https://neckpinch.com/authors/xi-zhang/New arXiv papers by Xi Zhang, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2411.12012arxiv:2411.12012Mon, 18 Nov 2024 00:00:00 GMTChenghong He, Di Wu, Xi Zhang — We prove that there is no nontrivial L²-integrable harmonic 1-form on noncompact complete gradient steady Ricci solitons or noncompact complete gradient shrinking Kähler-Ricci solitons. As an application, it can be used to distinguish certain flat vector bundles that arise from fundamental group representations into SL(r,ℂ).Ricci solitonsKähler–Ricci solitonsKähler geometryBoundaries & non-compact https://arxiv.org/abs/2406.08778arxiv:2406.08778Thu, 13 Jun 2024 00:00:00 GMTJiawei Liu, Shiyu Zhang, Xi Zhang — In this paper, we show the regularity and uniqueness of the twisted conical Kähler-Ricci flow running from a positive closed current with zero Lelong number, which extends the regularizing property of the smooth twisted Kähler-Ricci flow, known as Guedj-Zeriahi's existence theorem and Di Nezza-Lu's uniqueness theorem, to the conical singularity case.Ricci flowKähler–Ricci flowSingularitiesKähler geometryStability & uniqueness https://arxiv.org/abs/2406.04590arxiv:2406.04590Fri, 07 Jun 2024 00:00:00 GMTJiawei Liu, Xi Zhang — In this paper, we study the limit behavior of the conical Kähler-Ricci flow as its cone angle tends to zero. More precisely, we prove that as the cone angle tends to zero, the conical Kähler-Ricci flow converges to a unique Kähler-Ricci flow, which is smooth outside the divisor and admits cusp singularity along the divisor.Ricci flowKähler–Ricci flowSingularitiesKähler geometry https://arxiv.org/abs/2405.10495arxiv:2405.10495Fri, 17 May 2024 00:00:00 GMTChang Li, Huaiyu Zhang, Xi Zhang — In this paper, we study the rigidity of eigenvalues of shring Ricci solitons. It is known that the drifted Laplacian on shrinking Ricci solitons has discrete spectrum, its eigenvalues have a lower bound and a rigidity result holds. Firstly, we show that if the n^th eigenvalue is close to this lower bound, then the n-soliton must be the trivial Gaussian soliton ℝⁿ. Secondly, we show similar results for the (n-1)^th and (n-2)^th eigenvalue under a non-collapsing condition. Lastly, we give an alomost rigidity for the k^th eigenvalue with general k. Part of our results could be viewed as an soliton (could be noncompact) analog of Theorem 1.1 (which only holds for compact manifolds) in Peterson (Invent. Math. 138 (1999): 1-21).Ricci solitons(Non)collapsingStability & uniquenessBoundaries & non-compactEigenvalues & spectrum