Lower bounds on Ricci flow invariant curvatures and geometric applications

Abstract

We consider Ricci flow invariant cones C in the space of curvature operators lying between nonnegative Ricci curvature and nonnegative curvature operator. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to Ricci flow has its curvature operator which satsisfies R +ε I ∈ C at the initial time, then it satisfies R +Kε I ∈ C on some time interval depending only on the scalar curvature control.This allows us to link Gromov-Hausdorff convergence and Ricci flow convergence when the limit is smooth and R + I ∈ C along the sequence of initial conditions. Another application is a stability result for manifolds whose curvature operator is almost in C.Finally, we study the case where C is contained in the cone of operators whose sectional curvature is nonnegative. This allow us to weaken the assumptions of the previously mentioned applications. In particular, we construct a Ricci flow for a class of (not too) singular Alexandrov spaces.

Ricci flow starting from an embedded closed convex surface in $\mathbb{R}^3$

“…Before stating our results, we first recall Simon [18,19] and Richard's results [13,14]. One key point in [19] is the following estimates.…”

“…This is an approximation of the metric space by Ricci flow. Based on Simon's work, Richard [13,14] studied the existence and uniqueness of Ricci flow whose metric initial condition is a closed Alexandrov surface with curvature bounded from below, which gives a canonical smoothing of such surface via Ricci flow. The works of Simon and Richard are related to the Gromov-Hausdorff convergence.…”

“…The difference between Theorem 1.2 and Theorem 1.3 is that we remove the isometry between (M, d) and (X, d) in Theorem 1.2 when the metric initial condition (X, d) is an embedded closed convex surface in R 3 . This is due to the existence of smooth convex surfaces that approximate (X, d) in Hausdorff distance (Lemma 3.1) instead of the Gromov-Hausdorff convergence in [13,18,19]. In fact, removing the isometry is crucial for proving the uniqueness of such Ricci flow and then study the rigidity problem of closed convex surfaces (see our project in Section 4).…”

The commutative nonassociative algebra of metric curvature tensors

“…For background on the definition of * as in (5.14), its properties, and its role in the study of the Ricci flow, see also [20]. Some features of the algebra (MC(V * ), * ) are used implicitly in the study of the Ricci flow [2,4,5,17,18,21,32,33,34,41]. The algebraic perspective makes some of the manipulations used in such studies appear more natural and focuses attention on certain structural 1Due to a typographical error, its Theorem 2 is mislabeled as Theorem 3.…”

The commutative nonassociative algebra of metric curvature tensors

“…Some of the features of the algebra (MC(V * ), ⊛) are used implicitly in the study of the Ricci flow, e.g. in the work of R. Hamilton [27,28,29], G. Huisken [33], C. Böhm and B. Wilking [3], B. Wilking [58], S. Brendle [8,7,6], S. Brendle and R. Schoen [9], and T. Richard and H. Seshadri [42,43,44], and are, at least implicitly, well known to experts on the Ricci flow. The algebraic perspective makes some of the manipulations used in such studies appear more natural, and focuses attention on certain structural features, namely the invariance and nondegeneracy of the Killing type trace form and the identification of idempotent elements and the spectra of their left multiplication operators, that are perhaps not so self-evidently fundamental from the geometric perspective.…”