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Timothy Buttsworth

Publications and source records attributed to Timothy Buttsworth.

At least 19 recordsLinked to original sources
arXiv2026

Integral Gauss formula and the Poisson equation for the $G_2$-Laplacian

We produce a formula, analogous to the Gauss-Codazzi equation, which relates the geometry of a $G_2$-structure and its Hodge Laplacian to the geometry of the induced $SU(3)$-structure on an embedded hypersurface. As an application, we obtain necessary conditions for the solvability of the Poisson equation for (not necessarily closed) $G_2$-structures in a neighbourhood of this hypersurface. Next, we prove that our conditions are sufficient in the cohomogeneity one setting, assuming the symmetry group is compact and simple.

math.DG↗
arXiv2026

Incompressible Euler fluids on compact cohomogeneity one manifolds

Let $(M,\mathsf{g})$ be a connected and compact Riemannian manifold admitting an isometric action by a compact Lie group $G$ whose principal orbits have codimension one. We show that any $G$-invariant, smooth, and divergence-free vector field $u_0$ on $(M,\mathsf{g})$ initiates a $G$-invariant time-varying velocity-pressure pair $(u,p)$ which has time interval $\mathbb{R}$, is smooth, and solves the incompressible Euler fluid equations.

math.DG↗
arXiv2024

Computationally-assisted proof of a novel $\mathsf{O}(3)\times \mathsf{O}(10)$-invariant Einstein metric on $S^{12}$

We prove existence of a non-round Einstein metric $g$ on $S^{12}$ that is invariant under the usual cohomogeneity one action of $\mathsf{O}(3)\times\mathsf{O}(10)$ on $S^{12}\subset \mathbb{R}^{13}= \mathbb{R}^3\oplus \mathbb{R}^{10}$. The proof involves using several rigorous numerical analysis techniques to produce a Riemannian metric $\hat{g}$ which approximately satisfies the Einstein condition to known high precision, and then demonstrating that $\hat{g}$ can be perturbed into a true Einstein metric $g$.

math.DG↗
arXiv2024

Accurate transient heat flux from simple treatment of surface temperature distribution in the semi-infinite case

When the variations of surface temperature are measured both spatially and temporally, analytical expressions that correctly account for multi-dimensional transient conduction can be applied. To enhance the accessibility of these accurate multi-dimensional methods, expressions for converting between surface temperature and heat flux are presented as the sum of the one-dimensional component plus the multi-dimensional component. Advantage arises herein because potential numerical challenges are isolated within the one-dimensional component and practitioners are already familiar with well-established one-dimensional methods. The second derivative of the surface heat flux distribution scaled by the thermal diffusivity and the duration of the experiment delivers an approximation of the multi-dimensional conduction term. For the analysis of experiments in which multi-dimensional effects are significant, a simplified numerical approach in which the temperature within each pixel is treated as uniform is demonstrated. The approach involves convolution of temperature differences and pixel-based impulse response functions, followed by a summation of results across the region of interest, but there are no singularities that require special treatment in the multi-dimensional component. Recovery of heat flux distributions to within 1% is demonstrated for two-dimensional heat flux distributions discretized using several tens of elements, and for a three-dimensional distribution discretized using several hundred pixels. Higher accuracy can be achieved by using finer spatial resolution, but the level of discretization used herein is likely sufficient for practical applications since typical experimental uncertainties are much larger than 1%.

physics.ins-det↗
arXiv2023

Scalar curvature along Ebin geodesics

Let $M$ be a smooth, compact manifold and let $\mathcal{N}_{\mu}$ denote the set of Riemannian metrics on $M$ with smooth volume density $\mu$. For a given $g_0\in \mathcal{N}_{\mu}$, we show that if $\dim(M)\ge 5$, then there exists an open and dense subset $\mathcal{Y}_{g_0} \subset T_{g_0} \mathcal{N}_{\mu}$ (in the $C^{\infty}$ topology) so that for each $h\in \mathcal{Y}_{g_0}$, the $(\mathcal{N}_{\mu},L^2)$ Ebin geodesic $\gamma_h(t)$ with $\gamma_h(0)=g_0$ and $\gamma_h'(0)=h$ satisfies $\lim_{t \to \infty}$ $R(\gamma_h(t))=-\infty$, uniformly.

math.DG↗
arXiv2023

Ancient Ricci flows of bounded girth

For each $n\ge 3$, we construct a 'pancake-like', $O(2)\times O(n-1)$-invariant ancient Ricci flow with positive curvature operator and bounded "girth", and we determine its asymptotic limits backwards in time. This solution is new even in dimension three. The construction hinges on the Ricci flow invariance of certain conditions on the curvature and its spatial derivatives under this symmetry regime, whose proof does not follow from Hamilton's tensor maximum principle.

math.DG↗
arXiv2022

Canonical surgeries in rotationally invariant Ricci flow

We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of [32]. Results of Bamler-Kleiner [8] and Haslhofer [29] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared.

math.DG↗
arXiv2021

$SU(2)$-invariant steady gradient Ricci solitons on four-manifolds

Using center manifolds and topological degree theory, we construct a new family of complete, $SU(2)$-invariant and steady gradient Ricci solitons on the four-dimensional non-compact cohomogeneity one manifold with group diagram $\mathbb{Z}_4\subset U(1)\subset SU(2)$. We also provide simpler constructions of the existing $U(2)$-invariant steady and complete gradient solitons on the cohomogeneity one manifolds with group diagrams $\mathbb{Z}_n\subset U(1)\subset SU(2)$ for any $n\in \mathbb{N}$, including Appleton's non-collapsed solitons for $n\ge 3$.

