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Lorenzo Mazzieri - One of the best experts on this subject based on the ideXlab platform.
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Connected Sum construction for σk yamabe metrics
Journal of Geometric Analysis, 2013Co-Authors: Giovanni Catino, Lorenzo MazzieriAbstract:In this paper we produce families of Riemannian metrics with positive constant σk-curvature equal to \(2^{-k} {n \choose k}\) by performing the Connected Sum of two given compact nondegeneraten-dimensional solutions (M1,g1) and (M2,g2) of the (positive) σk-Yamabe problem, provided 2≤2k
nonlinear elliptic equation. -
Connected Sum Construction for σk-Yamabe Metrics
Journal of Geometric Analysis, 2011Co-Authors: Giovanni Catino, Lorenzo MazzieriAbstract:In this paper we produce families of Riemannian metrics with positive constant σk-curvature equal to \(2^{-k} {n \choose k}\) by performing the Connected Sum of two given compact nondegeneraten-dimensional solutions (M1,g1) and (M2,g2) of the (positive) σk-Yamabe problem, provided 2≤2k
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Connected Sum construction for $\sigma_k$-Yamabe metrics
arXiv: Differential Geometry, 2009Co-Authors: Giovanni Catino, Lorenzo MazzieriAbstract:In this paper we produce families of Riemannian metrics with positive constant $\sigma_k$-curvature equal to $2^{-k} {n \choose k}$ by performing the Connected Sum of two given compact {\em non degenerate} $n$--dimensional solutions $(M_1,g_1)$ and $(M_2,g_2)$ of the (positive) $\sigma_k$-Yamabe problem, provided $2 \leq 2k < n$. The problem is equivalent to solve a second order fully nonlinear elliptic equation.
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Generalized Connected Sum construction for scalar flat metrics
manuscripta mathematica, 2009Co-Authors: Lorenzo MazzieriAbstract:In this paper we construct constant scalar curvature metrics on the generalized Connected Sum $${M = M_1 \, \sharp_K \, M_2}$$ of two compact Riemannian scalar flat manifolds ( M _1, g _1) and ( M _2, g _2) along a common Riemannian submanifold ( K , g _ K ) whose codimension is ≥3. Here we present two constructions: the first one produces a family of “small” (in general nonzero) constant scalar curvature metrics on the generalized Connected Sum of M _1 and M _2. It yields an extension of Joyce’s result for point-wise Connected Sums in the spirit of our previous issues for nonzero constant scalar curvature metrics. When the initial manifolds are not Ricci flat, and in particular they belong to the (1_+) class in the Kazdan–Warner classification, we refine the first construction in order to produce a family of scalar flat metrics on M . As a consequence we get new solutions to the Einstein constraint equations on the generalized Connected Sum of two compact time symmetric initial data sets, extending the Isenberg–Mazzeo–Pollack gluing construction.
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Somme connesse generalizzate per problemi della geometria
2008Co-Authors: Lorenzo MazzieriAbstract:These last two decades the Connected Sum techniques, essentially based on analytical tools, are revealed to be a powerful instrument to understand solutions of several nonlinear problem issued from the geometry (constant scalar curvature metrics in Riemannian geometry, self-dual metrics, metrics with special holonomy group, extremal Kaehler metrics, Yang-Mills equations, minimal and constant mean curvature surfaces, Einstein metrics, etc.). Even tough the techniques which allows one to consider the Connected Sum at points for solutions of nonlinear PDE's are frequently used and deeply understood, the analogous techniques for Connected Sums along sub-manifolds have not been mastered yet. The main purpose of this thesis is to (partially) plug this gap by developing such techniques in the context of the constant scalar curvature metrics and the Einstein constraint equations in general relativity
Hungyu Yeh - One of the best experts on this subject based on the ideXlab platform.
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effective action from m theory on twisted Connected Sum g 2 manifolds
Communications in Mathematical Physics, 2018Co-Authors: Thaisa C Da C Guio, Hans Jockers, Albrecht Klemm, Hungyu YehAbstract:We study the four-dimensional low-energy effective $${\mathcal{N}=1}$$ supergravity theory of the dimensional reduction of M-theory on G 2-manifolds, which are constructed by Kovalev’s twisted Connected Sum gluing suitable pairs of asymptotically cylindrical Calabi–Yau threefolds X L/R augmented with a circle S 1. In the Kovalev limit the Ricci-flat G 2-metrics are approximated by the Ricci-flat metrics on X L/R and we identify the universal modulus—the Kovalevton—that parametrizes this limit. We observe that the low-energy effective theory exhibits in this limit gauge theory sectors with extended supersymmetry. We determine the universal (semi-classical) Kahler potential of the effective $${\mathcal{N}=1}$$ supergravity action as a function of the Kovalevton and the volume modulus of the G 2-manifold. This Kahler potential fulfills the no-scale inequality such that no anti-de-Sitter vacua are admitted. We describe geometric degenerations in X L/R , which lead to non-Abelian gauge symmetries enhancements with various matter content. Studying the resulting gauge theory branches, we argue that they lead to transitions compatible with the gluing construction and provide many new explicit examples of G 2-manifolds.
Giovanni Catino - One of the best experts on this subject based on the ideXlab platform.
