Asymptotic geometry of the Hitchin metric (Q1737557)

Let $\mathcal{M}$ be the moduli space of irreducible solutions of the Hitchin self-dula equations on \(\mathrm{SU}(2)\)-bundle $E$ of degree 0 over a compact Riemann surface$X$, modulo unitary gauge transformations. Let $\mathcal{B}$ be the space of holomorphic quadratic differentials. Then the map $\pi(A,\Phi)=\mathrm{det}\Phi$ is a proper surjective mapping from $\mathcal{M}$ to $\mathcal{B}$. In [\textit{R. Mazzeo} et al., Duke Math. J. 165, No. 12, 2227--2271 (2016; Zbl 1352.53018)], the authors introduced the free region $\mathcal{B}'\subset \mathcal{B}$ of quadratic differentials with only simple zeros and its inverse image $\mathcal{M}'\subset \mathcal{M}$ (cf. [\textit{T. Mochizuki}, J. Topol. 9, No. 4, 1021--1073 (2016; Zbl 1381.53047)]). Then introducing moduli space $\mathcal{M}_t$ of the scaled Hitchin equation \[ \mu_t(A,\Phi)=(F_A+t^2[\Phi\wedge\Phi^\ast], \bar{\partial}_A\Phi)=0, \] and the space of limiting configurations; the moduli space $\mathcal{M}_\infty$ of decoupled equations \[ F_{A_{\infty}}=0, \ [\Phi_\infty\wedge \phi_\infty^\ast]=0, \ \bar{\partial}_{A_\infty}\Phi_\infty=0, \] $\lim_{t\to\infty}\mathcal{M}'_t=\mathcal{M}'_\infty$ was shown. Here $\mathcal{M}'_t$, etc. are defined similar to $\mathcal{M}'$. The existence of diffeomorphism $\mathcal{F}:\mathcal{M}'_\infty\to \mathcal{M}'$ was also constructed. \par Beside Hitchin's hyperKähler metric $g_{L^2}$, $\mathcal{M}$ has a semiflat hyperKähler metric $g_{\mathrm{sf}}$. because $\mathcal{M}$ has the structure of algebraic integrable system (\S2.2). \textit{D. Gaiotto} et al. [Adv. Math. 234, 239--403 (2013; Zbl 1358.81150)], conjectured identifying the dilation parameter $t$ as a radial variable on $\mathcal{M}'$, the difference between $g_{L^2}$ and $g_{\mathrm{sf}}$ decays exponential order. Motivated these, main Theorems of this paper are \par Theorem 1.1. The pullback $\mathcal{F}^\ast g_{\mathrm{sf}}$ of the semiflat metric to $\mathcal{M}'_\infty$ is a renormalized $L^2$ metric on $\mathcal{M}'_\infty$. \par Theorem 1.2. There is a convergent series expansion \[ g_{L^2}=g_{\mathrm{sf}}+\sum_{j=0}^\infty t^{(4-j)/3}G_j+O(e^{-\beta t}), \] as $t\to\infty$, where each $G_j$ is a dilation invariant symmetric two-tensor. The rate $\beta>0$ of exponential decrease of the remainder is uniform in any closed dilation-invariant sector $\bar{\mathcal{W}}\subset \mathcal{M}'_\infty$ disjoint from $\pi_\infty^{-1}(\mathcal{B}\setminus\mathcal{B}')$. \par The authors remark soon after this paper haad been posted exponential decay of $g_{L^2}-g_{\mathrm{sf}}$ was proved [\textit{D. Dumas} and \textit{A. Nietzke}, ``Asymptotics of Hitchin's metric on Hitchin space'', Preprint, \url{arXiv:1802.07200}], [\textit{L. Fredrickson}, ``Exponential decay for aymptotic geometry of the Hitchin metric'', Preprint, \url{arXiv:1810.01554}]. Namely, the following theorem holds. \par Theorem. Fix a generic Higgs bundle $(\bar{\partial}_E, \phi)$ in $\mathcal{M}$, and a HIggs bundle variation $\Psi=(\eta,\phi)$. Consider the deformation $\psi_t=(\eta,t\phi)\in T_{(\bar{\partial}.t\phi)}\mathcal{M}$ over the ray $(\bar{\partial}_E, t\phi,h_t)$. As $t\to \infty$, the difference between Hitchin hyperKähler matric $g_{L^2}$ on $\mathcal{M}$ and the semiflat metric $g_{\mathrm{sf}}$ is exponentially decaying. In particular, there is some constant $\gamma>0$, such that \[ g_{L^2}(\psi_t,\psi_t)=g_{\mathrm{sf}}(\psi_t,\psi_t)+O(e^{-\gamma t}). \] This is obtained on the base of this paper. So to understand proof of exponential decay of $g_{L^2}-g_{\mathrm{sf}}$, to study this paper is useful. \par This paper also contains explanations on the geometry of $\mathcal{M}$ together with definitions of $g_{L^2}$ and $g_{\mathrm{sf}}$ (\S2). Then detailed exposition on limiting configurations is given in \S3, where Theorem 1.1 is proved via showing $L^2$ inner product of renormalized gauged tangent space premoduli space $\mathcal{PM}'_\infty$ correspond to the inner products for the semiflat metric. General strategy to prove Theorem 1.2 is explained in \S4., following the construction of approxiamte solutions in [\textit{D. Balduzzi}, Math. Res. Lett. 13, No. 5--6, 923--933 (2006; Zbl 1111.14026)]. Rest of this paper (\S5-\S10) is devoted to the proof of Theorem 1.2.