arXiv research

Study different types of G_2-structures on Lie groups.

In this note we study the conformal metrics of constant $Q$ curvature on closed locally conformally flat manifolds. We prove that for a closed locally conformally flat manifold of dimension $n\geq 5$ and with Poincarë exponent less than $\frac {n-4}2$, the set of conformal metrics of positive constant $Q$ and positive …

The article classifies G2-structures with conformally flat metrics.

In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_α$ were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold …

We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing $L^{2}-$form. Moreover, we prove vanishing theorems for closed and co-closed conformal Killing $L^{2}-$forms on some complete Riemanni…

Proves properties of 4-manifolds with scalar curvature constraints.

Local Lorentzian theorem preserves metrics or makes them flat.

We study locally conformally Berwald metrics on closed manifolds which are not globally conformally Berwald. We prove that the characterization of such metrics is equivalent to characterizing incomplete, simply-connected, Riemannian manifolds with reducible holonomy group whose quotient by a group of homotheties is clo…

Locally conformally product Lie algebras are characterized and constructed.

Study explores solvable Lie groups' actions on closed Lorentzian manifolds.

A locally conformally symplectic (LCS) form is an almost symplectic form $ω$ such that a closed one-form $θ$ exists with $dω= θ\wedge ω$. We present a version of the well-known result of Darboux and Weinstein in the LCS setting and give an application concerning Lagrangian submanifolds.

In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvatur…

The study quantizes energy for curves in symplectic manifolds.

We study the condition in which G2-structures are introduced by a non closed four-form, although they are satisfying locally conformal conditions.All solutions are found in the case when the Lee form of G2-structures is non-zero and gintroduces seven-dimensional Lie algebras, The main results are given in preposition1 …

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

We prove the existence of smooth closed hypersurfaces of prescribed mean curvature homeomorphic to $S^n$ for small $n, n\le6$, provided there are barriers.

Given a closed Riemannian manifold $(M, g_M)$ of dimension $n \geq 3$, we prove the existence of a conformally compact Einstein metric $g_{+}$ defined on a collar neighborhood $M \times (0,1]$ whose conformal infinity is $[g_M]$.

We characterize compact locally conformally Kähler (l.c.K.) manifolds under the assumption of a purely conformal, holomorphic circle action. As an application, we determine the structure of the compact l.c.K. manifolds with parallel Lee form. We introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffe…

The Weinstein conjecture is extended to a new class of manifolds.

New geometric structures defined in contact metric geometry.

Derives formulas for extrinsic Paneitz operator and $Q$-curvature in general dimensions.

We study conditions for which the mapping torus of a 6-manifold endowed with an $SU(3)$-structure is a locally conformal calibrated $G_2$-manifold, that is, a 7-manifold endowed with a $G_2$-structure $\varphi$ such that $d \varphi = - θ\wedge \varphi$ for a closed non-vanishing 1-form $θ$. Moreover, we show that if $(…

We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert-Einstein functional on such conformal classes, near homogeneous metrics. Both bifurcation and local…

Improved multivariate conformal prediction by standardizing residuals.

We introduce the concept of a Clifford-Weyl structure on a conformal manifold, which consists of an even Clifford structure parallel with respect to the tensor product of a metric connection on the Clifford bundle and a Weyl structure on the manifold. We show that the Weyl structure is necessarily closed except for som…

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

We study the conformal geometry of timelike curves in the (1+2)-Einstein universe, the conformal compactification of Minkowski 3-space defined as the quotient of the null cone of $\mathbb{R}^{2,3}$ by the action by positive scalar multiplications. The purpose is to describe local and global conformal invariants of time…

In this article, we found a connection between Brown-York mass and the first Dirichlet Eigenvalue of a Schrödingier type operator. In particular, we proved a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigated compactly conformal defo…

The paper introduces a new method to create stable ergodic actions on higher-dimensional manifolds.

We investigate the structure of conformal $C$-spaces,a class of Riemmanian manifolds which naturally arises as aconformal generalisation of the Einstein condition. A basic question is when such a structure is closed, or equivalently locally conformally Cotton. In dimension 4 we obtain a full answer to this question and…

In low dimensions, minimizers for the second conformal eigenvalue do not exist near the round sphere.

This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.

The paper studies equations in conformal geometry with gradient and existence results.

Locally conformal SKT structures are introduced and studied on Lie groups and their compact quotients.

The paper proves ellipsoids are the only centroaffine Tchebychev hyperovaloids.

Classifies Weyl structures on compact conformal manifolds with special holonomy.

In this paper, we study generic conformally flat hypersurfaces in the Euclidean $4$-space $\mathbb{R}^4$ using the framework of Möbius geometry. First, we classify locally the generic conformally flat hypersurfaces with closed Möbius form under the Möbius transformation group of $\mathbb{R}^4$. Such examples come from …

In this paper, we firstly extend Theorem 5.1.1 in \cite {Helein} due to Hélein to a rescaled branched conformal immersed sequence(c.f. Theorem 1.5). By virtue of this local convergence theorem, we study the blowup behavior of a sequence of branched conformal immersions of closed Riemannian surface in $\mathbb{R}^{n}$ w…

A locally metric connection on a smooth manifold $M$ is a torsion-free connection $D$ on $TM$ with compact restricted holonomy group $\mathrm{Hol}_0(D)$. If the holonomy representation of such a connection is irreducible, then $D$ preserves a conformal structure on $M$. Under some natural geometric assumption on the li…

The study characterizes and classifies specific types of manifolds using conformal and quasi-Einstein properties.

ABF-T-GLCP forecasts and calibrates uncertainty for multivariate nonstationary time series.

Study on conformal harmonic coordinates on manifolds, proving existence and properties.

Given a closed subset $\La$ of the open unit ball $B_1\subset \real^n$, $n \geq 3$, we will consider a complete Riemannian metric $g$ on $\bar{B_1} \setminus \La$ of constant scalar curvature equal to $n(n-1)$ and conformally related to the Euclidean metric. In this paper we prove that every closed Euclidean ball $\bar…

A locally conformally Kähler (LCK) manifold $M$ is one which is covered by a Kähler manifold $\tilde M$ with the deck transform group acting conformally on $\tilde M$. If $M$ admits a holomorphic flow, acting on $\tilde M$ conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vais…

In this paper, we employ a nonlocal $Q$-curvature flow inspired by Gursky-Malchiodi's work \cite{gur_mal} to solve the prescribed $Q$-curvature problem on a class of closed manifolds: For $n \geq 5$, let $(M^n,g_0)$ be a smooth closed manifold, which is not conformally diffeomorphic to the standard sphere, satisfying e…

An intrinsic definition in terms of conformal capacity is proposed for the conformal type of a Carnot--Carathéodory space (parabolic or hyperbolic). Geometric criteria of conformal type are presented. They are closely related to the asymptotic geometry of the space at infinity and expressed in terms of the isoperimetri…

Introduces conformal Bach flow and proves its well-posedness and backward uniqueness.

A locally conformally symplectic (LCS) form is an almost symplectic form $ω$ such that a closed one-form $θ$ exists with $dω=θ\wedgeω$. A fiber bundle with LCS fiber $(F, ω,θ)$ is called LCS if the transition maps are diffeomorphisms of $F$ preserving $ω$ (and hence $θ$). In this paper, we find conditions for the total…