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Register a new userRicci curvature integrals, local functionals, and the Ricci flow
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Consider a Riemannian manifold $(M^{m}, g)$ whose volume is the same as the standard sphere $(S^{m}, g_{round})$. If $p>frac{m}{2}$ and $int_{M} left{ Rc-(m-1)gright}_{-}^{p} dv$ is sufficiently small, we show that the normalized Ricci flow initiated from $(M^{m}, g)$ will exist immortally and converge to the standard sphere. The choice of $p$ is optimal.
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We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kahler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.
It is well known that the Kahler-Ricci flow on a Kahler manifold $X$ admits a long-time solution if and only if $X$ is a minimal model, i.e., the canonical line bundle $K_X$ is nef. The abundance conjecture in algebraic geometry predicts that $K_X$ must be semi-ample when $X$ is a projective minimal model. We prove that if $K_X$ is semi-ample, then the diameter is uniformly bounded for long-time solutions of the normalized Kahler-Ricci flow. Our diameter estimate combined with the scalar curvature estimate in [34] for long-time solutions of the Kahler-Ricci flow are natural extensions of Perelmans diameter and scalar curvature estimates for short-time solutions on Fano manifolds. We further prove that along the normalized Kahler-Ricci flow, the Ricci curvature is uniformly bounded away from singular fibres of $X$ over its unique algebraic canonical model $X_{can}$ if the Kodaira dimension of $X$ is one. As an application, the normalized Kahler-Ricci flow on a minimal threefold $X$ always converges sequentially in Gromov-Hausdorff topology to a compact metric space homeomorphic to its canonical model $X_{can}$, with uniformly bounded Ricci curvature away from the critical set of the pluricanonical map from $X$ to $X_{can}$.
In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In addition, for Ricci harmonic flow, we use the monotonicity of functional $ u_alpha$ to show the connection between finite-time singularity and shrinking Ricci harmonic soliton. At last, we explore the property of ancient solutions for Ricci harmonic flow.
We prove that for a solution $(M^n,g(t))$, $tin[0,T)$, where $T<infty$, to the Ricci flow with bounded curvature on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant $C$ on $M^ntimes [0,T)$, the curvature tensor stays uniformly bounded on $M^ntimes [0,T)$. Some other results are also presented.
We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than $-1$ can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than $-C$. Here the time of existence and the constant $C$ only depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other invariant curvature conditions, including positive biholomorphic curvature in the Kaehler case. We also get a local version of the main theorem. As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds).
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