Elementary spectral invariants and quantitative closing lemmas for contact three-manifolds (Q6632081)

The ECH capacities are a sequence of numerical invariants of symplectic four-manifolds which give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their basic properties currently requires Seiberg-Witten theory. The author [``An elementary alternative to ECH capacities'', Preprint, \url{arXiv:2201.03143}] defined a new sequence of symplectic capacities in four dimensions using only basic notions of holomorphic curves.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] introduces some basic definitions.\N\N\item[\S 3] defines a family of spectral invariants \(b_{k}\)\ of four-dimensional strong symplectic cobordisms, which is a precursor to the definition of the spectral invariants \(c_{k}\)\ of contact three-manifolds.\N\N\item[\S 4] defines the invariants, establishing the following theorem.\N\NTheorem. The invariants \(c_{k}\)\ have the following properties:\N\begin{itemize}\N\item[Conformality.] If \(r>0\)\ then\N\[\Nc_{k}\left( Y,r\lambda\right) =rc_{k}\left( Y,\lambda\right)\N\]\N\item[Increasing.]\N\[\N0=c_{0}\left( Y,\lambda\right) <c_{1}\left( Y,\lambda\right) \leq c_{2}\left( Y,\lambda\right) \leq\cdots\leq+\infty\N\]\N\item[Disjoint Union.] If \(\left( Y_{i},\lambda_{i}\right) \)\ is a closed contact three-manifold for \(i=1,\ldots,m\), then\N\[\Nc_{k}\left( \coprod\limits_{i=1}^{m}\left( Y_{i},\lambda_{i}\right) \right) =\underset{k_{1}+\cdots+k_{m}=k}{\max}\sum\limits_{i=1}^{m} c_{k}\left( Y_{i},\lambda_{i}\right)\N\]\N\item[Sublinearity.]\N\[\Nc_{k+l}\left( Y,\lambda\right) \leq c_{k}\left( Y,\lambda\right) +c_{l}\left( Y,\lambda\right)\N\]\N\item[Monotonicity.] If \(f:Y\rightarrow\mathbb{R}^{\geq0}\)\ then\N\[\Nc_{k}\left( Y,\lambda\right) \leq c_{k}\left( Y,e^{f}\lambda\right)\N\]\N\item[\(C^{0}\)-Continuity.] For fixed \(\left( Y,\lambda\right) \)\ and fixed \(k\), the map \(C^{\infty}\left( Y;\mathbb{R}\right) \rightarrow\mathbb{R} \)\ sending \[f\mapsto c_{k}\left( Y,e^{f}\lambda\right) \]is \(C^{0}\)-continuous.\N\item[Spectrality.] For given \(\left( Y,\lambda\right) \)\ and \(k\), if \(c_{k}\left( Y,\lambda\right) \)\ is finite, then there exists an orbit set \(\alpha\)\ such that\N\[\Nc_{k}\left( Y,\lambda\right) =\mathcal{A}\left( \alpha\right)\N\]\N\item[Liouville Domains.] If \(\left( Y,\lambda\right) \)\ is the boundary of a Liouville domain \(\left( X,\omega\right) \), then\N\[\Nc_{k}\left( Y,\lambda\right) \leq c_{k}^{\mathrm{Alt}}\left( Y,\lambda \right)\N\]\N\item[Sphere.]\N\[\Nc_{k}\left( \partial B^{4}\left( a\right) \right) =da\N\]\Nwhere \(d\)\ is the unique nonnegative integer such that\N\[\Nd^{2}+d\leq2k\leq d^{2}+3d\N\]\N\item[Asymptotic Lower Bound.]\N\[\N\liminf_{k\rightarrow\infty}\frac{c_{k}\left( Y,\lambda\right) ^{2}}{k}\geq2\mathrm{vol}\left( Y,\lambda\right)\N\]\N\item[Spectral Gap Closing Bound.] If \(k>0\)\ and \(c_{k}\left( Y,\lambda \right) \leq L<\infty\), then\N\[\N\mathrm{Close}^{L}\left( Y,\lambda\right) \leq c_{k}\left( Y,\lambda \right) -c_{k-1}\left( Y,\lambda\right)\N\]\N\end{itemize}\N\N\item[\S 5] establishes the following results:\N\NTheorem. Let \(Y\)\ be a compact star-shaped hypersurface in \(\mathbb{R}^{4}\)\ and suppose that \(Y\subset B^{4}\left( a\right) \). Then for every \(L>0\)\ we have\N\[\N\mathrm{Close}^{L}\left( Y\right) \leq\frac{2a}{\left[ La^{-1}\right] +3}\N\]\NTheorem. If \(a>1\)\ is irrational and \(L\geq a\), then we have\N\[\N\mathrm{Close}^{L}\left( \partial E\left( a,1\right) \right) =\min\left( am_{-}-n_{-},n_{+}-am_{+}\right)\N\]\Nwhere\N\begin{itemize}\N\item[(1)] \(m_{-}\) and \(n_{-}\) are relatively prime integers with \(m_{-}>0\)\ such that \(n_{-}/m_{-}\) is maximized to the constraints \(n_{-}lm_{-}<a\) and \(am_{-}\leq L\);\N\item[(2)] \(m_{+}\) and \(n_{+}\) are relatively prime integers with \(m_{+}>0\)\ such that \(n_{+}/m_{+}\) is minimized to the constraints \(n_{+}lm_{+}>a\) and \(n_{+-}\leq L\);\N\item[(3)]\N\[\NE\left( a,1\right) =\left\{ z\in\mathbb{C}^{2}\mid\frac{\pi\left\vert z_{1}\right\vert ^{2}}{a}+\pi\left\vert z_{2}\right\vert ^{2}\leq1\right\}\N\]\N\end{itemize}\N\NProposition.\N\[\N\mathrm{Box}^{L}\left( Y,\lambda\right) \leq\mathrm{Close}^{L}\left( Y,\lambda\right)\N\]\N\item[\S 6] establishes a relation between \(c_{k}\)\ and the ECH spectrum, leading to the following theorem, from which the general quantitative closing lemma follows.\N\NTheorem. Let \(\left( Y,\lambda\right) \)\ be any closed contact three-manifold. Then\N\[\N\lim_{k\rightarrow\infty}\frac{c_{k}\left( Y,\lambda\right) ^{2} }{k}\geq2\mathrm{vol}\left( Y,\lambda\right)\N\]\N\end{itemize}