On the first eigenvalue of the operator $-4\triangle +R$

8.1 Recall from [I,§1,2] that Ricci flow is the gradient flow for the first eigenvalue $\lambda$ of the operator $-4\triangle +R;$ moreover, $\frac{d}{dt}\lambda (t)\ge \frac{2}{3}\lambda ^2(t)$ and $\lambda (t)V^{\frac{2}{3}}(t)$ is non-decreasing whenever it is nonpositive. We would like to extend these inequalities to the case of Ricci flow with $\delta (t)$-cutoff. Recall that we immediately remove components with nonnegative scalar curvature.

Lemma. Given any positive continuous function $\xi(t)$ one can chose $\delta (t)$ in such a way that for any solution to the Ricci flow with $\delta (t)$-cutoff, with normalized initial data, and any surgery time $T_0,$ after which there is at least one component, where the scalar curvature is not strictly positive, we have an estimate $\lambda ^+(T_0)-\lambda ^-(T_0)\ge \xi(T_0)(V^+(T_0)-V^-(T_0)),$ where $V^-,V^+$ and $\lambda ^-,\lambda ^+$ are the volumes and the first eigenvalues of $-4\triangle +R$ before and after the surgery respectively.

Proof. Consider the minimizer $a$ for the functional

$$\int (4\vert\nabla a\vert^2+Ra^2)$$ (8.1)

under normalization $\int a^2=1,$ for the metric after the surgery on a component where scalar curvature is not strictly positive. Clearly is satisfies the equation

$$4\triangle a=Ra-\lambda ^-a$$ (8.2)

Observe that since the metric contains an $\epsilon$-neck of radius about $r(T_0),$ we can estimate $\lambda ^-(T_0)$ from above by about $r(T_0)^{-2}$.

Let $M_{cap}$ denote the cap, added by the surgery. It is attached to a long tube, consisting of $\epsilon$-necks of various radii. Let us restrict our attention to a maximal subtube, on which the scalar curvature at each point is at least $2\lambda ^-(T_0).$ Choose any $\epsilon$-neck in this subtube, say, with radius $r_0,$ and consider the distance function with range $[0,2\epsilon ^{-1}r_0],$ whose level sets $M_z$ are almost round two-spheres; let $M_z^+\supset M_{cap}$ be the part of $M,$ chopped off by $M_z.$ Then

$$\int_{M_z} -4aa_z=\int_{M_z^+}(4\vert\nabla a\vert^2+Ra^2-\lambda ^-a^2)>r_0^{-2}/2\int_{M_z^+}a^2$$

On the other hand,

$$\vert\int_{M_z}2aa_z-(\int_{M_z}a^2)_z\vert\le \mathrm{const}\cdot \int_{M_z}\epsilon r_0^{-1} a^2$$

These two inequalities easily imply that

$$\int_{M_0^+}a^2\ge \mathrm{exp}(\epsilon ^{-1}/10)\int_{M_{\epsilon ^{-1}r_0}^+}a^2$$

Now the chosen subtube contains at least about $-\epsilon ^{-1}\mathrm{log}(\lambda ^-(T_0)h^2(T_0))$ disjoint $\epsilon$-necks, where $h$ denotes the cutoff radius, as before. Since $h$ tends to zero with $\delta ,$ whereas $r(T_0),$ that occurs in the bound for $\lambda ^-,$ is independent of $\delta ,$ we can ensure that the number of necks is greater then $\mathrm{log}\ h,$ and therefore, $\int_{M_{cap}}a^2<h^6,$ say. Then standard estimates for the equation (8.2) show that $\vert\nabla a\vert^2$ and $Ra^2$ are bounded by $\mathrm{const}\cdot h$ on $M_{cap},$ which makes it possible to extend $a$ to the metric before surgery in such a way that the functional (8.1) is preserved up to $\mathrm{const}\cdot h^4.$ However, the loss of volume in the surgery is at least $h^3,$ so it suffices to take $\delta$ so small that $h$ is much smaller than $\xi.$

8.2 The arguments above lead to the following result

(a) If $(M,g_{ij})$ has $\lambda >0,$ then, for an appropriate choice of the cutoff parameter, the solution becomes extinct in finite time. Thus, if $M$ admits a metric with $\lambda >0$ then it is diffeomorphic to a connected sum of a finite collection of $\mathbb{S}^2\times\mathbb{S}^1$ and metric quotients of the round $\mathbb{S}^3.$ Conversely, every such connected sum admits a metric with $R>0,$ hence with $\lambda >0.$

(b) Suppose $M$ does not admit any metric with $\lambda >0,$ and let $\bar{\lambda }$ denote the supremum of $\lambda V^{\frac{2}{3}}$ over all metrics on this manifold. Then $\bar{\lambda }=0$ implies that $M$ is a graph manifold. Conversely, a graph manifold can not have $\bar{\lambda }<0.$

(c) Suppose $\bar{\lambda }<0$ and let $\bar{V}= (-\frac{2}{3}\bar{\lambda })^{\frac{3}{2}}.$ Then $\bar{V}$ is the minimum of $V,$ such that $M$ can be decomposed in connected sum of a finite collection of $\mathbb{S}^2\times\mathbb{S}^1,$ metric quotients of the round $\mathbb{S}^3,$ and some other components, the union of which will be denoted by $M',$ and there exists a (possibly disconnected) complete hyperbolic manifold, with sectional curvature $-1/4$ and volume $V,$ which can be embedded in $M'$ in such a way that the complement (if not empty) is a graph manifold. Moreover, if such a hyperbolic manifold has volume $\bar{V},$ then its cusps (if any) are incompressible in $M'.$

For the proof one needs in addition easily verifiable statements that one can put metrics on connected sums preserving the lower bound for scalar curvature [G-L], that one can put metrics on graph manifolds with scalar curvature bounded below and volume tending to zero [C-G], and that one can close a compressible cusp, preserving the lower bound for scalar curvature and reducing the volume, cf. [A,5.2]. Notice that using these results we can avoid the hyperbolic rigidity and minimal surface arguments, quoted in 7.3, which, however, have the advantage of not requiring any a priori topological information about the complement of the hyperbolic piece.

The results above are exact analogs of the conjectures for the Sigma constant, formulated by Anderson [A], at least in the nonpositive case.