- On harmonic and biharmonic maps from gradient Ricci solitons(arXiv)
Author : Volker Branding
Abstract : We study harmonic and biharmonic maps from gradient Ricci solitons. We derive a number of analytic and geometric conditions under which harmonic maps are constant and which force biharmonic maps to be harmonic. In particular, we show that biharmonic maps of finite energy from the two-dimensional cigar soliton must be harmonic.
2.O(2)-symmetry of 3D steady gradient Ricci solitons (arXiv)
Author : Yi Lai
Abstract : For any 3D steady gradient Ricci soliton with positive curvature, we prove that it must be isometric to the Bryant soliton if it is asymptotic to a ray. Otherwise, it is asymptotic to a sector and hence a flying wing. We show that all 3D flying wings are O(2)-symmetric. Therefore, all 3D steady gradient Ricci solitons are O(2)-symmetric.
3. On noncompact warped product Ricci solitons(arXiv)
Author : Valter Borges
Abstract : The goal of this article is to investigate complete noncompact warped product gradient Ricci solitons. Nonexistence results, estimates for the warping function and for its gradient are proven. When the soliton is steady or expanding these nonexistence results generalize to a broader context certain pde estimates and rigidity obtained when studying warped product Einstein manifolds. When the soliton is shrinking, it is presented a nonexistence theorem with no counterpart in the Einstein case, which is proved using properties of the first eigenvalue of a weighted Laplacian.
4. The Stability of Generalized Ricci Solitons(arXiv)
Author : Kuan-Hui Lee
Abstract : In this paper, I compute the second variation formula of the generalized Einstein-Hilbert functional and prove that a Bismut-flat, Einstein manifold is linearly stable under some curvature assumption. In the last part of the paper, I prove that dynamical stability and the linear stability are equivalent on a steady gradient generalized Ricci soliton (g,H,f) which generalizes the result done by Kröncke, Haslhofer, Sesum, Raffero and Vezzoni.