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Register a new userLocally conformal calibrated $G_2$-manifolds
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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 $d varphi = - theta wedge varphi$ for a closed non-vanishing 1-form $theta$. Moreover, we show that if $(M, varphi)$ is a compact locally conformal calibrated $G_2$-manifold with $mathcal{L}_{theta^{#}} varphi =0$, where ${theta^{#}}$ is the dual of $theta$ with respect to the Riemannian metric $g_{varphi}$ induced by $varphi$, then $M$ is a fiber bundle over $S^1$ with a coupled $SU(3)$-manifold as fiber.
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We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity $(M^n , [h])$ of a Poincare-Einstein manifold $(X^{n+1} , g^+ )$ with either $n = 2$ or $n geq 3$ and $(M^n , [h])$ is locally flat - namely $(M, h)$ is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits also a local situation and a global one. Furthermore the latter global situation includes the case of conformal infinities of Poincare-Einstein manifolds of dimension either 2 or of dimension greater than $2$ and which are locally flat, and hence the minimizing technique of Aubin- Schoen in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau which is not known to hold. Using the algebraic topological argument of Bahri-Coron, we bypass the latter positive mass issue and show that any conformal infinity of a Poincare-Einstein manifold of dimension either $n = 2$ or of dimension $n geq 3$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature.
Given $(bar{M},Omega)$ a calibrated Riemannian manifold with a parallel calibration of rank $m$, and $M^m$ an immersed orientable submanifold with parallel mean curvature $H$ we prove that if $cos theta$ is bounded away from zero, where $theta$ is the $Omega$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. In the particular case $M$ is complete with $Ricc^Mgeq 0$ we may replace the boundedness condition on $cos theta$ by $cos thetageq Cr^{-beta}$, when $rto +infty$, where $ 0leqbeta <1 $ and $C > 0$ are constants and $r$ is the distance function to a point in $M$. Our proof is surprisingly simple and extends to a very large class of submanifolds in calibrated manifolds, in a unified way, the problem started by Heinz and Chern of estimating the mean curvature of graphic hypersurfaces in Euclidean spaces. It is based on a estimation of $|H|$ in terms of $costheta$ and an isoperimetric inequality. We also give some conditions to conclude $M$ is totally geodesic. We study some particular cases.
We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product.
It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their rational cohomology algebra already -- an as prominent as particular property in rational homotopy theory. Special interest now lies on exceptional holonomy $G_2$ and $Spin(7)$. In this article we provide a method of how to confirm that the famous Joyce examples of holonomy $G_2$ indeed are formal spaces; we concretely exert this computation for one example which may serve as a blueprint for the remaining Joyce examples (potentially also of holonomy $Spin(7)$). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly Kahler manifolds. A connection between these two results can be found in the fact that both special holonomy and nearly Kahler naturally generalize compact Kahler manifolds, whose formality is a classical and celebrated theorem by Deligne-Griffiths-Morgan-Sullivan.
The goal of this article is to investigate nontrivial $m$-quasi-Einstein manifolds globally conformal to an $n$-dimensional Euclidean space. By considering such manifolds, whose conformal factors and potential functions are invariant under the action of an $(n-1)$-dimensional translation group, we provide a complete classification when $lambda=0$ and $mgeq 1$ or $m=2-n.$
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