The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers
Fei He, Jianyu Ou
TL;DR
The paper studies holomorphic functions and forms on complete gradient Kähler Ricci shrinkers, establishing finiteness and quantitative dimension bounds for spaces of polynomial-growth objects via spectral data of the $f$-Laplacian $\Delta_f$. By relating ancient caloric solutions to $f$-heat flow and exploiting an eigenfunction expansion, it proves a finite decomposition for holomorphic functions with growth rate $d$ and yields a sharp linear-growth bound $\dim_{\mathbb{C}} \mathcal{O}_1 \le m+1$, with equality only on the Gaussian soliton and a corresponding splitting rigidity. Under curvature and radial-scalar curvature assumptions, the work provides a polynomial upper bound $\dim_{\mathbb{C}} \mathcal{O}_d \le C d^{2m-1}$, extending finiteness results to explicit growth-dependent estimates; it further shows that holomorphic $(p,0)$-forms also have finite dimension controlled by the spectrum of $\Delta^d_f$, with bounds that reflect curvature data. Collectively, these results connect the complex analytic structure on shrinking solitons to the Bakry-Émery spectrum, offering both sharp linear bounds and growth-driven dimension estimates with potential rigidity consequences.
Abstract
We study polynomial growth holomorphic functions and forms on complete gradient shrinking Ricci solitons. By relating to the spectral data of the $f$-Laplacian, we show that the dimension of the space of polynomial growth holomorphic functions or holomorphic $(p,0)$-forms are finite. In particular, a sharp dimension estimate for the space of linear growth holomorphic functions was obtained. Under some additional curvature assumption, we prove a sharp estimate for the frequency of polynomial growth holomorphic functions, which was used to obtain dimension upper bound as a power function of the polynomial order.