ﻻ يوجد ملخص باللغة العربية
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).
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 $M^ntimes [0,T)$,
We introduce a new curvature flow which matches with the Ricci flow on metrics and preserves the almost Hermitian condition. This enables us to use Ricci flow to study almost Hermitian manifolds.
Motivated by the recent work of Chu-Lee-Tam on the nefness of canonical line bundle for compact K{a}hler manifolds with nonpositive $k$-Ricci curvature, we consider a natural notion of {em almost nonpositive $k$-Ricci curvature}, which is weaker than
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
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