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Connected Sum construction for σk yamabe metrics
Journal of Geometric Analysis, 2013Co-Authors: Giovanni Catino, Lorenzo MazzieriAbstract:In this paper we produce families of Riemannian metrics with positive constant σk-curvature equal to \(2^{-k} {n \choose k}\) by performing the Connected Sum of two given compact nondegeneraten-dimensional solutions (M1,g1) and (M2,g2) of the (positive) σk-Yamabe problem, provided 2≤2k
nonlinear elliptic equation. -
Connected Sum Construction for σk-Yamabe Metrics
Journal of Geometric Analysis, 2011Co-Authors: Giovanni Catino, Lorenzo MazzieriAbstract:In this paper we produce families of Riemannian metrics with positive constant σk-curvature equal to \(2^{-k} {n \choose k}\) by performing the Connected Sum of two given compact nondegeneraten-dimensional solutions (M1,g1) and (M2,g2) of the (positive) σk-Yamabe problem, provided 2≤2k
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Connected Sum construction for $\sigma_k$-Yamabe metrics
arXiv: Differential Geometry, 2009Co-Authors: Giovanni Catino, Lorenzo MazzieriAbstract:In this paper we produce families of Riemannian metrics with positive constant $\sigma_k$-curvature equal to $2^{-k} {n \choose k}$ by performing the Connected Sum of two given compact {\em non degenerate} $n$--dimensional solutions $(M_1,g_1)$ and $(M_2,g_2)$ of the (positive) $\sigma_k$-Yamabe problem, provided $2 \leq 2k < n$. The problem is equivalent to solve a second order fully nonlinear elliptic equation.
Andreas P Braun - One of the best experts on this subject based on the ideXlab platform.
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Spin(7)-manifolds as generalized Connected Sums and 3d \( \mathcal{N}=1 \) theories
Journal of High Energy Physics, 2018Co-Authors: Andreas P Braun, Sakura Schafer-namekiAbstract:M-theory on compact eight-manifolds with Spin(7)-holonomy is a framework for geometric engineering of 3d $$ \mathcal{N}=1 $$ gauge theories coupled to gravity. We propose a new construction of such Spin(7)-manifolds, based on a generalized Connected Sum, where the building blocks are a Calabi-Yau four-fold and a G2-holonomy manifold times a circle, respectively, which both asymptote to a Calabi-Yau three-fold times a cylinder. The generalized Connected Sum construction is first exemplified for Joyce orbifolds, and is then used to construct examples of new compact manifolds with Spin(7)-holonomy. In instances when there is a K3-fibration of the Spin(7)-manifold, we test the spectra using duality to heterotic on a T 3-fibered G2-holonomy manifold, which are shown to be precisely the recently discovered twisted-Connected Sum constructions.
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towards generalized mirror symmetry for twisted Connected Sum g 2 manifolds
Journal of High Energy Physics, 2018Co-Authors: Andreas P Braun, Michele Del ZottoAbstract:We revisit our construction of mirror symmetries for compactifications of Type II superstrings on twisted Connected Sum G2 manifolds. For a given G2 manifold, we discuss evidence for the existence of mirror symmetries of two kinds: one is an autoequivalence for a given Type II superstring on a mirror pair of G2 manifolds, the other is a duality between Type II strings with different chiralities for another pair of mirror manifolds. We clarify the role of the B-field in the construction, and check that the corresponding massless spectra are respected by the generalized mirror maps. We discuss hints towards a homological version based on BPS spectroscopy. We provide several novel examples of smooth, as well as singular, mirror G2 backgrounds via pairs of dual projecting tops. We test our conjectures against a Joyce orbifold example, where we reproduce, using our geometrical methods, the known mirror maps that arise from the SCFT worldsheet perspective. Along the way, we discuss non-Abelian gauge symmetries, and argue for the generation of the Affleck-Harvey-Witten superpotential in the pure SYM case.
Karen Uhlenbeck - One of the best experts on this subject based on the ideXlab platform.
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Connected Sum constructions for constant scalar curvature metrics
arXiv: Differential Geometry, 1995Co-Authors: Rafe Mazzeo, Daniel Pollack, Karen UhlenbeckAbstract:We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making `analytic' Connected Sums. In particular, we can easily construct complete metrics of constant positive scalar curvature on the complement of certain configurations of an even number of points on the sphere, which is a special case of Schoen's \cite{S1} well-known, difficult construction. Applications of this construction produces metrics with prescribed asymptotics. In particular, we produce metrics with cylindrical ends, the simplest type of asymptotic behaviour. Solutions on the complement of an infinite number of points are also constructed by an iteration of our construction.
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Connected Sum constructions for constant scalar curvature metrics
Topological Methods in Nonlinear Analysis, 1995Co-Authors: Rafe Mazzeo, Daniel Pollack, Karen UhlenbeckAbstract:We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making "analytic" Connected Sums. In particular, we can easily construct complete metrics of constant positive scalar curvature on the complement of certain configurations of an even number of points on the sphere, which is a special case of Schoen's [S1] well-known, difficult construction. Applications of this construction produces metrics with prescribed asymptotics. In particular, we produce metrics with cylindrical ends, the simplest type of asymptotic behaviour. Solutions on the complement of an infinite number of points are also constructed by an iteration of our construction.