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Register a new userA Comparison Theorem and A Sharp Bound via the Ricci Flow
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Authors
Jun Ling
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We prove a comparison theorem for the compact surfaces with negative Euler characteristic via the Ricci flow.
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We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of smooth compact manifolds with smooth metrics, is tangen- tial to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of smooth manifolds with rough metrics with sufficiently regular heat kernels. On an appropriate non- singular open region, we provide a family of metric tensors evolving in time and provide a regularity theory for this flow in terms of the regularity of the heat kernel. When the rough metric induces a metric measure space satisfying a Riemannian Curvature Dimension condition, we demonstrate that the distance induced by the flow is identical to the evolving distance metric defined by Gigli and Mantegazza on appropriate admissible points. Consequently, we demonstrate that a smooth compact manifold with a finite number of geometric conical singularities remains a smooth manifold with a smooth metric away from the cone points for all future times. Moreover, we show that the distance induced by the evolving metric tensor agrees with the flow of RCD(K, N) spaces defined by Gigli-Mantegazza.
We construct a class of monotonic quantities along the normalized Ricci flow on closed n-dimensional manifolds.
This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and Kahler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and Calabi-Yau structures as `canonical metrics in generalized Riemannian and complex geometry. The generalized Ricci flow is introduced as a tool for constructing such metrics, and extensions of the fundamental Hamilton/Perelman regularity theory of Ricci flow are proved. These results are refined in the setting of generalized complex geometry, where the generalized Ricci flow is shown to preserve various integrability conditions, taking the form of pluriclosed flow and generalized Kahler-Ricci flow. This leads to global convergence results, and applications to complex geometry. A purely mathematical introduction to the physical idea of T-duality is given, and a discussion of its relationship to generalized Ricci flow.
In this paper we establish new geometric and analytic bounds for Ricci flows, which will form the basis of a compactness, partial regularity and structure theory for Ricci flows in [Bam20a, Bam20b]. The bounds are optimal up to a constant that only depends on the dimension and possibly a lower scalar curvature bound. In the special case in which the flow consists of Einstein metrics, these bounds agree with the optimal bounds for spaces with Ricci curvature bounded from below. Moreover, our bounds are local in the sense that if a bound depends on the collapsedness of the underlying flow, then we are able to quantify this dependence using the pointed Nash entropy based only at the point in question. Among other things, we will show the following bounds: Upper and lower volume bounds for distance balls, dependence of the pointed Nash entropy on its basepoint in space and time, pointwise upper Gaussian bound on the heat kernel and a bound on its derivative and an $L^1$-Poincare inequality. The proofs of these bounds will, in part, rely on a monotonicity formula for a notion, called variance of conjugate heat kernels. We will also derive estimates concerning the dependence of the pointed Nash entropy on its basepoint, which are asymptotically optimal. These will allow us to show that points in spacetime that are nearby in a certain sense have comparable pointed Nash entropy. Hence the pointed Nash entropy is a good quantity to measure local collapsedness of a Ricci flow Our results imply a local $varepsilon$-regularity theorem, improving a result of Hein and Naber. Some of our results also hold for super Ricci flows.
In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let $g(t)$ be a smooth complete solution to the Ricci flow on $mathbb{R}^{3}$, with the canonical Euclidean metric $E$ as initial data, then $g(t)$ is trivial, i.e. $g(t)equiv E$.
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