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Ronan J. Conlon

Publications and source records attributed to Ronan J. Conlon.

At least 19 recordsLinked to original sources
arXiv2026

Examples of complete Calabi--Yau metrics on affine smoothings of irregular toric Calabi--Yau cones

We present infinitely many new examples of affine Calabi--Yau manifolds of Euclidean volume growth and quadratic curvature decay, whose tangent cones at infinity are irregular and have smooth links. In the process, we demonstrate (and provide the relevant computer code) how to explicitly compute the Reeb field, the Minkowski summand cone, and its maximal decompositions for a given toric Calabi--Yau cone with smooth link from the data of its toric diagram. In complex dimension three, the component corresponding to a maximal Minkowski decomposition into lattice summands is a smoothing component if and only if every summand is either a primitive lattice segment or a unimodular lattice triangle. Furthermore, we propose an effective strategy to generate smoothable Calabi--Yau cones from a given non-smoothable one by taking Minkowski sums of certain toric diagrams, and provide an example to illustrate the method.

math.DG↗
arXiv2026

A family of Kähler flying wing steady Ricci solitons

In $1996$, H.-D. Cao constructed a $U(n)$-invariant steady gradient Kähler-Ricci soliton on $\mathbb{C}^{n}$ and asked whether every steady gradient Kähler-Ricci soliton of positive curvature on $\mathbb{C}^{n}$ 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 $U(1)\times U(n-1)$-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 $\mathbb{C}^{n}$ for $n\geq3$, thereby answering Cao's question in the negative for $n\geq3$. 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 $\mathbb{P}^{n}$ endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by $2$ on real $(1,\,1)$-forms.

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arXiv2026

Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

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.

math.DG↗
arXiv2026

Warped quasi-asymptotically conical Calabi-Yau metrics

We construct many new examples of complete Calabi-Yau metrics of maximal volume growth on certain smoothings of Cartesian products of Calabi-Yau cones with smooth cross-sections. A detailed description of the geometry at infinity of these metrics is given in terms of a compactification by a manifold with corners obtained through the notion of weighted blow-up for manifolds with corners. A key analytical step in the construction of these Calabi-Yau metrics is to derive good mapping properties of the Laplacian on some suitable weighted Hölder spaces. Our methods also produce singular Calabi-Yau metrics with an isolated conical singularity modelled on a Calabi-Yau cone distinct from the tangent cone at infinity, in particular yielding a transition behavior between different Calabi-Yau cones as conjectured by Yang Li. This is used to exhibit many examples where the tangent cone at infinity does not uniquely specify a complete Calabi-Yau metric with exact Kähler form.

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arXiv2026

A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

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.

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arXiv2025

An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for $t=0$

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.

math.DG↗
arXiv2025

Non-collapsed finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

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 $\mathcal{O}_{\mathbb{P}^1}(-1)\to\mathbb{P}^{1}$.

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arXiv2024

An Aubin continuity path for shrinking gradient Kähler-Ricci solitons

Let $D$ be a toric Kähler-Einstein Fano manifold. We show that any toric shrinking gradient Kähler-Ricci soliton on certain toric blowups of $\mathbb{C}\times D$ 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.

math.DG↗
arXiv2022

On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

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 $\operatorname{R}_{g}$ whose soliton vector field $X$ has an integral curve along which $\operatorname{R}_{g}\not\to0$ is biholomorphic to either $\mathbb{C}\times\mathbb{P}^{1}$ 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.

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arXiv2022

Classification of asymptotically conical Calabi-Yau manifolds

A Riemannian cone $(C, g_C)$ is by definition a warped product $C = \mathbb{R}^+ \times L$ with metric $g_C = dr^2 \oplus r^2 g_L$, where $(L,g_L)$ is a compact Riemannian manifold without boundary. We say that $C$ is a Calabi-Yau cone if $g_C$ is a Ricci-flat Kähler metric and if $C$ admits a $g_C$-parallel holomorphic volume form; this is equivalent to the cross-section $(L,g_L)$ being a Sasaki-Einstein manifold. In this paper, we give a complete classification of all smooth complete Calabi-Yau manifolds asymptotic to some given Calabi-Yau cone at a polynomial rate at infinity. As a special case, this includes a proof of Kronheimer's classification of ALE hyper-Kähler $4$-manifolds without twistor theory.

