant scalar curvature. For that, we use weighted Sobolev embeddings.

Let $(M,g)$ be a compact Riemannian manifold of dimension $n \geq 3$. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to $g$ and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.

Study finds infinite sign-changing solutions for a specific equation on manifolds.

For a closed Riemannian manifold of dimension $n\geq 3$ and a subgroup $G$ of the isometry group, we define and study the $G-$equivariant second Yamabe constant and we obtain some results on the existence of $G-$invariant nodal solutions of the Yamabe equation.

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.

Researchers found solutions to a complex equation on spheres, overcoming a key difficulty.

We compute the Yamabe invariants for a new infinite class of closed $4$-dimensional manifolds by using a "twisted" version of the Seiberg-Witten equations, the $\mathrm{Pin}^-(2)$-monopole equations. The same technique also provides a new obstruction to the existence of Einstein metrics or long-time solutions of the no…

New flow on compact manifolds solves Yamabe equation.

Among all conformal classes of Riemannian metrics on ${\Bbb CP}_2$, that of the Fubini-Study metric is shown to have the largest Yamabe constant. The proof, which involves perturbations of the Seiberg-Witten equations, also yields new results on the total scalar curvature of almost-Kähler 4-manifolds.

Paper proves existence of minimum energy solutions in 5D contact spin manifolds.

For a sequence of blow up solutions of the Yamabe equation on non-locally confonformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points…