Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz

Hyperkähler manifolds with circle actions and the Gibbons-Hawking Ansatz

We show that a complete simply-connected hyperkaehler 4-manifold with an isometric triholomorphic circle action is obtained from the Gibbons-Hawking ansatz with some suitable harmonic function.

The Gibbons-Hawking ansatz in generalized Kähler geometry

We derive a local ansatz for generalized Kähler surfaces with nondegenerate Poisson structure and a biholomorphic $S^1$ action which generalizes the classic Gibbons-Hawking ansatz for invariant hyperKähler manifolds, and allows for the choice of one arbitrary function. By imposing the generalized Kähler-Ricci soliton equation, or equivalently the equations of type IIB string theory, the construction becomes rigid, and we classify all complete solutions with the smallest possible symmetry group.

具有无限拓扑类型的Gibbons-Hawking方案的复杂结构

Complex structures of the Gibbons-Hawking ansatz with infinite topological type

本文研究了由Gibbons-Hawking方案产生的具有无限拓扑类型的超kähler四流形的复杂结构，证明其与C^3中的超曲面或解析表面的双全形同，并扩展了LeBrun的工作到可数多个穿孔的情况。

椭圆分析在坍缩的引力瞬子模型中的应用：基于Gibbons-Hawking ansatz

Elliptic analysis on collapsing gravitational instantons modelled using the Gibbons-Hawking ansatz

本文研究了在无穷远处坍缩到三维平直空间的引力瞬子族的椭圆正则性估计，证明了这些估计的常数可对整个度量族统一选择，并应用于拉普拉斯算子，证明其为Fredholm算子，并确定其作为同构映射的加权空间范围。

An analogue of the Gibbons-Hawking Ansatz for quaternionic Kähler spaces

We show that the geometry of $4n$-dimensional quaternionic Kähler spaces with a locally free $\mathbb{R}^{n+1}$-action admits a Gibbons-Hawking-like description based on the Galicki-Lawson notion of quaternionic Kähler moment map. This generalizes to higher dimensions a four-dimensional construction, due to Calderbank and Pedersen, of self-dual Einstein manifolds with two linearly independent commuting Killing vector fields. As an application, we use this new Ansatz to give an explicit equivariant completion of the twistor space construction of the local c-map proposed by Roček, Vafa and Vando…

微分同胚自然的MDL泛函到带Gibbons-Hawking-York边界项的Einstein-Hilbert泛函的Γ-收敛

$Γ$-convergence of a diffeomorphism-natural MDL functional to Einstein-Hilbert with Gibbons-Hawking-York boundary term

本文证明了微分同胚自然的离散MDL型泛函到带Gibbons-Hawking-York边界项的Einstein-Hilbert作用量的Γ-收敛，确定了内部和边界密度，分析了余项阶数并提供了可复现的校准协议，为相关数值方法奠定基础。

A simple proof of the recent generalisations of Hawking's black hole topology theorem

A key result in four dimensional black hole physics, since the early 1970s, is Hawking's topology theorem asserting that the cross-sections of an "apparent horizon", separating the black hole region from the rest of the spacetime, are topologically two-spheres. Later, during the 1990s, by applying a variant of Hawking's argument, Gibbons and Woolgar could also show the existence of a genus dependent lower bound for the entropy of topological black holes with negative cosmological constant. Recently Hawking's black hole topology theorem, along with the results of Gibbons and Woolgar, has been g…

Classification of 't Hooft instantons over multi-centered gravitational instantons

This work presents a classification of all smooth 't Hooft-Jackiw-Nohl-Rebbi instantons over Gibbons-Hawking spaces. That is, we find all smooth SU(2) Yang-Mills instantons over these spaces which arise by conformal rescalings of the metric with suitable functions. Since the Gibbons-Hawking spaces are hyper-Kahler gravitational instantons, the rescaling functions must be positive harmonic. By using twistor methods we present integral formulae for the kernel of the Laplacian associated to these spaces. These integrals are generalizations of the classical Whittaker-Watson formula. By the aid of …

Calabi-Yau (p+1)-folds from p-folds

We establish the general formalism for constructing metrics of Calabi-Yau (p+1)-folds in terms of that of a p-fold by adding a complex-line bundle. We present a few explicit low-lying examples. We further consider holomorphic linearization and obtain the six-dimensional analogue of the Gibbons-Hawking instanton. Whilst the Kahler potential for the Gibbons-Hawking instanton is given by the harmonic function of a three-dimensional flat space, for the generalized solution it is related to the harmonic functions of certain three-dimensional non-flat spaces that are direct products of R and two-dim…

Limiting behavior of local Calabi-Yau metrics

We use a generalization of the Gibbons-Hawking ansatz to study the behavior of certain non-compact Calabi-Yau manifolds in the large complex structure limit. This analysis provides an intermediate step toward proving the metric collapse conjecture for toric hypersurfaces and complete intersections.

新的卡拉比-扬度量 Taub-NUT 类型在 C^{N+1}

New Calabi-Yau Metrics of Taub-NUT Type on C^{N+1}

本文提出了一种在 C^{N+1} 上构造非平坦卡拉比-扬度量的方法，通过广义 Gibbons-Hawking 假设和粘合程序克服了体积形式缺陷导致的奇点问题。

Mean curvature flow in a Ricci flow background

Following work of Ecker, we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties and its time derivative under Perelman's modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton's Harnack expression for the mean curvature flow in Euclidean space. We also derive the evolution equations for the second fundamental form and the mean curvature, under a mean curvature flow in a Ricci flow background. In the case of a gradient Ricci soliton background, we discuss mean curvature solitons an…