math.DG↗
arXiv2019

Classification results for expanding and shrinking gradient Kähler-Ricci solitons

We first show that a Kähler cone appears as the tangent cone of a complete expanding gradient Kähler-Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model (on which the soliton lives). This allows us to classify two-dimensional complete expanding gradient Kähler-Ricci solitons with quadratic curvature decay with derivatives. We then show that any two-dimensional complete shrinking gradient Kähler-Ricci soliton whose scalar curvature tends to zero at infinity is, up to pullback by an element of $GL(2,\,\mathbb{C})$, either the flat Gaussian shrinking soliton on $\mathbb{C}^{2}$ or the $U(2)$-invariant shrinking gradient Kähler-Ricci soliton of Feldman-Ilmanen-Knopf on the blowup of $\mathbb{C}^{2}$ at one point. Finally, we show that up to pullback by an element of $GL(n,\,\mathbb{C})$, the only complete shrinking gradient Kähler-Ricci soliton with bounded Ricci curvature on $\mathbb{C}^{n}$ is the flat Gaussian shrinking soliton and on the total space of $\mathcal{O}(-k)\to\mathbb{P}^{n-1}$ for $0<k<n$ is the $U(n)$-invariant example of Feldman-Ilmanen-Knopf. In the course of the proof, we establish the uniqueness of the soliton vector field of a complete shrinking gradient Kähler-Ricci soliton with bounded Ricci curvature in the Lie algebra of a torus. A key tool used to achieve this result is the Duistermaat-Heckman theorem from symplectic geometry. This provides the first step towards understanding the relationship between complete shrinking gradient Kähler-Ricci solitons and algebraic geometry.

math.DG↗
arXiv2019

Quasi-asymptotically conical Calabi-Yau manifolds

We construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). We do so by first providing a natural compactification of QAC-spaces by manifolds with fibred corners and by giving a definition of QAC-metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on QAC-spaces developed by the second author and Mazzeo, we can in many instances obtain Kähler QAC-metrics having Ricci potential decaying sufficiently fast at infinity. This allows us to obtain QAC Calabi-Yau metrics in the Kähler classes of these metrics by solving a corresponding complex Monge-Ampère equation.

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arXiv2016

Expanding Kähler-Ricci solitons coming out of Kähler cones

We give necessary and sufficient conditions for a Kähler equivariant resolution of a Kähler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kähler-Ricci soliton. In particular, it follows that for any $n\in\mathbb{N}_{0}$ and for any negative line bundle $L$ over a compact Kähler manifold $D$, the total space of the vector bundle $L^{\oplus (n+1)}$ admits a unique AC expanding gradient Kähler-Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if $c_{1}(K_{D}\otimes(L^{*})^{\otimes (n+1)})>0$. This generalises the examples already known in the literature. We further prove a general uniqueness result and show that the space of certain AC expanding gradient Kähler-Ricci solitons on $\mathbb{C}^{n}$ with positive curvature operator on $(1,\,1)$-forms is path-connected.

math.DG↗
arXiv2014

Asymptotically conical Calabi-Yau metrics on quasi-projective varieties

Let X be a compact Kahler orbifold without \C-codimension-1 singularities. Let D be a suborbifold divisor in X such that D \supset Sing(X) and -pK_X = q[D] for some p, q \in \N with q > p. Assume that D is Fano. We prove the following two main results. (1) If D is Kahler-Einstein, then, applying results from our previous paper, we show that each Kahler class on X\D contains a unique asymptotically conical Ricci-flat Kahler metric, converging to its tangent cone at infinity at a rate of O(r^{-1-ε}) if X is smooth. This provides a definitive version of a theorem of Tian and Yau. (2) We introduce new methods to prove an analogous statement (with rate O(r^{-0.0128})) when X = Bl_{p}P^3 and D = Bl_{p_1,p_2}P^2 is the strict transform of a smooth quadric through p in P^3. Here D is no longer Kahler-Einstein, but the normal S^1-bundle to D in X admits an irregular Sasaki-Einstein structure which is compatible with its canonical CR structure. This provides the first example of an affine Calabi-Yau manifold of Euclidean volume growth with irregular tangent cone at infinity.

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