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Thomas Murphy

Publications and source records attributed to Thomas Murphy.

At least 19 recordsLinked to original sources
arXiv2026

Twisted scalar curvature as a moment map

We develop the moment map theory of the twisted scalar curvature of a K\"ahler metric. Primarily, we introduce a coupled system of equations on a holomorphic submersion intertwining the twisted scalar curvature of a K\"ahler metric on the base and the fibrewise scalar curvature of a relatively K\"ahler metric on the total space. This resulting system can be viewed as producing the natural coupled metric geometry of holomorphic submersions, and we show that this system appears canonically as a moment map. The approach generalises to foliations, where we prove similar results.

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arXiv2025

3-symmetric spaces, Ricci solitons, and homogeneous structures

The full classification of Riemannian $3$-symmetric spaces is presented. Up to Riemannian products the main building blocks consist in (possibly symmetric) spaces with semisimple isometry group, nilpotent Lie groups of step at most $2$ and spaces of type III and IV. For the most interesting family of examples, the Type III spaces, we produce an explicit description including results concerning the moduli space of all $3$-symmetric metrics living on a given Type III space. Each moduli space contains a unique distinguished point corresponding to an (almost-K\"ahler) expanding Ricci soliton metric. For certain classes of 3-symmetric metrics there are many different groups acting transitively and isometrically on a fixed Riemannian 3-symmetric space. The construction of expanding Ricci solitons on spaces of Type III is also shown to generalize to \emph{any} effective representation of a simple Lie group of non-compact type, yielding a very general construction of homogeneous Ricci solitons. We also give a procedure to compute the isometry group of any Ambrose--Singer space.

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arXiv2023

Riemannian Geometry and Molecular Similarity II: K\"ahler Quantization

Shape-similarity between molecules is a tool used by chemists for virtual screening, with the goal of reducing the cost and duration of drug discovery campaigns. This paper reports an entirely novel shape descriptor as an alternative to the previously described RGMolSA descriptors \cite{cole2022riemannian}, derived from the theory of Riemannian geometry and K\"ahler quantization (KQMolSA). The treatment of a molecule as a series of intersecting spheres allows us to obtain the explicit \textit{Riemannian metric} which captures the geometry of the surface, which can in turn be used to calculate a Hermitian matrix $\mathbb{M}$ as a directly comparable surface representation. The potential utility of this method is demonstrated using a series of PDE5 inhibitors considered to have similar shape. The method shows promise in its capability to handle different conformers, and compares well to existing shape similarity methods. The code and data used to produce the results are available at: \url{https://github.com/RPirie96/KQMolSA}.

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arXiv2021

Rigidity of $SU_n$-type symmetric spaces

We prove that the bi-invariant Einstein metric on $SU_{2n+1}$ is isolated in the moduli space of Einstein metrics, even though it admits infinitesimal deformations. This gives a non-K\"ahler, non-product example of this phenomenon adding to the famous example of $\mathbb{CP}^{2n}\times\mathbb{CP}^{1}$ found by Koiso. We apply our methods to derive similar solitonic rigidity results for the K\"ahler--Einstein metrics on `odd' Grassmannians. We also make explicit a connection between non-integrable deformations and the dynamical instability of metrics under Ricci flow.

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arXiv2020

Bounding the invariant spectrum when the scalar curvature is non-negative

On compact Riemannian manifolds with a large isometry group we investigate the invariant spectrum of the ordinary Laplacian. For either a toric Kaehler metric, or a rotationally-symmetric metric on the sphere, we produce upper bounds for all eigenvalues of the invariant spectrum assuming non-negative scalar curvature.

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arXiv2018

Compact Hermitian symmetric spaces, coadjoint orbits, and the dynamical stability of the Ricci flow

Using a stability criterion due to Kr\"oncke, we show, providing ${n\neq 2k}$, the K\"ahler--Einstein metric on the Grassmannian $Gr_{k}(\mathbb{C}^{n})$ of complex $k$-planes in an $n$-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Kr\"oncke and Knopf--Sesum on the instability of the Fubini--Study metric on $\mathbb{CP}^{n}$ for $n>1$. The key to the proof is using the description of Grassmannians as certain coadjoint orbits of $SU(n)$. We are also able to prove that Kr\"oncke's method will not work on any of the other compact, irreducible, Hermitian symmetric spaces.

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arXiv2018

Numerically destabilizing minimal discs

When calculating the index of a minimal surface, the set of smooth functions on a domain with compact support is the standard setting to describe admissible variations. We show that the set of admissible variations can be widened in a geometrically meaningful manner by considering the difference of area functional, leading to a more general notion of index. This allows us to produce explicit examples of destabilizing perturbations for the fundamental Scherk surface and dihedral Enneper surfaces. In the case of dihedral Enneper surfaces we show that both the classical and our modified index can be explicitly determined.

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arXiv2017

Random Manifolds have no Totally Geodesic Submanifolds

For $n\geq 4$ we show that generic closed Riemannian $n$-manifolds have no nontrivial totally geodesic submanifolds, answering a question of Spivak. An immediate consequence is a severe restriction on the isometry group of a generic Riemannian metric. Both results are widely believed to be true, but we are not aware of any proofs in the literature.

