arXiv research

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

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

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

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

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

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.

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…

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…

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

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 …

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

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

Study on rigidity of special Riemannian manifolds.

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold with boundary. We consider the Yamabe type problem \begin{equation} \left\{ \begin{array}{ll} -Δ_{g}u+au=0 & \text{ on }M \\ \partial_νu+\frac{n-2}{2}bu= u^{{n\over n-2}\pm\varepsilon} & \text{ on }\partial M \end{array}\right. \end{equation} where $a\in C^1(…

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.

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

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

New solutions found for Yamabe problem on spheres with foliations.

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

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 …

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

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…

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…

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

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 …

Study rigidity on CR Yamabe equation on Sasakian manifolds.

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

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

The conformal properties of complex Finsler metrics are studied. We give a characterization of a compact complex Finsler manifold to be globally conformal Kähler. The critical points of the total holomorphic curvature and total Ricci curvature in the volume preserved conformal classes are studied. The stability of crit…

Solves Neumann problem on CR manifold boundary.

New flow on compact manifolds solves Yamabe equation.

Sharp inequality for compactifying Poincaré-Einstein manifolds.

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

The article provides conditions for unobstructedness of ASD manifolds.

Study solves Yamabe problems on metric measure spaces with or without 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…

Inequality found for a specific equation on 5D manifolds.

This paper classifies solitons under specific tensor conditions.

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positiv…

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

Smooth solutions found for a specific type of Yamabe problem.

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 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…

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

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

We illustrate an example of a generic, positive function K on a Riemannian manifold to be conformally prescribed as the scalar curvature, for which the corresponding Yamabe type L2-gradient flow exhibits non compact flow lines, while a slight modification of it is compact.