The viability of theories with matter coupled to the Ricci scalar
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f(R) gravity
Ricci Flow
This paper studies normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically -1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature -1. A relative estimate of Green's function is proved as a tool.
Ricci Flow
Constant (computer programming)
Yamabe flow
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Ricci Flow
Operator (biology)
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In this note, we give a new proof for Perelman's scalar curvature and diameter estimates for the K\"ahler-Ricci flow on Fano manifolds. The proof relies on a new Harnack estimate for a special family of functions in space-time. Our new approach initiates the work in \cite{JST23a} for general finite time solutions of the K\"ahler-Ricci flow.
Ricci Flow
Fano plane
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Parameterized post-Newtonian formalism
Equivalence principle (geometric)
Einstein field equations
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We show that a uniform local bound for the curvature operator can be derived from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. As a consequence, we obtain new compactness theorems for the Ricci flow and Ricci soliton without assuming any bounds on the curvature operator. In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly unknown. In particular, a Shi-type estimate for Ricci curvature is derived when the derivative of Ricci curvature is controlled by the derivative of scalar curvature.
Ricci Flow
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We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with $|Ric|\leq K$ must satisfy $|Rm|\leq Ct^{-1}$ for all $t\in (0,T]$, where $C$ depends only on the dimension $n$ and $T$ depends on $K$ and the injectivity radius $inj_{g(t)}$. In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly unknown. In particular, another Shi-type estimate for Ricci curvature is derived when the derivative of Ricci curvature is controlled by the derivative of scalar curvature.
Ricci Flow
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Abstract Let ( M, g ( t )) be a compact Riemannian manifold and the metric g ( t ) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δ φ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δ φ is the Witten Laplacian operator, φ ∈ C 2 ( M ), and R is the scalar curvature with respect to the metric g ( t ). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.
Ricci Flow
Geometric flow
Riemannian manifold
Manifold (fluid mechanics)
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Ricci Flow
Yamabe flow
Geometric flow
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We prove that the magnitude of the derivative of Ricci curvature can be uniformly controlled by the bounds of Ricci curvature and injectivity radius along the Ricci flow.As a consequence, a precise uniform local bound of curvature operator can be constructed from local bounds of Ricci curvature and injectivity radius among all n-dimensional Ricci flows.In particular, we show that every Ricci flow with |Ric| ≤ K must satisfy |Rm| ≤ Ct -1 for all t ∈ (0, T ], where C depends only on the dimension n, and T depends on K and the injectivity radius inj g(t) .In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly unknown.In particular, another Shi-type estimate for Ricci curvature is derived when the derivative of Ricci curvature is controlled by the derivative of scalar curvature.
Ricci Flow
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Parameterized post-Newtonian formalism
Brans–Dicke theory
Scalar field theory
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