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arXiv2016

Destabilising compact warped product Einstein manifolds

The linear stability of warped product Einstein metrics as fixed points of the Ricci flow is investigated. We generalise the results of Gibbons, Hartnoll and Pope and show that in sufficiently low dimensions, all warped product Einstein metrics are unstable. By exploiting the relationship between warped product Einstein metrics, quasi-Einstein metrics and Ricci solitons, we introduce a new destabilising perturbation (the Ricci variation) and show that certain infinite families of warped product Einstein metrics will be unstable in high dimensions.

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arXiv2015

Complex Riemannian Foliations of open K\"ahler manifolds

Classification results for complex Riemannian foliations are obtained. For open subsets of irreducible Hermitian symmetric spaces of compact type, where one has explicit control over the curvature tensor, we completely classify such foliations by studying the infinitesimal model associated to the canonical connection. We also establish results for symmetric spaces of non-compact type and a general rigidity result for any irreducible K\"ahler manifold.

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arXiv2015

Approximating Ricci solitons and quasi-Einstein metrics on toric surfaces

We present a general numerical method for investigating prescribed Ricci curvature problems on toric K\"ahler manifolds. This method is applied to two generalisations of Einstein metrics, namely Ricci solitons and quasi-Einstein metrics. We begin by recovering the Koiso--Cao soliton and the L\"u--Page--Pope quasi-Einstein metrics on $\mathbb{CP}^{2}\sharp\overline{\mathbb{CP}}^{2}$ (in both cases the metrics are known explicitly). We also find numerical approximations to the Wang--Zhu soliton on $\mathbb{CP}^{2}\sharp 2\overline{\mathbb{CP}}^{2}$ (here the metric is not known explicitly). Finally, a substantial numerical investigation of the quasi-Einstein equation on $\mathbb{CP}^{2}\sharp 2\overline{\mathbb{CP}}^{2}$ is conducted. In this case it is an open problem as to whether such metrics exist on this manifold. We find metrics that solve the quasi-Einstein equation to the same degree of accuracy as the approximations to the Wang--Zhu soliton solve the Ricci soliton equation.

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arXiv2015

Bounding the first invariant eigenvalue of toric K\"ahler manifolds

We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^{1}$-invariant metrics on $\mathbb{CP}^{1}$ to general toric K\"ahler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metrics on complex projective space $\mathbb{C}P^{n}$ is exhibited. We derive an analogous bound in the case when the metric is extremal and a detailed study is made of the accuracy of the bound in the case of Calabi's extremal metrics on $\mathbb{C}P^{2}\sharp -\mathbb{C}P^{2}$.

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arXiv2015

Conformally K\"ahler geometry and quasi-Einstein metrics

We prove that the quasi-Einstein metrics found by L\"u, Page and Pope on $\mathbb{C}P^{1}$-bundles over Fano K\"ahler-Einstein bases are conformally K\"ahler and that the K\"ahler class of the conformal metric is a multiple of the first Chern class. A detailed study of the lowest-dimensional example of such metrics on $\mathbb{C}P^{2}\sharp \overline{\mathbb{C}P}^{2}$ using the methods developed by Abreu and Guillemin for studying toric K\"ahler metrics is given. Our methods yield, in a unified framework, proofs of the existence of the Page, Koiso-Cao and L\"u-Page-Pope metrics on $\mathbb{C}P^{2}\sharp \overline{\mathbb{C}P}^{2}$. Finally, we investigate the properties that similar quasi-Einstein metrics would have if they also exist on the toric surface $\mathbb{C}P^{2}\sharp 2 \overline{\mathbb{C}P}^{2}$.

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arXiv2014

Numerical approximations to extremal toric K\"ahler metrics with arbitrary K\"ahler class

We develop new algorithms for approximating extremal toric K\"ahler metrics. We focus on an extremal metric on $\mathbb{CP}^{2}\sharp2\overline{\mathbb{CP}}^{2}$, which is conformal to an Einstein metric (the Chen-LeBrun-Weber metric). We compare our approximation to one given by Bunch and Donaldson and compute various geometric quantities. In particular, we demonstrate a small eigenvalue of the scalar Laplacian of the Einstein metric which gives a numerical evidence that the Einstein metric is conformally unstable under the Ricci flow.

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arXiv2013

Rigidity Results for Hermitian-Einstein manifolds

A differential operator introduced by A. Gray on the unit sphere bundle of a K\"ahler-Einstein manifold is studied. A lower bound for the first eigenvalue of the Laplacian for the Sasaki metric on the unit sphere bundle of a K\"ahler-Einstein manifold is derived. Some rigidity theorems classifying complex space forms amongst compact Hermitian surfaces and the product of two projective lines amongst all K\"ahler-Einstein surfaces are then derived.

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arXiv2012

On the spectrum of the Page and the Chen-LeBrun-Weber metrics

We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen-LeBrun-Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the calculation of the invariant part of the spectrum for both metrics. We go on to discuss an application of these bounds to the linear stability of the metrics. We also give numerical evidence to suggest that the bounds for both metrics are extremely close to the actual eigenvalue.

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