Special Lagrangian submanifolds of the nearly Kaehler 6-sphere

In this paper, we study Lagrangian submanifolds of the nearly K?ahler 6-sphere S 6 (1). It is well known that such submanifolds, which are 3-dimensional, are always minimal and admit a symmetric cubic form. Following an idea of Bryant, developed in the study of Lagrangian submanifolds of C 3 , we then investigate those Lagrangian submanifolds which at each point admit an isometry preserving this cubic form. We obtain that all such Lagrangian submanifolds can be obtained starting from complex curves in S 6 (1) or from holomorphic curves in CP 2 (4). As a corollary we classify the Lagrangian submanifolds which admit a Sasakian structure which is compatible with the induced metric. This last result generalizes theorems obtained by Deshmukh and ElHadi

Special classes of three dimensional affine hyperspheres characterized by properties of their cubic form

It is well known that locally strongly convex ane hyperspheres can be determined as solutions of dierential equations of Monge-Ampere type. The global properties of those solutions are well understood. However, due to the nature of the Monge-Ampere equation, not much isknown about local solutions, particularly if the dimension of the hypersurface is greater then 2. By the fundamental theorem, ane hyperspheres are completely determined by their metric h and their dierence tensor K which together build the symmetric cubic form C . Following an idea of Bryant [1],we want toinvestigate ane hyperspheres for which at each point there exist isometries with respect to h preserving this cubic form. The rst non-trivial case is the case that M is 3-dimensional which is also the case which is investigated further in this paper

From certain minimal surfaces in the 5-sphere to minimal Lagrangian submanifolds of the 3-dimensional complex projective space

In a previous paper it was shown how to associate with a Lagrangian submanifold satisfying Chen's equality in 3-dimensional complex projective space, a minimal surface in the 5-sphere with ellipse of curvature is a circle. In this paper we focus on the reverse construction

Transforms for minimal surfaces in the 5-sphere.

We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how to associate to such a surface a corresponding ruled minimal Lagrangian submanifold of complex projective 3-space, which gives the converse of a construction considered in a previous paper, and illustrate this explicitly in the case of bipolar minimal surfaces

Surfaces with prescribed Weingarten-Operator

We investigate pairs of surfaces in Euclidean 3-space with the same Weingarten operator in case that one surface is given as surface of revolution. Our local and global results complement global results on ovaloids of revolution from [S-V-W-W]

Presupernova Evolution of Differentially Rotating Massive Stars Including Magnetic Fields

As a massive star evolves through multiple stages of nuclear burning on its way to becoming a supernova, a complex, differentially rotating structure is set up. Angular momentum is transported by a variety of classic instabilities, and also by magnetic torques from fields generated by the differential rotation. We present the first stellar evolution calculations to follow the evolution of rotating massive stars including, at least approximately, all these effects, magnetic and non-magnetic, from the zero-age main sequence until the onset of iron-core collapse. The evolution and action of the magnetic fields is as described by Spruit 2002 and a range of uncertain parameters is explored. In general, we find that magnetic torques decrease the final rotation rate of the collapsing iron core by about a factor of 30 to 50 when compared with the non-magnetic counterparts. Angular momentum in that part of the presupernova star destined to become a neutron star is an increasing function of main sequence mass. That is, pulsars derived from more massive stars will rotate faster and rotation will play a more dominant role in the star's explosion. The final angular momentum of the core is determined - to within a factor of two - by the time the star ignites carbon burning. For the lighter stars studied, around 15 solar masses, we predict pulsar periods at birth near 15 ms, though a factor of two range is easily tolerated by the uncertainties. Several mechanisms for additional braking in a young neutron star, especially by fall back, are also explored.Comment: 32 pages, 3 figures (8 eps files), submitted to Ap

Producing Crack-Free, High Density M2 HSS Parts by Selective Laser Melting: Pre-Heating the Baseplate

Cracks and delamination, resulting from residual stresses are a barrier in the world of Additive Manufacturing and Selective Laser Melting (SLM) that prohibits the use of many metals in this field. By preheating the baseplate, thermal gradients are lowered and stresses can be reduced. In this work, some initial tests were performed with M2 Tool Steel. Results show that pre-heating enables the production of dense M2 parts. The influence of pre-heating on density and mechanical and physical properties is investigated. The paper shows many promising results for the production of SLM parts in materials that are very sensitive to crack formation and delamination. When using a pre-heating of 200°C, crack-free parts were produced with a relative density of 99.8%.Mechanical Engineerin

Almost Contact Lagrangian Submanifolds of Nearly Kaehler 6-Sphere

For a Lagrangian submanifold M of S 6 with nearly Kaehler structure, we provide conditions for a canonically induced almost contact metric structure on M by a unit vector field, to be Sasakian. Assuming M contact metric, we show that it is Sasakian if and only if the second fundamental form annihilates the Reeb vector field ξ, furthermore, if the Sasakian submanifold M is parallel along ξ, then it is the totally geodesic 3-sphere. We conclude with a condition that reduces the normal canonical almost contact metric structure on M to Sasakian or cosymplectic structure