math.DG↗
arXiv2021

$SO(2)\times SO(3)$-invariant Ricci solitons and ancient flows on $\mathbb{S}^4$

Consider the standard action of $SO(2)\times SO(3)$ on $\mathbb{R}^5=\mathbb{R}^2\oplus \mathbb{R}^3$. We establish the existence of a uniform constant $\mathcal{C}>0$ so that any $SO(2)\times SO(3)$-invariant Ricci soliton on $\mathbb{S}^4\subset \mathbb{R}^5$ with Einstein constant $1$ must have Riemann curvature and volume bounded by $\mathcal{C}$, and injectivity radius bounded below by $\frac{1}{\mathcal{C}}$. This observation, coupled with basic numerics, gives strong evidence to suggest that the only $SO(2)\times SO(3)$-invariant Ricci solitons on $\mathbb{S}^4$ are round. We also encounter the so-called `pancake' ancient solution of the Ricci flow.

math.DG↗
arXiv2020

Prescribing Ricci curvature on a Product of Spheres

We prove an existence result for the prescribed Ricci curvature equation for certain doubly warped product metrics on $\mathbb{S}^{d_1+1}\times \mathbb{S}^{d_2}$, where $d_i \geq 2$. If $T$ is a metric satisfying certain curvature assumptions, we show that $T$ can be scaled independently on the two factors so as to itself be the Ricci tensor of some metric.

math.DG↗
arXiv2020

Deducing Flux from Single Point Temperature History when Relative Spatial Variation of Flux is Prescribed

Surface heat transfer in convective and radiative environments is sometimes measured by recording the surface temperature history in a transient experiment and interpreting this surface temperature with the aid of a suitable model for transient conduction within the substrate. The semi-infinite one-dimensional model is often adopted, and several well-developed techniques for application of this model to surface temperature data are available. However, when a spatial variation of heat flux exists across the surface, the application of the semi-infinite one-dimensional approach may not always be a reasonable approximation. In this paper we introduce a method for treatment of the measured surface temperature history that is more accurate than the semi-infinite one-dimensional approximation when substrate lateral conduction is significant and the relative spatial distribution of the flux is known a priori. This new method uses the so-called \textit{Neumann heat kernel}, which evolves a temperature over an insulated domain with unit energy initially deposited at a specified point. A useful impulse response function is formed by integrating this Neumann heat kernel against the spatial variation of flux over the surface of the domain. Neumann heat kernels are constructed for the solid box, cylinder, and sphere. By applying the heat kernel result for the sphere to the analysis of a convective experiment using hemispherical-nosed probes, we demonstrate how the theoretical results enhance the practical analysis of transient surface temperature measurements. The current approach is superior to former methods relying on semi-empirical approximations because the multi-dimensional heat conduction within the substrate is modelled with greater fidelity using the heat kernel analysis.

physics.ins-det↗
arXiv2019

The Prescribed Ricci Curvature Problem for Homogeneous Metrics

The prescribed Ricci curvature problem consists in finding a Riemannian metric $g$ on a manifold $M$ such that the Ricci curvature of $g$ equals a given $(0,2)$-tensor field $T$. We survey the recent progress on this problem in the case where $M$ is a homogeneous space.

math.DG↗
arXiv2019

Local Stability of Einstein Metrics Under the Ricci Iteration

We provide a sufficient condition for the local stability of closed Einstein manifolds of positive Ricci curvature under the Ricci iteration in terms of the spectrum of the Lichnerowicz Laplacian acting on divergence-free tensor fields. We use this result to consider the stability of several Einstein manifolds under the Ricci iteration, including symmetric spaces of compact type.

math.DG↗
arXiv2018

On the Ricci iteration for homogeneous metrics on spheres and projective spaces

We study the Ricci iteration for homogeneous metrics on spheres and complex projective spaces. Such metrics can be described in terms of modifying the canonical metric on the fibers of a Hopf fibration. When the fibers of the Hopf fibration are circles or spheres of dimension 2 or 7, we observe that the Ricci iteration as well as all ancient Ricci iterations can be completely described using known results. The remaining and most challenging case is when the fibers are spheres of dimension 3. On the 3-sphere itself, using a result of Hamilton on the prescribed Ricci curvature equation, we establish existence and convergence of the Ricci iteration and confirm in this setting a conjecture on the relationship between ancient Ricci iterations and ancient solutions to the Ricci flow. In higher dimensions we obtain sufficient conditions for the solvability of the prescribed Ricci curvature equation as well as partial results on the behavior of the Ricci iteration.

math.DG↗
arXiv2018

Cohomogeneity-One Quasi-Einstein Metrics

Let $G/H$ be a connected, simply connected homogeneous space of a compact Lie group $G$. We study $G$-invariant quasi-Einstein metrics on the cohomogeneity one manifold $G/H\times (0,1)$ imposing the so-called monotypic condition on $G/H$. We obtain estimates on the rate of blow-up for these metrics near a singularity under a mild assumption on $G/H$. Next, we demonstrate that we can find quasi-Einstein metrics satisfying arbitrary $G$-invariant Dirichlet conditions.

math.DG↗
arXiv2017

The Prescribed Ricci Curvature Problem on Three-Dimensional Unimodular Lie Groups

Let G be a three-dimensional unimodular Lie group, and let T be a left-invariant symmetric (0, 2)-tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair (g, c) consisting of a left-invariant Riemannian metric g and a positive constant c such that Ric(g) = cT, where Ric(g) is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in almost all cases, there exists at most one positive constant c such that Ric(g) = cT is solvable for some left-invariant Riemannian metric g.

math.DG↗