Vincent Minerbe
We prove positive mass theorems on ALF manifolds, i.e. complete noncompact manifolds that are asymptotic to a circle fibration over a Euclidean base, with fibers of asymptotically constant length.
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Ricci Flow on ALF manifolds
本文证明Ricci流保持ALF结构，定义重整化泛函$\lambda_{\mathrm{ALF}}$，建立变分与线性稳定性概念，并证明共形Kähler非超Kähler例子沿Ricci流动态不稳定，且$\lambda_{\mathrm{ALF}}$符号与正相对质量相关。
In this paper, we want to prove positive mass theorems for ALF and ALG manifolds with model spaces $\mathbb R^{n-1}\times \mathbb S^1$ and $\mathbb R^{n-2}\times \mathbb T^2$ respectively in dimensions no greater than $7$ (Theorem \ref{ALFPMT0}). { Different from the compatibility condition for spin structure in \cite[Theorem 2]{minerbe2008a}, we show that some type of incompressible condition for $\mathbb S^1$ and $\mathbb T^2$ is enough to guarantee the nonnegativity of the mass.} As in the asymptotically flat case, we reduce the desired positive mass theorems to those ones concerning non-ex…
A Comparison Theorem For the Mass of ALE and ALF Toric 4-Manifolds
研究通过均衡几何建立ALE和ALF托ric 4流形质量下界，结合引力瞬子的性质，提出总质量概念并证明刚性条件。
We propose an infinite-dimensional generalization of Kronheimer's construction of families of hyperkahler manifolds resolving flat orbifold quotients of $\mathbb{R}^4$. As in [Kro89], these manifolds are constructed as hyperkahler quotients of affine spaces. This leads to a study of \emph{singular equivariant instantons} in various dimensions. In this paper, we study singular equivariant Nahm data to produce the family of $D_2$ asymptotically locally flat (ALF) manifolds as a deformation of the flat orbifold $(\mathbb{R}^3 \times S^1)/Z_2$. We furthermore introduce a notion of stability for Na…
In this paper, we introduce the notions of ALF Weyl connection and of associated mass, and we prouve the positive mass theorem for the ALF Weyl structures.
We compute the equivariant elliptic genera of several classes of ALE and ALF manifolds using localization in gauged linear sigma models. In the sigma model computation the equivariant action corresponds to chemical potentials for U(1) currents and the elliptic genera exhibit interesting pole structure as a function of the chemical potentials. We use this to decompose the answers into polar terms that exhibit wall crossing and universal terms. We compare our results to previous results on the large radius limit of the Taub-NUT elliptic genus and also discuss applications of our results to count…
We study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is $\mathrm{R}^4$ or $\mathrm{R}^3 \times \mathbb{S}^1$ and the structure group is $\mathrm{SU} (2)$.
Let N_0 = C^2/H be an isolated quotient singularity with H in U (2) a finite subgroup. We show that for any Q-Gorenstein smoothings of N_0 a nearby fiber admits ALE Ricci-flat Kahler metrics in any Kahler class. Moreover, we generalize Kronheimer's results on hyperkahler 4-manifolds, by giving an explicit classification of the ALE Ricci-flat Kahler surfaces. We construct ALF Ricci-flat Kahler metrics on the above non-simply connected manifolds. These provide new examples of ALF Ricci-flat Kahler 4-manifolds, with cubic volume growth and cyclic fundamental group at infinity.
We study inverse scattering for $Δ_g+V$ on $(X,g)$ a conformally compact manifold with metric $g,$ with variable sectional curvature $-\alf^2(y)$ at the boundary and $V\in C^\infty(X)$ not vanishing at the boundary. We prove that the scattering matrix at a fixed energies $(λ_1,$ $λ_2)$ in a suitable subset of $\mc$, determines $\alf,$ and the Taylor series of both the potential and the metric at the boundary.
We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4-dimensional ALF hyperkähler metrics. We apply our construction to asymptotically conical G2 metrics arising from self-dual Einstein 4-orbifolds with positive scalar curvature. As …
Complete non-compact $\operatorname{Spin}(7)$-manifolds from $T^2$-bundles over AC Calabi Yau manifolds
本研究提出在渐近锥形Calabi-Yau流形的主T²丛上构造完备非紧Spin(7)流形的新方法，得到渐近T²纤维化锥形的新型无穷远几何，并构造出无穷多微分同胚型及首个完备环形Spin(7)流形实例。
Mass Lower Bounds for Asymptotically Locally Flat Manifolds
该研究针对渐近局部平坦流形建立正质量型定理，证明了特定拓扑条件下非负数量曲率对应非负质量，还得到四维带U(1)作用情形的质量下界，首次揭示渐进端非平凡拓扑对质量的影响。
In this paper we prove that over an asymptotically locally flat (ALF) Riemannian four-manifold the energy of an "admissible" SU(2) Yang--Mills is always integer. This result sharpens the previously known energy identity for such Yang--Mills instantons over ALF geometries. Furthermore we demonstrate that this statement continues to hold for the larger gauge group U(2). Finally we make the observation that there might be a natural relationship between 4 dimensional Yang--Mills theory over an ALF space and 2 dimensional conformal field theory. This would provide a further support for the existenc…
Instability of conformally Kähler, Einstein metrics
本文证明了非半共形平坦、具有非负标量曲率的共形凯勒紧致或ALF爱因斯坦4-流形的不稳定性，并应用于所有已知非超凯勒引力瞬子及Chen-Lebrun-Weber度量。
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