arXiv research

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

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

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

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…

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

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…

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.

Motivated by recent progress on a spinorial analogue of the Yamabe problem in the geometric literature, we study a conformally invariant spinor field equation on the $m$-sphere, $m\geq2$. Via variational methods and the spinorial Weierstraß representation, we study the problem of prescribing mean curvature for the imme…

Study spinorial Yamabe problem on product manifolds, proving spike layer solutions.

Paper proves a spinorial version of Aubin's estimate for the Yamabe problem.

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

This work deals with the conformal transformations in six-dimensional spinorial formalism. Several conformally invariant equations are obtained and their geometrical interpretation are worked out. Finally, the integrability conditions for some of these equations are established. Moreover, in the course of the article, …

We describe the different classes of $\mathrm{Spin(7)}$ structures in terms of spinorial equations. We relate them to the spinorial description of $\mathrm{G}_2$ structures in some geometrical situations. Our approach enables us to analyze invariant $\mathrm{Spin(7)}$ structures on quasi abelian Lie algebras.

Refines spinorial Sobolev inequality on sphere, proving stability and new properties of Killing spinors.

We associate to a compact spin manifold M a real-valued invariant τ(M) by taking the supremum over all conformal classes over the infimum inside each conformal class of the first positive Dirac eigenvalue, normalized to volume 1. This invariant is a spinorial analogue of Schoen's $σ$-constant, also known as the smooth …

Paper introduces new fractional Dirac operator and Q-curvature.

New spinorial field equation reveals geometric properties of Sasaki manifolds.

Proves principles and estimates for initial data sets in Einstein equations.

Let $(M,g,\si)$ be a compact Riemannian spin manifold of dimension $\geq 2$. For any metric $\tilde g$ conformal to $g$, we denote by $\tildeλ$ the first positive eigenvalue of the Dirac operator on $(M,\tilde g,\si)$. We show that $$\inf_{\tilde{g} \in [g]} \tildeλ\Vol(M,\tilde g)^{1/n} \leq (n/2) \Vol(S^n)^{1/n}.$$ T…

The concept of pure spinor is generalized, giving rise to the notion of pure subspaces, spinorial subspaces associated to isotropic vector subspaces of non-maximal dimension. Several algebraic identities concerning the pure subspaces are proved here, as well as some differential results. Furthermore, the freedom in the…

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, …

Let (M^n,g) be a Riemannian spin manifold. The basic equations in supergravity models of type IIa string theory with 4-form flux involve a 3-form T, a 4-form F, a spinorial covariant derivative \nabla depending on \nabla^g, T, F, and a \nabla-parallel spinor field Ψ. We classify and construct many explicit families of …

We present a uniform description of $\mathrm{SU}(3)$-structures in dimension $6$ as well as $G_2$-structures in dimension $7$ in terms of a characterising spinor and the spinorial field equations it satisfies. We apply the results to hypersurface theory to obtain new embedding theorems, and give a general recipe for bu…

We construct exact solutions of the Einstein-Dirac equation, which couples the gravitational field with an eigenspinor of the Dirac operator via the energy-momentum tensor. For this purpose we introduce a new field equation generalizing the notion of Killing spinors. The solutions of this spinorial field equation are c…

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 $η$-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.