arXiv research

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…

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

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

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

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

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 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…

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 …

The paper proves conditions under which certain geometric structures are rigid.

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

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 on solutions of Yamabe-type equations on projective spaces.

The paper establishes lower bounds for the relative volume of Poincaré-Einstein manifolds.

New solutions found for Yamabe problem on spheres with foliations.

New findings on static near horizon geometries and quasi-Einstein manifolds, including rigidity results for negative cosmological constant.

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…

The study explores metrics with constant curvature on compact manifolds.

We study a conformal flow for compact Riemannian manifolds of dimension greater than two with boundary. Convergence to a scalar-flat metric with constant mean curvature on the boundary is established in dimensions up to seven, and in any dimensions if the manifold is spin or if it satisfies a generic condition.

Let (M,g) be a compact Riemannian manifold with boundary. This paper addresses the Yamabe-type problem of finding a conformal scalar-flat metric on M, which has the boundary as a constant mean curvature hypersurface. When the boundary is umbilic, we prove an existence theorem that finishes some remaining cases of this …

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

Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. This involves a blow-up analysis of a Yamabe-type equation with critical Sobolev e…

Let $G=(V,E)$ be a finite connected weighted graph, and assume $1\leqα\leq p\leq q$. In this paper, we consider the following $p$-th Yamabe type equation $-Δ_pu+hu^{q-1}=λfu^{α-1}.$ on $G$, where $Δ_p$ is the $p$-th discrete graph Laplacian, $h\leq0$ and $f>0$ are real functions defined on all vertices of $G$. Instea…

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

Study finds solutions for complex problems on non-compact manifolds.

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

Our aim in this paper is to study local rigidity for metrics defined on a compact manifold $M$ with boundary satisfying constant scalar curvature on $M$ and constant mean curvature on $\partial M$. We present some geometrical hypotheses ensuring local rigidity for both, the general Riemannian and the warped metric case…

Study degenerate solutions on product of spheres using bifurcation theory.

Proves product metrics are Yamabe metrics under small flat torus conditions.

In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension $n+1>3$. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold $(X,g)$ is less than $\ndemi -1$ if and only …

The abstract discusses nonuniqueness results for specific Riemannian invariants.

Study on rigidity of special Riemannian manifolds.

Solves Neumann problem on CR manifold boundary.

In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary. The last problem will be called Escobar-Riemann mapping type problem. For this purpose, we consider the generalization o…

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

The paper proves stability for scalar-flat metrics on manifolds with boundary.

Let (M, g) be an (n + 1)-dimensional asymptotically locally hyperbolic (ALH) manifold with a conformal compactification whose conformal infinity is ($\partial$M, [$γ$]). We will first observe that Ch(M, g) $\le$ n, where Ch(M, g) is the Cheeger constant of M. We then prove that, if the Ricci curvature of M is bounded f…

Simplified Obata-Vétois argument for Einstein manifolds with nonnegative scalar curvature.

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…

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

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…

Study on eigenvalue rate of geodesic balls in asymptotically hyperbolic Einstein manifolds.

Study rigidity on CR Yamabe equation on Sasakian manifolds.

We complement a recent work on the stability of fixed points of the CMC-Einstein-$Λ$ flow. In particular, we modify the utilized gauge for the Einstein equations and remove a restriction on the fixed points whose stability we are able to prove by this method, and thereby generalize the stability result. In addition, we…

We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.

New flow on compact manifolds solves Yamabe equation.

Sharp inequality for compactifying Poincaré-Einstein manifolds.

The study classifies warped products with harmonic curvature on surfaces, showing two possibilities for the metric.

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 $C^1$). To solve this problem, we start t…