Junming Xie — Neckpinch

Kählerity of complete almost-Kähler gradient shrinking Ricci solitons

Tue, 01 Sep 2026 00:00:00 GMT

Junming Xie — In this paper, we prove that any complete, compact or noncompact, almost-Kähler gradient shrinking Ricci soliton is Kähler in arbitrary even dimension. Among other applications, combining our result with the classification of complete gradient shrinking Kähler-Ricci solitons in complex dimension two, we obtain a full classification of complete almost-Kähler gradient shrinking Ricci solitons in real dimension four.

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.

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].

Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

Sun, 31 Aug 2025 00:00:00 GMT

Junming Xie — In this paper, we investigate the convexity of mean convex asymptotically conical self-expanders to the mean curvature flow in ℝⁿ⁺¹. Specifically, for n≥ 3, we show that any n-dimensional complete mean convex self-expander asymptotic to mean convex and weakly convex cones must be strictly convex.

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.

Convexity of 2-convex translating and expanding solitons to the mean curvature flow in ℝⁿ⁺¹

Fri, 25 Nov 2022 00:00:00 GMT

Junming Xie, Jiangtao Yu — In this paper, inspired by the work of Spruck-Xiao [27] and based partly on a result of Derdziński [11], we prove the convexity of complete 2-convex translating and expanding solitons to the mean curvature flow in ℝⁿ⁺¹. More precisely, for n≥ 3, we show that any n-dimensional complete 2-convex translating solitons are convex, and any n-dimensional complete 2-convex self-expanders asymptotic to (strictly) mean convex cones are convex.

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.

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.