论文标题
在圆环和球体上接触结构和Beltrami场
Contact structures and Beltrami fields on the torus and the sphere
论文作者
论文摘要
我们介绍了（圆形）3-Sphere和（Flat）3螺道的新的显式紧密和明确的接触结构，其环境度量较弱。我们的证明是基于使用合适的雅各比或三角多项式家庭的非捕捞卷曲本田的构建。结果，我们表明，Etnyre，Komendarczyk和Massot（2012）的接触球理定理对猜想的弱兼容度量不满意。我们还通过表明在3个球体上的任何接触形式承认兼容的度量标准是等轴测图，最高为常数因子，以达到标准（紧密的）接触形式，这还建立了紧密接触结构的几何刚度。
We present new explicit tight and overtwisted contact structures on the (round) 3-sphere and the (flat) 3-torus for which the ambient metric is weakly compatible. Our proofs are based on the construction of nonvanishing curl eigenfields using suitable families of Jacobi or trigonometric polynomials. As a consequence, we show that the contact sphere theorem of Etnyre, Komendarczyk and Massot (2012) does not hold for weakly compatible metric as it was conjectured. We also establish a geometric rigidity for tight contact structures by showing that any contact form on the 3-sphere admitting a compatible metric that is the round one is isometric, up to a constant factor, to the standard (tight) contact form.
