https://neckpinch.com/authors/fei-he/New arXiv papers by Fei He, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2607.21959arxiv:2607.21959Fri, 24 Jul 2026 00:00:00 GMTFei He — We study the spectrum of the f-Laplacian on complete gradient Ricci shrinkers. Upper and lower bounds for the k-th eigenvalue are established in terms of the volume growth rate. Both bounds are shown to be sharp in the exponent. The method extends to f-Laplace-type operators on vector bundles; as an application we obtain explicit upper bounds for the Betti numbers.Ricci solitonsEigenvalues & spectrum https://arxiv.org/abs/2605.04476arxiv:2605.04476Wed, 06 May 2026 00:00:00 GMTFei He — This paper proves several topological results for smooth gradient Ricci shrinkers. We establish upper bounds for the Betti numbers, a vanishing theorem for cohomology, and a dichotomy for the number of ends. We also prove a full Hodge theorem for a large class of shrinkers. The methods are based on weighted L² cohomology and extend to self-shrinkers of the mean curvature flow.Mean curvature flowRicci solitonsSelf-shrinkers & translators https://arxiv.org/abs/2411.12171arxiv:2411.12171Tue, 19 Nov 2024 00:00:00 GMTFei He, Jianyu Ou — We prove an upper bound for the dimension of the linear space of holomorphic functions with polynomial growth on gradient Kähler Ricci shrinkers with bounded curvature. The upper bound is given as a power function of the growth rate. Similar results hold for holomorphic (p, 0)-forms, and holomorphic sections of the pluri-anticanonical line bundle K_M^(−q). We also prove the existence of holomorphic sections of K_M^(−q) with polynomial growth when the Kähler Ricci shrinker is asymptotically conical, provided q is sufficiently large; as an application, we show that the Kodaira map constructed using such sections is a holomorphic embbedding into a complex projective space.Ricci solitonsKähler geometryBoundaries & non-compact https://arxiv.org/abs/2401.02685arxiv:2401.02685Fri, 05 Jan 2024 00:00:00 GMTFei He, Jianyu Ou — 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 an almost 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.Ricci solitonsKähler geometryEigenvalues & spectrum