In this article, we prove a Kähler extension theorem for real Kähler submanifolds of codimension 4 and rank at least 5. Our main theorem states that such a manifold is a holomorphic hypersurface in another real Kähler submanifold of codimension 2. This generalizes a result of Dajczer and Gromoll in 1997 which states that any real Kähler submanifolds of codimension 3 and rank at least 4 admits a Kähler extension.

A three dimensional finite element model (FEM) was established to simulate the temperature distribution, flow activity, and deformation of the melt pool of selective laser melting (SLM) AZ91D magnesium alloy powder. The latent heat in phase transition, Marangoni effect, and the movement of laser beam power with a Gaussian energy distribution were taken into account. The influence of the applied linear laser power on temperature distribution, flow field, and the melt-pool dimensions and shape, as well as resultant densification activity, was investigated and is discussed in this paper. Large temperature gradients and high cooling rates were observed during the process. A violent flow occurred in the melt pool, and the divergent flow makes the melt pool wider and longer but shallower. With the increase of laser power, the melt pool’s size increases, but the shape becomes longer and narrower. The width of the melt pool in single-scan experiment is acquired, which is in good agreement with the results predicted by the simulation (with error of 1.49%). This FE model provides an intuitive understanding of the complex physical phenomena that occur during SLM process of AZ91D magnesium alloy. It can help to select the optimal parameters to improve the quality of final parts and reduce the cost of experimental research.

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Abstract:In 1991, Dajczer and Rodriguez proved in [10] that a complete minimal real Kähler submanifold of codimension 2, if with complex dimension > 2, would be either holomorphic, or a cylinder, or complex ruled. In this article, we generalize their result to real analytic complete real Kähler submanifolds of codimension 4. The conclusion is that such the submanifold, if with complex dimension > 4, would be either partially holomorphic, or a cylinder, or a twisted cylinder in the sense that the complex relative nullity foliation is contained in a strictly larger holomorphic foliation, whose leaves are cylinders. We also examine the question of when such a submanifold is complex ruled.

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