AI 中文总结
该研究证明了方砖铺成的曲面、正六边形和正八边形得到的平移曲面可通过单值严格递增的连续形变成为最大化平移曲面，还得到另一类具有该性质的平移曲面。
AI 中文摘要
单位面积的平移曲面被称为“最大化”的，若它在同一层的所有曲面中使最短鞍连接的长度最大。我们研究非最大化平移曲面是否可在单值严格单调增长的情况下连续形变为最大化平移曲面。尽管由于单值函数存在局部而非全局最大值，这类形变一般不存在，但我们证明对于方砖铺成的曲面、由正六边形得到的平移曲面以及正八边形，这类形变存在。对每种情况，我们明确构造一条连续形变路径，沿该路径单值严格递增地指向最大化平移曲面。最后，上述构造得到另一类平移曲面，这类曲面允许连续的单值增长形变指向最大化曲面。
英文摘要
A unit-area translation surface is called \emph{maximal} if it maximizes the length of the shortest saddle connection among all surfaces in the same stratum. We investigate whether a non-maximal translation surface can be continuously deformed into a maximal one while the systole increases strictly monotonically. Although such a deformation does not exist in general due to the existence of local but not global maxima of the systole function, we prove that it exists for square-tiled surfaces and translation surfaces obtained from regular hexagons and regular octagons. In each case, we explicitly construct a continuous deformation to a maximal translation surface along which the systole is strictly increasing. Finally, the preceding construction yields another family of translation surfaces admitting continuous systole-increasing deformations to maximal surfaces.
Comments23 pages, 15 figures
