On Lagrangian Catenoids

2007 ◽  
Vol 50 (3) ◽  
pp. 321-333 ◽  
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.

scholarly journals Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 303-319
Author(s):  
Yoshihiro Ohnita
Keyword(s):  
Symmetric Space ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Lagrangian Submanifold ◽  
Riemannian Symmetric Space ◽  
Canonical Embedding ◽  
Real Form ◽  
Isotropy Representation ◽  
Totally Geodesic

AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

scholarly journals Prescribed Schouten Tensor in Locally Conformally Flat Manifolds

2019 ◽  
Vol 74 (4) ◽  
Author(s):  
Marcos Tulio Carvalho ◽  
Mauricio Pieterzack ◽  
Romildo Pina
Keyword(s):  
Sufficient Conditions ◽  
Conformally Flat ◽  
Symmetric Tensors ◽  
Conformally Flat Manifolds ◽  
Dimensional Group ◽  
Schouten Tensor ◽  
Locally Conformally Flat ◽  
Flat Manifolds ◽  
Necessary And Sufficient ◽  
Complete Metrics

Abstract We consider the pseudo-Euclidean space $$({\mathbb {R}}^n,g)$$(Rn,g), with $$n \ge 3$$n≥3 and $$g_{ij} = \delta _{ij} \varepsilon _{i}$$gij=δijεi, where $$\varepsilon _{i} = \pm 1$$εi=±1, with at least one positive $$\varepsilon _{i}$$εi and non-diagonal symmetric tensors $$T = \sum \nolimits _{i,j}f_{ij}(x) dx_i \otimes dx_{j} $$T=∑i,jfij(x)dxi⊗dxj. Assuming that the solutions are invariant by the action of a translation $$(n-1)$$(n-1)- dimensional group, we find the necessary and sufficient conditions for the existence of a metric $$\bar{g}$$g¯ conformal to g, such that the Schouten tensor $$\bar{g}$$g¯, is equal to T. From the obtained results, we show that for certain functions h, defined in $$\mathbb {R}^{n}$$Rn, there exist complete metrics $$\bar{g}$$g¯, conformal to the Euclidean metric g, whose curvature $$\sigma _{2}(\bar{g}) = h$$σ2(g¯)=h.

scholarly journals Normal curvature of minimal submanifolds in a sphere

1997 ◽  
Vol 39 (1) ◽  
pp. 29-33
Author(s):  
Sharief Deshmukh
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Minimal Submanifolds ◽  
Normal Curvature ◽  
Minimal Submanifold ◽  
Rigidity Result ◽  
Totally Geodesic ◽  
Veronese Surface ◽  
Arbitrary Codimension ◽  
Image Position

Simons [5] has proved a pinching theorem for compact minimal submanifolds in a unit sphere, which led to an intrinsic rigidity result. Sakaki [4] improved this result of Simons for arbitrary codimension and has proved that if the scalar curvature S of the minimal submanifold Mn of Sn+P satisfiesthen either Mn is totally geodesic or S= 2/3 in which case n = 2 and M2 is the Veronese surface in a totally geodesic 4-sphere. This result of Sakaki was further improved by Shen [6] but only for dimension n=3, where it is shown that if S>4, then M3 is totally geodesic (cf. Theorem 3, p. 791).

scholarly journals On an inequality of Oprea for Lagrangian submanifolds

2009 ◽  
Vol 7 (1) ◽  
Author(s):  
Franki Dillen ◽  
Johan Fastenakels
Keyword(s):  
Complex Space ◽  
Space Form ◽  
Lagrangian Submanifolds ◽  
Lagrangian Submanifold ◽  
Complex Space Form ◽  
Totally Geodesic

AbstractWe show that a Lagrangian submanifold of a complex space form attaining equality in the inequality obtained by Oprea in [8], must be totally geodesic.

scholarly journals Totally real submanifolds in a complex projective space

1999 ◽  
Vol 22 (1) ◽  
pp. 205-208 ◽  
Author(s):  
Liu Ximin
Keyword(s):  
Projective Space ◽  
Scalar Curvature ◽  
Ricci Curvature ◽  
Complex Projective Space ◽  
Minimal Submanifold ◽  
Real Submanifolds ◽  
Totally Geodesic ◽  
Totally Real ◽  
Complex Projective

In this paper, we establish the following result: LetMbe ann-dimensional complete totally real minimal submanifold immersed inCPnwith Ricci curvature bounded from below. Then eitherMis totally geodesic orinf  r≤(3n+1)(n−2)/3, whereris the scalar curvature ofM.

scholarly journals Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 from minimal surfaces in 𝕊3

2018 ◽  
Vol 149 (03) ◽  
pp. 655-689 ◽  
Author(s):  
Burcu Bektaş ◽  
Marilena Moruz ◽  
Joeri Van der Veken ◽  
Luc Vrancken
Keyword(s):  
Minimal Surface ◽  
Minimal Surfaces ◽  
Lagrangian Submanifolds ◽  
Large Family ◽  
Maximal Rank ◽  
Totally Geodesic ◽  
Nearly Kähler

AbstractWe study non-totally geodesic Lagrangian submanifolds of the nearly Kähler 𝕊3 × 𝕊3 for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms of the so-called angle functions and that such Lagrangian submanifolds are closely related to minimal surfaces in 𝕊3. Indeed, starting from an arbitrary minimal surface, we can construct locally a large family of such Lagrangian immersions, including one exceptional example. We also show that locally all such Lagrangian submanifolds can be obtained in this way.

scholarly journals Some results on null hypersurfaces in $(LCS)$-manifolds

2019 ◽  
Author(s):  
Samuel Ssekajja
Keyword(s):  
Vector Field ◽  
Ricci Tensor ◽  
Characteristic Vector ◽  
Null Hypersurface ◽  
Conformally Flat ◽  
Totally Geodesic ◽  
Null Hypersurfaces ◽  
Characteristic Vector Field

We prove that a Lorentzian concircular structure $ (LCS)$-manifold does not admit any null hypersurface which is tangential or transversal to its characteristic vector field. Due to the above, we focus on its ascreen null hypersurfaces, and show that such hypersurfaces admit a symmetric Ricci tensor. Furthermore, we prove that there is no any totally geodesic ascreen null hypersurfaces of a conformally flat $(LCS)$-manifold.

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