arXiv research

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

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

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

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

Inequality found for a specific equation on 5D manifolds.

Study rigidity on CR Yamabe equation on Sasakian manifolds.

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

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 …

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

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

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

Study degenerate solutions on product of spheres using bifurcation theory.

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

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

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

Sharp decay found for solutions of a specific equation in Lie groups.

Defines contact structures on Heisenberg groups for geometric interpretation.

New flow on compact manifolds solves Yamabe equation.

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

In this paper we consider the functional whose critical points are solutions of the fractional CR Yamabe type equation on the sphere. We firstly study the behavior of the Palais-Smale sequences characterizing the bubbling phenomena and therefore we prove a multiplicity type result by showing the existence of infinitely…

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

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.

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…

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…

The paper constructs solutions to a critical Dirac equation on spheres.

Global well-posedness and asymptotic convergence for vacuum Einstein's equations proved.

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

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…

In the work of Ammann, Dahl and Humbert it has turned out that the Yamabe invariant on closed manifolds is a bordism invariant below a certain threshold constant. A similar result holds for a spinorial analogon. These threshold constants are characterized through Yamabe-type equations on products of spheres with rescal…

The paper studies $*$-$η$-Ricci-Yamabe solitons on $α$-Cosymplectic manifolds.

The mixed scalar curvature of a foliated Riemannian manifold, i.e., an averaged mixed sectional curvature, has been considered by several geometers. We explore the Yamabe type problem: to prescribe the constant mixed scalar curvature for a foliation by a conformal change of the metric in normal directions only. For a h…

This paper studies non-compactness in spinorial Yamabe-type problems on manifolds.

The paper solves spinorial Yamabe-type problems on spheres, with applications in geometry.

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

We construct new type II ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to a tower of two spheres. Their curvature operator changes sign. We allow two time-dependent parameters in our ansatz. We use perturbation theory, via fixed point arguments,…

Optimal Liouville theorem for half-Euclidean space equations.

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 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…

We consider a closed cohomogeneity one Riemannian manifold $(M^n,g)$ of dimension $n\geq 3$. If the Ricci curvature of $M$ is positive, we prove the existence of infinite nodal solutions for equations of the form $-Δ_g u + λu = λu^q$ with $λ>0$, $q>1$. In particular for a positive Einstein manifold which is of cohomog…

Let (M,g) be a compact manifold of dimension n greater or equals to 3. We suppose that g is a given metric in a precised Sobolev space and there is a point P in M and d>o such that g is smooth on the ball B(P,d). We define the second Yamabe invariant with singularities a the minimum of the second eigenvalue of the sing…

In this paper we consider Yamabe type problem for higher order curvatures on manifolds with totally geodesic boundaries. We prove local gradient and second derivative estimates for solutions to the fully nonlinear elliptic equations associated with the problems.

Solves nonlinear problems on metric structures through eigenvalue counting.

Constructs singular Yamabe solutions via equivariant reduction.

Paper proves uniqueness of Type II Yamabe metrics on manifolds.

We consider a closed Riemannian manifold $(M^n ,g)$ of dimension $n\geq 3$ and study positive solutions of the equation $-Δ_g u + λu = λu^q$, with $λ>0$, $q>1$. If $M$ supports a proper isoparametric function with focal varieties $M_1$, $M_2$ of dimension $d_1 \geq d_2$ we show that for any $q<\frac{ n-d_2+2 }{n - d_2…

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, …

Study of conformal logarithmic Laplacian on sphere, connecting Yamabe problems and Sobolev spaces.