Rigidity of gradient Ricci Solitons

We define a gradient Ricci soliton to be rigid if it is a flat bundle $N\times_{\Gamma}\mathbb{R}^{k}$ where $N$ is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.Comment: 16 page

Warped product Einstein metrics over spaces with constant scalar curvature

In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natural curvature conditions that characterize the rigid examples in higher dimensions.Comment: 38 pages, 1 appendi

On the classification of warped product Einstein metrics

In this paper we take the perspective introduced by Case-Shu-Wei of studying warped product Einstein metrics through the equation for the Ricci curvature of the base space. They call this equation on the base the $m$-Quasi Einstein equation, but we will also call it the $(\lambda,n+m)$-Einstein equation. In this paper we extend the work of Case-Shu-Wei and some earlier work of Kim-Kim to allow the base to have non-empty boundary. This is a natural case to consider since a manifold without boundary often occurs as a warped product over a manifold with boundary, and in this case we get some interesting new canonical examples. We also derive some new formulas involving curvatures which are analogous to those for the gradient Ricci solitons. As an application, we characterize warped product Einstein metrics when the base is locally conformally flat.Comment: 29 pages. Minor changes and references updated. Submitted versio

Warped product Einstein metrics on homogeneous spaces and homogeneous Ricci solitons

In this paper we consider connections between Ricci solitons and Einstein metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton admits an Einstein one-dimensional extension if the soliton derivation can be chosen to be normal. Using our previous work on warped product Einstein metrics, we show that every normal semi-algebraic Ricci soliton also admits a $k$-dimensional Einstein extension for any $k\geq 2$. We also prove converse theorems for these constructions and some geometric and topological structure results for homogeneous warped product Einstein metrics. In the appendix we give an alternative approach to semi-algebraic Ricci solitons which naturally leads to a definition of semi-algebraic Ricci solitons in the non-homogeneous setting.Comment: 28 pages. supersedes the earlier version of arXiv:1110.245

On the Classification of Gradient Ricci Solitons

We show that the only complete shrinking gradient Ricci solitons with vanishing Weyl tensor are quotients of the standard ones. This gives a new proof of the Hamilton-Ivey-Perel\u27man classification of 3-dimensional shrinking gradient solitons. We also prove a classification for expanding gradient Ricci solitons with constant scalar curvature and suitably decaying Weyl tensor

The relationship of marine stratus to synoptic conditions

The marine stratus which persistently covered most of the eastern Pacific Ocean, had large clear areas during the FIRE Intensive Field Operations (IFO) in 1987. Clear zones formed inside the large oceanic cloud mass on almost every day during the IFO. The location and size of the clear zones varied from day to day implying that they were related to dynamic weather conditions and not to oceanic conditions. Forecasting of cloud cover for aircraft operations during the IFO was directed towards predicting when and where the clear and broken zones would form inside the large marine stratus cloud mass. The clear zones often formed to the northwest of the operations area and moved towards it. However, on some days the clear zones appeared to form during the day in the operations area as part of the diurnal cloud burn off. The movement of the clear zones from day to day were hard to follow because of the large diurnal changes in cloud cover. Clear and broken cloud zones formed during the day only to distort in shape and fill during the following night. The field forecasters exhibited some skill in predicting when the clear and broken cloud patterns would form in the operations area. They based their predictions on the analysis and simulations of the models run by NOAA's Numeric Meteorological Center. How the atmospheric conditions analyzed by one NOAA/NMC model related to the cloud cover is discussed

Warped Product Rigidity

In this paper we study the space of solutions to an overdetermined linear system involving the Hessian of functions. We show that if the solution space has dimension greater than one, then the underlying manifold has a very rigid warped product structure. This warped product structure will be used to study warped product Einstein structures in our paper The space of virtual solutions to the warped product Einstein equation