In 3D printing, it is critical to use as few as possible supporting materials for efficiency and material saving. Multiple model decomposition methods and multi-DOF (degrees of freedom) 3D printers have been developed to address this issue. However, most systems utilize model decomposition and multi-DOF independently. Only a few existing approaches combine the two, i.e. partitioning the models for multi-DOF printing. In this paper, we present a novel model decomposition method for multidirectional 3D printing, allowing consistent printing with the least cost of supporting materials. Our method is based on a global optimization that minimizes the surface area to be supported for a 3D model. The printing sequence is determined inherently by minimizing a single global objective function. Experiments on various complex 3D models using a five-DOF 3D printer have demonstrated the effectiveness of our approach.

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In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. %either a totally geodesic disk or a spherical cap. Our result also provides a proof of a conjecture proposed by Sternberg-Zumbrun in {\it J Reine Angew Math 503 (1998), 63--85}. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces with free boundary in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.

We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $\alpha'$. Actually we obtain at the same time a solution of the open case $\alpha'>0$, an improved solution of the known case $\alpha'<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $\Gamma_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space of a semisimple algebraic group G. We define two families of algebraic subvarieties of the associated partial flag variety, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of m × n real matrices whose image is not contained in any subvariety coming from these two families, Dirichlet’s theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner’s theorem on measure rigidity for unipotent flows, and linearization technique.