Chiara Guidi, Ali Maalaoui, Vittorio Martino
In this paper we consider the coupled system given by the first variation of the conformal Dirac-Einstein functional. We will show existence of solutions by means of perturbation methods.
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In this paper we study the Palais-Smale sequences of the conformal Dirac-Einstein problem. After we characterize the bubbling phenomena, we prove an Aubin type result leading to the existence of a positive solution. Then we show the existence of infinitely many solutions to the problem provided that the underlying manifold exhibits certain symmetries.
In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold will have at most finite fundamental group. Under the further assumption that the renormalized volume is relatively large, we conclude that the conformally compact Einstein 4-manifold is diffeomorphic to $B^4$ and its conformal infinity is diffeomorphic to $S^3$.
Conformally Invariant Dirac Equation with Non-Local Nonlinearity
本文研究受4维共形爱因斯坦-狄拉克问题启发的共形不变狄拉克方程，通过分析能量泛函的性质、刻画基态解、证明Aubin型不等式及研究线性扰动，确保了问题解的存在性并得到相关结果。
Conformal deformation of a Riemannian metric via an Einstein-Dirac parabolic flow
本文提出一种基于狄拉克-爱因斯坦泛函的约束梯度流，用于形变自旋流形上的黎曼度量，并证明了光滑解的局部适定性，为爱因斯坦-狄拉克问题研究奠定基础。
In this paper we consider the Hilbert-Einstein-Dirac functional, whose critical points are pairs, metrics-spinors, that satisfy a system coupling the Riemannian and the spinorial part. Under some assumptions, on the sign of the scalar curvature and the diameter, we prove a compactness result for this class of pairs, in dimension three and four. This can be seen as the equivalent of the study of compactness of sequences of Einstein manifolds as in \cite{And0,N}. Indeed, we study the compactness of sequences of critical points of the Hilbert-Einstein-Dirac functional which is an extension of the…
Conformally critical metrics and optimal bounds for Dirac eigenvalues on spin surfaces
本文研究了在固定共形类下Dirac算子本征值的极小化问题，推导了闭合流形上极小解存在的条件，并推导了二维球面上Dirac算子的等周不等式及其共形谱特征。
Uniqueness of conformal metrics with constant Q-curvature on closed Einstein manifolds
本文在具有正数量曲率且不与标准球面共形微分同胚的闭黎曼流形上，证明了具有4阶常Q-曲率的共形度量只能是常数倍的原度量，从而建立了该类度量的唯一性。
A Conformally Invariant Dirac-type Equation on Compact Spin Manifolds: the Effect of the Geometry
本文研究了紧致辛流形上的共形不变狄拉克型方程，证明了在非球面情况下Aubin型不等式严格成立，并为四维共形狄拉克-爱因斯坦问题提供了地面态存在性结果。
In this paper we study Dirac-Einstein equations on manifolds with boundary, restricted to a conformal class with constant boundary volume, under chiral bag boundary conditions for the Dirac operator. We characterize the bubbling phenomenon, also classifying ground state bubbles. Finally, we prove an Aubin-type inequality and a related existence result.
In this article, we give nonexistence and nonuniqueness results for the vacuum Einstein conformal constraint equations in the far-from-CMC case and also show that in some cases the equations of the conformal method for positive Yamabe metrics and with TT-tensor $σ$ = 0 have a non-trivial solution, and thus answer a question by D. Maxwell.
Conformal scattering theory for the Dirac field on Kerr spacetime
本研究借助共形几何方法，在Dirac场逐点衰减假设下构建Kerr时空上自旋1/2无质量Dirac方程的共形散射理论，且该构建在逐点衰减已证实的Schwarzschild及极慢转动Kerr黑洞外部区域有效。
The aim of this work is to prove by a perturbation method the existence of solutions of the coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state. We relate the solutions of our equations to those of the nonlinear Choquard equation and we show that the nondegenerate solution of Choquard's equation generates solutions for Einstein-Dirac equations.
We determine the structure of conformal powers of the Dirac operator on Einstein {\it Spin}-manifolds in terms of the product formula for shifted Dirac operators. The result is based on the techniques of higher variations for the Dirac operator on Einstein manifolds and spectral analysis of the Dirac operator on the associated Poincaré-Einstein metric, and relies on combinatorial recurrence identities related to the dual Hahn polynomials.
The aim of this Note is to prove by a perturbation method the existence of solutions of the coupled Einstein-Dirac-Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state and with the electromagnetic coupling constant $(\frac{e}{m})^2<1$. We show that the nondegenerate solution of Choquard's equation generates a branch of solutions of the Einstein-Dirac-Maxwell equations.
We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac-Einstein equations on $\mathbb{R}^3$, which appear in the bubbling analysis of conformal Dirac-Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin-Talenti functions, while the spinorial part is the conformal image of $-\frac{1}{2}$-Killing spinors on the round sphere $\mathbb{S}^3$.
Poincaré-Einstein 4-manifolds with conformally Kähler geometry
本文研究共形类含Kähler度量的4维Poincaré-Einstein流形，将Einstein方程约化为Toda型方程，针对$\u212d^1$作用情形构建Dirichlet边值问题并证明存在唯一性，非微扰构造出共形无穷远为非正Yamabe型的无穷多族新度量。
We construct a conformal scattering theory for Dirac fields in the interior of a Reissner-Nordström-like black hole, between the black hole event horizon and the Cauchy horizon. The main result is a resolution of the characteristic Cauchy problem for the Dirac equation on the horizons by solving a system of wave equations and re-interpreting its solution as the components of the required Dirac solution.
We prove that any compact complex surface with positive first Chern class admits an Einstein metric which is conformally related to a Kaehler metric. The key new ingredient is the existence of such a metric on the blow-up of the complex projective plane at two distinct points.
A conformal lower bound of weighted Dirac eigenvalues on manifolds with boundary
本文研究了带有边界的流形上加权狄拉克本征值的共形下界，并证明了在特定条件下等号成立的条件。
In this article, we make a generalization of classical fixed point theorems by using the concept of half-continuity and then apply it to improve the nonuniqueness result for solutions to the vacuum Einstein conformal equations shown by the author in arxiv.org/abs/1507.01081
In this summary article, we review and discuss the non-existence of stationary black hole solutions for the Einstein-Dirac-Maxwell equations.
Time-periodic solutions of the conformally invariant wave equation on the Einstein cylinder
本文研究爱因斯坦柱面上三次共形波动方程的时间周期解，结合数值与微扰方法发现其形成复杂的分岔图样，为探索负宇宙学常数下爱因斯坦方程的时间周期解迈出第一步。
We study a static, spherically symmetric system of (2j+1) massive Dirac particles, each having angular momentum j, j=1,2,..., in a classical gravitational and SU(2) Yang-Mills field. We show that for any black hole solution of the associated Einstein-Dirac-Yang/Mills equations, the spinors must vanish identically outside of the event horizon.
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