In 2023, H. Brezis [2] published a list of his “favorite open problems”, which he described as challenges he had “raised throughout his career and has resisted so far”. In this paper, we shall provide a partial answer to this question by presenting the existence of signchanging solutions to the equation whenever the parameter is small enough. Our construction is based on the building blocks of Del Pino-Musso-Pacard-Pistoia sign-changing solutions to Yamabe problem.

We prove the existence of a homogeneous singular solution of the critical equation $$-Δu = u^{\frac{Q+2}{Q-2}}$$ on the Heisenberg group $H^n$, where $Q$ is the \textit{homogeneous dimension}. In order to do this, we introduce a suitable concept of normal curvature for hypersurfaces. Furthermore we study the bifurcation of non-homogeneous solutions from the homogeneous one.

Abstract In this paper, we study the multiplicity of solutions to the Yamabe problem on product manifolds. Specifically, we consider the product of a compact Riemannian manifold without boundary and with zero scalar curvature, and a compact Riemannian manifold with boundary, also with zero scalar curvature and constant mean curvature on the boundary. Our first objective is to demonstrate the existence of a sequence of bifurcation instants for a family of solutions to the Yamabe problem on the product manifold, using classical bifurcation theory. Our second objective is to classify a finite number of degeneracy instants that remained undecidable called neutral instants, using equivariant bifurcation techniques.

Some results on the weighted Yamabe problem with or without boundary

Abstract We introduce a sequence of conformally invariant scalar curvature quantities, defined along the conformal infinity of a conformally compact (CC) manifold, that measure the failure of a CC metric to have constant negative scalar curvature in the interior, i.e. its failure to solve the singular Yamabe problem. Indeed, these ‘CC boundary curvature scalars’ compute canonical expansion coefficients for singular Yamabe metrics. Residues of their poles yield obstructions to smooth solutions to the singular Yamabe problem and thus, in particular, give an alternate derivation of generalized Willmore invariants. Moreover, in a given dimension, the critical CC boundary scalar characterizes the image of a Dirichlet-to-Neumann map for the singular Yamabe problem. We give explicit formulæ for the first five CC boundary curvature scalars required for a global study of four dimensional singular Yamabe metrics, as well as asymptotically de Sitter spacetimes.

Abstract Blowup analysis is crucial in studying many geometric PDEs, such as the Yamabe problem. In contrast to single equations, the blowup analysis of systems of differential equations reveals new phenomena. Toda systems, which extend the Liouville equation to systems based on simple Lie algebras, exhibit a broader range of blowup behaviors, presumed to be related to the Weyl group. We construct concrete examples of solutions to the Toda system that demonstrate blowup masses corresponding to elements of the Weyl group.

Infinitely many solutions for a boundary Yamabe problem

Transformation optics establishes an equivalence relationship between gradient media and curved space, unveiling intrinsic geometric properties of gradient media. However, this approach based on curved spaces is concentrated on two-dimensional manifolds, namely, curved surfaces. In this Letter, we establish an intrinsic connection between three-dimensional manifolds and three-dimensional gradient media in transformation optics by leveraging the Yamabe problem and Ricci scalar curvature-a measure of spatial curvature in manifolds. The invariance of the Ricci scalar under conformal mappings is proven. Our framework is validated through the analysis of representative conformal optical lenses.

On a perturbed Yamabe problem with mixed boundary conditions

abstract: The finite-rank Lieb-Thirring inequality provides an estimate on a Riesz sum of the $N$ lowest eigenvalues of a Schr\"odinger operator $-\Delta-V(x)$ in terms of an $L^p(\mathbb{R}^d)$ norm of the potential $V$. We prove here the existence of an optimizing potential for each $N$, discuss its qualitative properties and the Euler--Lagrange equation (which is a system of coupled nonlinear Schr\"odinger equations) and study in detail the behavior of optimizing sequences. In particular, under the condition $\gamma>\max\{0,2-d/2\}$ on the Riesz exponent in the inequality, we prove the compactness of all the optimizing sequences up to translations. We also show that the optimal Lieb-Thirring constant cannot be stationary in $N$, which sheds a new light on a conjecture of Lieb-Thirring. In dimension $d=1$ at $\gamma=3/2$, we show that the optimizers with $N$ negative eigenvalues are exactly the Korteweg-de Vries $N$-solitons and that optimizing sequences must approach the corresponding manifold. Our work also covers the critical case $\gamma=0$ in dimension $d\geq3$ (Cwikel-Lieb-Rozenblum inequality) for which we exhibit and use a link with invariants of the Yamabe problem.

Abstract A version of the singular Yamabe problem in smooth domains in a closed manifold yields complete conformal metrics with negative constant scalar curvatures. In this paper, we study the blow-up phenomena of Ricci curvatures of these metrics on domains whose boundary is close to a certain limit set of a lower dimension. We will characterize the blow-up set according to the Yamabe invariant of the underlying manifold. In particular, we will prove that all points in the lower dimension part of the limit set belong to the blow-up set on manifolds not conformally equivalent to the standard sphere and that all but one point in the lower dimension part of the limit set belong to the blow-up set on manifolds conformally equivalent to the standard sphere. In certain cases, the blow-up set can be the entire manifold. We will demonstrate by examples that these results are optimal.

Abstract In this paper we investigate the existence of singular solutions to the conformal Dirac–Einstein system. Because of its conformal invariance, there are many similarities with the classical construction of singular solutions for the Yamabe problem. We construct here a family of singular solutions, on the three dimensional sphere, having exactly two singularities.

We prove the compactness of the set of solutions to the CR Yamabe problem on a compact strictly pseudoconvex CR manifold of dimension three whose blow-up manifolds at every point have positive p-mass. As a corollary, we deduce that compactness holds for CR-embeddable manifolds which are not CR-equivalent to [Formula: see text]. The theorem is proved by blow-up analysis.

Abstract We show that the GJMS operators of a special Einstein product factor as a composition of second‐ and fourth‐order differential operators. In particular, our formula applies to the Riemannian product . We also show that there is an integer such that if , then for any special Einstein product , the Green's function for the GJMS operator of order is positive. As a result, these products give new examples of closed Riemannian manifolds for which the ‐Yamabe problem is solvable.

Asymptotic behavior of complete conformal metric near singular boundary

We consider the classical geometric problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a n−dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if n = 3 all the blow-up points are isolated and simple. In this work we prove that, for a linear perturbation, this is not true anymore in low dimensions 4 ≤ n ≤ 7. In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and is a local minimizer of the norm of the trace-free second fundamental form of the boundary.

The source of a whole vast literature, the Yamabe problem stands as an impressive reminder of the power of asking the right question.

Abstract We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 {d>\frac{n-2}{2}} . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.

We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type. Our first existence and uniqueness result concerns those such manifolds which are asymptotically locally hyperbolic. In this context, our result requires only a partial $C^2$ decay of the metric, namely the full decay of the metric in $C^1$ and the decay of the scalar curvature. In particular, no decay of the Ricci curvature is assumed. In our second result we establish that a local volume ratio condition, when combined with negativity of the scalar curvature at infinity, is sufficient for existence of a solution. Our volume ratio condition appears tight. This paper is based on the DPhil thesis of the first author.

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow