Rigidity of Homogeneous Gradient Soliton Metrics and Related Equations.
In this talk I’ll discuss structure results for homogeneous Riemannian spaces that support a non-constant solution to two general classes of equations involving the Hessian of a function. Our results generalize earlier rigidity results for gradient Ricci solitons and warped product Einstein metrics. In particular, they provide results for homogeneous gradient solitons of any invariant curvature flow and give a new structure result for homogeneous conformally Einstein metrics. This is joint work with P. Petersen of UCLA.
Classification of Ground States for Critical Dirac Equations.
In this talk I will present a classification result for nonlinear Dirac equations with critical nonlinearities on the Euclidean space. They appear naturally in conformal spin geometry and in variational problems related to critical Dirac equations on spin manifolds. Moreover, two-dimensional critical Dirac equations recently attracted a considerable attention as effective equations for wave propagation in honeycomb structures. Exploiting the conformal invariance of the problem ground state solutions can be classified, in analogy with the well-known result for the Yamabe equation. This is a joint work with Andrea Malchiodi (SNS, Pisa) and Ruijun Wu (SISSA, Trieste).
Conformally Kähler Einstein-Maxwell metrics on blow-ups.
Conformally Kähler Hermitian metrics of constant Riemannian scalar curvature and J-invariant Ricci are called conformally Kähler Einstein-Maxwell metrics. In this talk, we discuss deformations and possible constructions of such metrics on blow-ups. This is a joint work in progress with Abdellah Lahdili.
Asymptotically hyperbolic (AH) manifolds arise naturally in general relativity as a spacelike hypersurface in an asymptotically Minkowski spacetime or an asymptotically AdS spacetime. The mass of AH manifolds is a geometric invariant that measures its deviation from hyperbolic space. In this talk, we present a new mass formula using large coordinate horospheres. The formula is stated as a limit of the (weighted) total difference of mean curvatures on large coordinate horospheres. We remark on a few geometric implications of the formula including scalar curvature rigidity of AH manifolds. This talk is based on joint work with Pengzi Miao.