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Register a new userOn gradient Ricci solitons with constant scalar curvature
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We use the theory of isoparametric functions to investigate gradient Ricci solitons with constant scalar curvature. We show rigidity of gradient Ricci solitons with constant scalar curvature under some conditions on the Ricci tensor, which are all satisfied if the manifold is curvature homogeneous. This leads to a complete description of four- and six-dimensional Kaehler gradient Ricci solitons with constant scalar curvature.
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We show that the scalar curvature of a steady gradient Ricci soliton satisfying that the ratio between the square norm of the Ricci tensor and the square of the scalar curvature is bounded by one half, is boundend from below by the hyperbolic secant of one half the distance function from a fixed point.
In this paper, we study constant weighted mean curvature hypersurfaces in shrinking Ricci solitons. First, we show that a constant weighted mean curvature hypersurface with finite weighted volume cannot lie in a region determined by a special level set of the potential function, unless it is the level set. Next, we show that a compact constant weighted mean curvature hypersurface with a certain upper bound or lower bound on the mean curvature is a level set of the potential function. We can apply both results to the cylinder shrinking Ricci soliton ambient space. Finally, we show that a constant weighted mean curvature hypersurface in the Gaussian shrinking Ricci soliton (not necessarily properly immersed) with a certain assumption on the integral of the second fundamental form must be a generalized cylinder.
In this paper, we extend the work of Cao-Chen [9] on Bach-flat gradient Ricci solitons to classify $n$-dimensional ($nge 5$) complete $D$-flat gradient steady Ricci solitons. More precisely, we prove that any $n$-dimensional complete noncompact gradient steady Ricci soliton with vanishing $D$-tensor is either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof extends to the shrinking case and the expanding case as well.
In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension $geq 4$. We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of such Ricci flows converges, after passing to a subsequence, to a metric space that is smooth away from a set of codimension $geq 4$. In the course of the proof, we will also establish $L^{p < 2}$-curvature bounds on time-slices of such flows.
We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete classification in dimension three.
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