https://neckpinch.com/authors/ronan-conlon/New arXiv papers by Ronan Conlon, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.02164arxiv:2610.02164Thu, 01 Oct 2026 00:00:00 GMTRonan J. Conlon, Alix Deruelle — We establish the uniqueness of the soliton vector field of a complete shrinking gradient Kähler-Ricci soliton up to pushforward by biholomorphism.Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniqueness https://arxiv.org/abs/2609.16733arxiv:2609.16733Tue, 15 Sep 2026 00:00:00 GMTCharles Cifarelli, Ronan Conlon, Max Hallgren, Junsheng Zhang — For any volume-collapsing finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface, we show the flow satisfies a Type I curvature bound and classify the corresponding tangent flows. Combined with previous results, this shows that any finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface is of Type I.Ricci flowKähler–Ricci flowSingularities(Non)collapsingKähler geometry https://arxiv.org/abs/2607.26054arxiv:2607.26054Tue, 28 Jul 2026 00:00:00 GMTRonan J. Conlon, Alix Deruelle — We show that any complete shrinking gradient Kähler-Ricci soliton on a resolution of a Kähler cone is necessarily asymptotically conical. From a result of Esparza, it then follows that up to pullback by biholomorphism, there exists at most one complete shrinking gradient Kähler-Ricci soliton on such a resolution. This confirms a special case of the uniqueness part of a conjecture of Song-Zhang. Some other consequences are also discussed.Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2603.25082arxiv:2603.25082Thu, 26 Mar 2026 00:00:00 GMTRichard H. Bamler, Eric Chen, Ronan J. Conlon — We construct an example of an asymptotically conical (AC) non-Kähler expanding gradient Ricci soliton that has a Kähler tangent cone at infinity. This yields an example of a Kähler cone that can be desingularised by a smooth AC expanding gradient Ricci soliton but not by a smooth AC expanding gradient Kähler–Ricci soliton.Ricci solitonsKähler–Ricci solitonsKähler geometryBoundaries & non-compact https://arxiv.org/abs/2512.18137arxiv:2512.18137Fri, 19 Dec 2025 00:00:00 GMTIvin Babu, Ronan J. Conlon, Alix Deruelle — We show that any toric asymptotically conical shrinking gradient Kähler-Ricci soliton on an anti-canonically polarised resolution of a Kähler cone satisfies a complex Monge-Ampère equation. We then set up an Aubin continuity path to solve the resulting equation and show that it has a solution at the initial value of the path parameter in the toric case. This we do by implementing another continuity method. Finally, we prove openness of the initial value of the path parameter independent of the toricity.Ricci solitonsKähler–Ricci solitonsKähler geometryBoundaries & non-compact https://arxiv.org/abs/2502.19804arxiv:2502.19804Thu, 27 Feb 2025 00:00:00 GMTRonan J. Conlon, Max Hallgren, Zilu Ma — We show that any non-collapsed finite time singularity of the Ricci flow on a compact Kähler surface is of Type I. Combined with a previous result of the first author, Cifarelli, and Deruelle, it follows that any such singularity is modeled on the shrinking Ricci soliton of Feldman-Ilmanen-Knopf on the total space of the line bundle 𝓞_(ℙ¹)(−1)→ℙ¹.Ricci flowRicci solitonsSingularities(Non)collapsingKähler geometry https://arxiv.org/abs/2403.04089arxiv:2403.04089Wed, 06 Mar 2024 00:00:00 GMTPak-Yeung Chan, Ronan J. Conlon, Yi Lai — In 1996, H.-D. Cao constructed a U(n)-invariant steady gradient Kähler-Ricci soliton on ℂⁿ and asked whether every steady gradient Kähler-Ricci soliton of positive curvature on ℂⁿ is necessarily U(n)-invariant (and hence unique up to scaling). Recently, Apostolov-Cifarelli answered this question in the negative for n=2. Here, we construct a family of U(1)× U(n-1)-invariant, but not U(n)-invariant, complete steady gradient Kähler-Ricci solitons with strictly positive curvature operator on real (1, 1)-forms (in particular, with strictly positive sectional curvature) on ℂⁿ for n≥3, thereby answering Cao's question in the negative for n≥3. This family of steady Ricci solitons interpolates between Cao's U(n)-invariant steady Kähler-Ricci soliton and the product of the cigar soliton and Cao's U(n-1)-invariant steady Kähler-Ricci soliton. This provides the Kähler analog of the Riemannian flying wings construction of Lai. In the process of the proof, we also demonstrate that the almost diameter rigidity of ℙⁿ endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by 2 on real (1, 1)-forms.Ricci solitonsKähler–Ricci solitonsAncient solutionsSphere theoremsKähler geometryStability & uniqueness https://arxiv.org/abs/2206.10785arxiv:2206.10785Wed, 22 Jun 2022 00:00:00 GMTRichard H. Bamler, Charles Cifarelli, Ronan J. Conlon, Alix Deruelle — We prove the existence of a unique complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on the blowup of ℂ×ℙ¹ at one point. This completes the classification of such solitons in two complex dimensions.Ricci solitonsKähler–Ricci solitonsSingularitiesSurfacesKähler geometryScalar curvature https://arxiv.org/abs/2205.08482arxiv:2205.08482Tue, 17 May 2022 00:00:00 GMTCharles Cifarelli, Ronan J. Conlon, Alix Deruelle — Let D be a toric Kähler-Einstein Fano manifold. We show that any toric shrinking gradient Kähler-Ricci soliton on certain toric blowups of ℂ× D satisfies a complex Monge-Ampère equation. We then set up an Aubin continuity path to solve this equation and show that it has a solution at the initial value of the path parameter. This we do by implementing another continuity method.Ricci solitonsKähler–Ricci solitonsEinstein metricsSingularitiesKähler geometryKähler–Einstein & K-stability https://arxiv.org/abs/2203.04380arxiv:2203.04380Tue, 08 Mar 2022 00:00:00 GMTCharles Cifarelli, Ronan J. Conlon, Alix Deruelle — We show that the underlying complex manifold of a complete non-compact two-\linebreak dimensional shrinking gradient Kähler-Ricci soliton (M, g, X) with soliton metric g with bounded scalar curvature R_g whose soliton vector field X has an integral curve along which R_gnot→0 is biholomorphic to either ℂ×ℙ¹ or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.Ricci flowKähler–Ricci flowRicci solitonsKähler–Ricci solitonsSingularitiesKähler geometryScalar curvatureBoundaries & non-compact