On the structure of almost einstein manifolds

By G. Tian, Bing Wang. 2012

61 citations recorded by the source corpus.

In this paper, we study the structure of the limit space of a sequence of almost Einstein manifolds, which are generalizations of Einstein manifolds. Roughly speaking, such manifolds are the initial manifolds of some normalized Ricci flows whose scalar curvatures are almost constants over space-time in the L1-sense, Ricci curvatures are bounded from below at the initial time. Under the non-collapsed condition, we show that the limit space of a sequence of almost Einstein manifolds has most properties which is known for the limit space of Einstein manifolds. As applications, we can apply our structure results to study the properties of Kahler manifolds.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/afdb1cbafc0821a5206a55b73e1a99d8a93c50f1

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

The local entropy along Ricci flow---Part A: the no-local-collapsing theorems (2017, background): “inequality (25) of Tian-Wang [44]), which implies that 2φ log φ is a bounded function on Ω.”
Space of Ricci Flows I (2009, background): “Similar argument was also used in the distance estimate of the work of Tian and the second named author [109].”
Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds (2013, unknown): “results in [44] and his paper [5], the above conditions are also equivalent to the condition that the sup of (normalized) Perelman’s -functional is equal to nVol(X). Using the terminology in [2] and [36], we could say that K-semistable Fano manifold is the same as almost Kahler-Einstein Fano manifol”
Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2\pi\ and completion of the main proof (2013, methodology): “e are new diﬃculties to overcome. This is done in Section 2 below. We give the second 3 proof in Section 3: this uses Ricci ﬂow and a variant of Perelman’s pseudolocality theorem due to Tian and Wang [41]. In the ﬁrst approach, we only need to approximate the metrics ωi by smooth Kahler metrics with”
On the Kahler Ricci flow on projective manifolds of general type (2015, background): “ map extends to the Gromov-Hausdorﬀ limit space Φ∞ : (X∞,d∞) → (Xcan,ωFS), which is Lipschitz continuous. Under our assumption that the Ricci curvature is uniformly bounded below, Tian-Wang’s theory ([24]) on the structure of limit of almost Ka¨hler Einstein manifolds implies that the singular set i”
The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality (2012, methodology): “The above conjecture is related to the recent result in [40], where the Kähler–Ricci flow is combined with the continuity method to produce a limiting Einstein metric space when R.”
K‐Stability and Kähler‐Einstein Metrics (2012, background): “10 Then it follows from theorem 2 in [52] (also Theorem B.”
Gromov–Hausdorff limits of Kähler manifolds with Ricci curvature bounded below (2018, methodology): “Since for the initial metric B(p, δ1) is Gromov-Hausdorff close to the Euclidean ball, we can again apply the result of Cavalletti-Mondino [2] and the pseudolocality theorem (alternatively we could apply the version of the pseudolocality theorem proved by Tian-Wang [46]).”