Huai-Dong Cao — Neckpinch

A note on the classification of four-dimensional gradient steady and expanding Ricci solitons

Sun, 29 Mar 2026 00:00:00 GMT

Huai-Dong Cao, Junming Xie — In this note, we study the classification of four-dimensional complete gradient steady and expanding Ricci solitons. Specifically, under the asymptotically cylindrical (respectively, asymptotically conical) assumption, we classify gradient steady (respectively, expanding) Ricci solitons with half-harmonic Weyl curvature. In addition, we obtain a partial classification of four-dimensional gradient expanding Ricci solitons with half-nonnegative isotropic curvature.

On curvature estimates for four-dimensional gradient Ricci solitons

Tue, 07 Oct 2025 00:00:00 GMT

Huai-Dong Cao — In this survey paper, we analyse and compare the recent curvature estimates for three types of 4-dimensional gradient Ricci solitons, especially between Ricci shrinkers [58] and expanders [17]. In addition, we provide some new curvature estimates for 4-dimensional gradient steady Ricci solitons, including the sharp curvature estimate |Rm|≤ C R for gradient steady Ricci solitons with positive Ricci curvature (see Theorem 1.1).

Curvature pinching of asymptotically conical gradient expanding Ricci solitons

Mon, 06 Oct 2025 00:00:00 GMT

Huai-Dong Cao, Junming Xie — In this paper, we investigate curvature pinching phenomena in complete non-compact asymptotically conical gradient expanding Ricci solitons and establish several Hamilton-Ivey type curvature pinching estimates. These results are parallel to those known for shrinking and steady Ricci solitons. In particular, we prove a three-dimensional Hamilton-Ivey type curvature pinching theorem: any three-dimensional non-compact gradient Ricci expander, which is asymptotic to a cone with positive scalar curvature, must have positive sectional curvature. Furthermore, we formulate a general method and apply it to obtain analogues of several additional known generalized Hamilton-Ivey type curvature pinching results for ancient solutions. Among these is a curvature pinching estimate for four-dimensional asymptotically conical Ricci expanders with uniformly positive isotropic curvature, analogous to a result for four-dimensional gradient steady solitons due to Brendle [8].

Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature

Thu, 28 Mar 2024 00:00:00 GMT

Huai-Dong Cao, Junming Xie — This is a sequel to our paper [24], in which we investigated the geometry of 4-dimensional gradient shrinking Ricci solitons with half positive (nonnegative) isotropic curvature. In this paper, we mainly focus on 4-dimensional gradient steady Ricci solitons with nonnegative isotropic curvature (WPIC) or half nonnegative isotropic curvature (half WPIC). In particular, for 4D complete ancient solutions with WPIC, we are able to prove the 2-nonnegativity of the Ricci curvature and bound the curvature tensor Rm by |Rm|\leq R. For 4D gradient steady solitons with WPIC, we obtain a classification result. We also give a partial classification of 4D gradient steady Ricci solitons with half WPIC. Moreover, we obtain a preliminary classification result for 4D complete gradient expanding Ricci solitons with WPIC. Finally, motivated by the recent work [59], we improve our earlier results in [24] on 4D gradient shrinking Ricci solitons with half PIC or half WPIC, and also provide a characterization of complete gradient Kaehler-Ricci shrinkers in complex dimension two among 4-dimensional gradient Ricci shrinkers.

Linear stability of compact shrinking Ricci solitons

Tue, 04 Apr 2023 00:00:00 GMT

Huai-Dong Cao, Meng Zhu — In this paper, we continue investigating the second variation of Perelman's ν-entropy for compact shrinking Ricci solitons. In particular, we improve some of our previous work in "H.-D. Cao and M. Zhu, Math. Ann. 353 (2012), No. 3, 747-763", as well as the more recent work in "M. Mehrmohamadi and A. Razavi, arXiv:2104.08343", and obtain a necessary and sufficient condition for a compact shrinking Ricci soliton to be linearly stable. Our work also extends similar results of Hamilton, Ilmanen and the first author in "arXiv:math.DG/0404165" (see also "H.-D. Cao and C. He, J. Reine Angew. Math. 2015 (2015), no. 709, 229-246.") for positive Einstein manifolds to the compact shrinking Ricci soliton case.

Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature

Wed, 16 Nov 2022 00:00:00 GMT

Huai-Dong Cao, Junming Xie — In this paper, we investigate the geometry of 4-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature (half PIC) or half nonnegative isotropic curvature. Our first main result is a certain form of curvature estimates for such Ricci shrinkers, including a quadratic curvature lower bound estimate for noncompact ones with half PIC. As a consequence, we obtain a new and more direct proof of the classification result, first observed by Li-Ni-Wang [35], for gradient shrinking Kähler-Ricci solitons of complex dimension two with nonnegative isotropic curvature. Moreover, based on a strong maximum principle argument, we classify 4-dimensional complete gradient shrinking Ricci solitons with half nonnegative isotropic curvature (except the half PIC case). Finally, we treat the half PIC case under an additional assumption on the Ricci tensor.

A Weil-Petersson Type Metric on the Space of Fano Kaehler-Ricci Solitons

Thu, 27 Oct 2022 00:00:00 GMT

Huai-Dong Cao, Xiaofeng Sun, Yingying Zhang — In this paper we define a Weil-Petersson type metric on the space of shrinking Kaehler-Ricci solitons and prove a necessary and sufficient condition on when it is independent of the choices of Kaehler-Ricci soliton metrics. We also show that the Weil-Petersson metric is Kaehler when it defines a metric on the Kuranishi space of small deformations of Fano Kaehler-Ricci solitons. Finally, we establish the first and second order deformation of Fano Kähler-Ricci solitons and show that, essentially, the first effective term in deforming Kaehler-Ricci solitons leads to the Weil-Petersson metric.

Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio

Thu, 24 Mar 2022 00:00:00 GMT

Huai-Dong Cao, Tianbo Liu, Junming Xie — Let (Mⁿ, g, f), n≥ 5, be a complete gradient expanding Ricci soliton with nonnegative Ricci curvature Rc≥ 0. In this paper, we show that if the asymptotic scalar curvature ratio of (Mⁿ, g, f) is finite (i.e., limsup_(r→ ∞) R r²< ∞), then the Riemann curvature tensor must have at least sub-quadratic decay, namely, limsup_(r→ ∞) |Rm| r^α< ∞ for any 0<α<2.

Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

Thu, 18 Nov 2021 00:00:00 GMT

Huai-Dong Cao, Tianbo Liu — In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set K). More precisely, we prove that the norm of the curvature tensor Rm and its covariant derivative ∇Rm can be bounded by the scalar curvature R by |Rm|≤ Cₐ R^a and |∇Rm| ≤ Cₐ R^a (on Mbackslash K), for any 0≤ a <1 and some constant Cₐ >0. Moreover, if the scalar curvature has at most polynomial decay at infinity, then |Rm| ≤ C R (on Mbackslash K). As an application, it follows that that if a 4-dimensional complete gradient expanding Ricci soliton (M⁴, g, f) has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, and C^(1,α) asymptotic cones at infinity (0 < α< 1) according to Chen-Deruelle [20].

Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture

Sun, 03 Dec 2006 00:00:00 GMT

Huai-Dong Cao, Xi-Ping Zhu — In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré conjecture due to Hamilton and Perelman.