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Extra resources for 2-dimensional simple factors of J 0(N)

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As ǫk rk2 → 0, the monotonicity of V˜ implies that V˜k (tk ) ≤ V˜k (ǫk rk2 ) for large k, which is a contradiction. 26. 3 The distortion of distances under Ricci flow can be estimated in terms of the Ricci tensor. We first mention a crude estimate. 1. 2) distt1 (x0 , x1 ) ≥ e−(n−1)K(t1 −t0 ) . distt0 (x0 , x1 ) 46 BRUCE KLEINER AND JOHN LOTT Proof. 3) d d L(γ) = dt dt a dγ dγ , ds ds 0 a ds = − Ric 0 dγ dγ , ds ds ds dγ ds ≥ − (n − 1) K L(γ). 4) L(γ) t1 t0 ≥ e−(n−1)K(t1 −t0 ) . The lemma follows by taking γ to be a minimal geodesic at time t1 between x0 and x1 .

17. 6) In this section we do some preliminary calculations leading up to the computation of the second variation of L. A remark about the notation : L is a function of a point q and a time τ . e. differentiation while keeping q fixed. e. simultaneously varying both the point and the time. If q is not in the time-τ L-cut locus of p, let γ : [0, τ ] → M be the unique minimizing Lgeodesic from p to q, with length L(q, τ ). 3) gives γ˜ (s, τ ) = c(s). 2) (∇L)(q, τ ) = 2 = 2 √ √ τ X(τ), Y (τ ) . 3) |∇L|2 (q, τ ) = 4 τ |X(τ )|2 = − 4 τ R(q) + 4 τ R(q) + |X(τ )|2 .

3(a)) Suppose that Ric(x, t0 ) ≤ (n − 1) K on Bt0 (x0 , r0 ). 19) dt − △d ≥ − (n − 1) 2 K r0 + r0−1 3 48 BRUCE KLEINER AND JOHN LOTT at time t = t0 , outside of Bt0 (x0 , r0 ). The inequality must be understood in the barrier sense (see Section 23) if necessary. 27. 2 This section is concerned with a localized version of the no-local-collapsing theorem. 2, says that noncollapsing propagates forward in time and to a larger distance scale. 1. 1. (cf. 1) A Ricci flow solution is said to be κ-collapsed at (x0 , t0 ), on the scale r > 0, if | Rm |(x, t) ≤ r −2 for all (x, t) ∈ Bt0 (x0 , r) × [t0 − r 2 , t0 ], but vol(Bt0 (x0 , r 2 )) ≤ κr n .

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2-dimensional simple factors of J 0(N) by Wang X.

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