Minimal lagrangian submanifolds of Kähler-einstein manifolds

AbstractWe study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics $$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$ ( g a , { J α } α = 1 3 ) on the domain Y of the standard quaternion space $${\mathbb {H}}^n$$ H n one of which, say $$(g_a,J_1)$$ ( g a , J 1 ) is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie group$${{\mathcal {M}}}$$ M to obtain quaternionic Hermitian metrics on the quotient Y of X by $${\mathbb {R}}^3$$ R 3 .

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Linear Boundary Port Hamiltonian Systems defined on Lagrangian submanifolds

IFAC-PapersOnLine ◽

10.1016/j.ifacol.2020.12.1526 ◽

2020 ◽

Vol 53 (2) ◽

pp. 7734-7739

Author(s):

Bernhard Maschke ◽

Arjan van der Schaft

Keyword(s):

Hamiltonian Systems ◽

Lagrangian Submanifolds ◽

Linear Boundary

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Bootstrap bounds on closed Einstein manifolds

Journal of High Energy Physics ◽

10.1007/jhep10(2020)069 ◽

2020 ◽

Vol 2020 (10) ◽

Author(s):

James Bonifacio ◽

Kurt Hinterbichler

Keyword(s):

Compact Riemannian Manifold ◽

Tree Level ◽

Conformal Field Theories ◽

Einstein Manifolds ◽

Consistency Conditions ◽

Conformal Field ◽

Geometric Data ◽

Conformal Bootstrap ◽

Laplacian Operators ◽

Kaluza Klein

Abstract A compact Riemannian manifold is associated with geometric data given by the eigenvalues of various Laplacian operators on the manifold and the triple overlap integrals of the corresponding eigenmodes. This geometric data must satisfy certain consistency conditions that follow from associativity and the completeness of eigenmodes. We show that it is possible to obtain nontrivial bounds on the geometric data of closed Einstein manifolds by using semidefinite programming to study these consistency conditions, in analogy to the conformal bootstrap bounds on conformal field theories. These bootstrap bounds translate to constraints on the tree-level masses and cubic couplings of Kaluza-Klein modes in theories with compact extra dimensions. We show that in some cases the bounds are saturated by known manifolds.

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On Lagrangian Catenoids

Canadian Mathematical Bulletin ◽

10.4153/cmb-2007-031-4 ◽

2007 ◽

Vol 50 (3) ◽

pp. 321-333 ◽

Cited By ~ 1

Author(s):

David E. Blair

Keyword(s):

General Problem ◽

Lagrangian Submanifolds ◽

Minimal Submanifold ◽

Conformally Flat ◽

Totally Geodesic ◽

Schouten Tensor ◽

Multiplicity One

AbstractRecently I. Castro and F.Urbano introduced the Lagrangian catenoid. Topologically, it is ℝ × Sn–1 and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ℂn is foliated by round (n – 1)-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ℂn. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

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An Obata-type theorem on compact Einstein manifolds with boundary

Geometriae Dedicata ◽

10.1007/s10711-021-00598-y ◽

2021 ◽

Author(s):

Kazuo Akutagawa

Keyword(s):

Boundary Condition ◽

Smooth Boundary ◽

Type Theorem ◽

Type Inequality ◽

Einstein Metric ◽

Manifolds With Boundary ◽

Einstein Manifolds ◽

Conformal Class ◽

Totally Geodesic ◽

Yamabe Constant

AbstractWe show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

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