论文标题
3D凸触点表格和Ruelle不变
3d Convex Contact Forms And The Ruelle Invariant
论文作者
论文摘要
令$ x \ subset \ mathbb {r}^4 $为具有光滑边界$ y $的凸域。我们使用$ y $的外部曲率与自然reeb流的外部曲率$ \ text {ru}（y）$在$ y $上流量的关系，以证明存在常数$ c> c> 0 $ $ y $独立于$ y $，因此\ [c <\ frac {c <\ frac {c {\ frac {\ ru} {y} {y） \ text {sys}（y）<c \]这里$ \ text {sys}（y）$是收缩比，即$ y $ y $ y $ y $的最低时期的平方，分别划分为$ x $的两倍。然后，我们在$ s^3 $上动态构建凸触点表格，以使用Abbondandolo-Bramham-Hryniewicz-Salomão的方法违反了这种界限。这些是动态凸的第一个示例，即$ 3 $ -SPHERES，并非严格地与凸边界$ y $接触。
Let $X \subset \mathbb{R}^4$ be a convex domain with smooth boundary $Y$. We use a relation between the extrinsic curvature of $Y$ and the Ruelle invariant $\text{Ru}(Y)$ of the natural Reeb flow on $Y$ to prove that there exist constants $C > c > 0$ independent of $Y$ such that \[c < \frac{\text{Ru}(Y)^2}{\text{vol}(X)} \cdot \text{sys}(Y) < C\] Here $\text{sys}(Y)$ is the systolic ratio, i.e. the square of the minimal period of a closed Reeb orbit of $Y$ divided by twice the volume of $X$. We then construct dynamically convex contact forms on $S^3$ that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salomão. These are the first examples of dynamically convex contact $3$-spheres that are not strictly contactomorphic to a convex boundary $Y$.
