标题： 拟正规性与三维流形中最小面积曲面的PL逼近 显示英文标题

标题： Quasi Normality and PL Approximation of Least Area Surfaces in 3-Manifolds

摘要： 本文提出了最小面积与在欧几里得或双曲$3$-流形中嵌入的正常曲面之间的关系。引入了一种放松版的正常曲面，称为准正常曲面，并表明在适当条件下，每个嵌入的最小面积曲面相对于足够精细的流形$3$的脂肪三角剖分都是准正常的。此外，还表明当这样的精细三角剖分中的最小面积曲面与四面体的交集，即使不像正常曲面那样简单，也是行为良好的。最后，表明当将最小面积曲面视为准正常曲面时，会生成一系列称为平坦关联曲面的分片平坦曲面，并且该序列收敛于给定的最小面积曲面并近似其面积。 显示英文摘要

摘要： This} paper presents relations between least area and normal surfaces, embedded in either a Euclidean or hyperbolic $3$-manifold. A relaxed version of normal surfaces, termed quasi-normal, is introduced, and it is shown that under appropriate conditions, every embedded least area surface is quasi-normal with respect to a fine enough fat triangulation of the $3$-manifold. In addition, it is shown that the intersections of a least area surface with the tetrahedra of such fine enough triangulation, even when not as simple as in the case of normal surfaces, are also well behaved. Finally, it is shown that a least area surface, when considered as a quasi normal surface, gives rise to a sequence of piecewise flat surfaces termed as flat-associated surfaces, and this sequence converges to the given least area surface and approximates its area.