A note on Morse’s index theorem for Perelman’s L-length

This is essentially a note on Section 7 of Perelman’s first paper on Ricci flow. We list some basic properties of the index form for Perelman’s L-length, which are analogous to the ones in Riemannian case (with fixed metric), and observe that Morse’s index theorem for Perelman’s L-length holds. As a corollary we get the finiteness of the number of the L-conjugate points along a finite L-geodesic. In his ground-breaking work [6] on Ricci flow Perelman introduced L-length, LJacobi field among many other important innovations. For more details see [2],[3], and [5]. Here we’ll add some notes on Section 7 of this paper of Perelman’s. We list some basic properties of the index form for Perelman’s L-length, which are analogous to the ones in Riemannian case (with fixed metric, cf.[1], [4] and [7] ), and observe that Morse’s index theorem for Perelman’s L-length holds. The main idea of the proof is the same as that of the fixed metric case, but one needs to be careful when the τ -interval is [0, τ̄ ] (see in particular the proof of the Key Lemma below). As a corollary we get the finiteness of number of the L-conjugate points along a finite L-geodesic. Throughout this note we assume (M, g(τ)), where (gij)τ = 2Rij , is a (backwards) Ricci flow, which is complete for each τ -slice and has uniformly bounded curvature operator on an interval [τ1, τ2]. Recall Perelman’s L-length L(γ)= τ2 τ1 √ τ(R(γ(τ)) + |γ̇(τ)|2)dτ for a curve γ(τ) (τ1 ≤ τ ≤ τ2) in M . Definition 1 Let p ∈ M , v ∈ TpM , and γv be the L-geodesic with γv(0) = p, limτ→0 √ τ γ̇v(τ) = v. We say q = γv(τ̄ ) is a L-conjugate point of p along the Lgeodesic γv if v is a critical point of the Lτ̄ -exponential map Lτ̄exp. Here, (following the notation in [2],) Lτ̄exp(v)=Lexpv(τ̄) (=γv(τ̄)). Definition 2(Perelman) A vector field along a L-geodesic γ is called L-Jacobi field, if it is the variation vector field of a one parameter family of L-geodesics γs with γ0 = γ . The equation for a L-Jacobi field U along a L-geodesic γv(τ) ( 0 < τ1 ≤ τ ≤ τ2) is ( see, for example, [2]) ∇X∇XU−R(X,U)X−1/2∇U (∇R)+2(∇URic)(X)+2Ric(∇XU)+1/(2τ)∇XU = 0. (Here and below, X(τ) = γ̇v(τ). Moreover Ric(Y ) here means Ric(Y, ·) in Perelman [6]. ) One can easily extend this to the case τ1 = 0. (See [2].) Remark 1 As in the Riemannian case (with fixed metric) q = γv(τ̄) is a Lconjugate point of p = γv(0) along the L-geodesic γv(τ)(0 ≤ τ ≤ τ̄) if and only if there is a nontrivial L-Jacobi field U along γv with U(0) = U(τ̄ ) = 0.