Pseudolocality theorem

Theorem 10.1   For every $\alpha>0$ there exist $\delta>0,\epsilon>0$ with the following property. Suppose we have a smooth solution to the Ricci flow $(g_{ij})_t=-2R_{ij}, 0\le t\le (\epsilon r_0)^2,$ and assume that at $t=0$ we have $R(x)\ge -r_0^{-2}$ and $Vol(\partial\Omega)^n\ge(1-\delta)c_nVol(\Omega)^{n-1}$ for any $x,\Omega\subset B(x_0,r_0),$ where $c_n$ is the euclidean isoperimetric constant. Then we have an estimate $\vert Rm\vert(x,t)\le \alpha t^{-1}+(\epsilon r_0)^{-2}$ whenever $0<t\le (\epsilon r_0)^2, d(x,t)=$dist$_t(x,x_0)<\epsilon r_0.$

Thus, under the Ricci flow, the almost singular regions (where curvature is large) can not instantly significantly influence the almost euclidean regions. Or , using the interpretation via renormalization group flow, if a region looks trivial (almost euclidean) on higher energy scale, then it can not suddenly become highly nontrivial on a slightly lower energy scale.

Proof. It is an argument by contradiction. The idea is to pick a point $(\bar{x},\bar{t})$ not far from $(x_0,0)$ and consider the solution $u$ to the conjugate heat equation, starting as $\delta$-function at $(\bar{x},\bar{t}),$ and the corresponding nonpositive function $v$ as in 9.3. If the curvatures at $(\bar{x},\bar{t})$ are not small compared to $\bar{t}^{-1}$ and are larger than at nearby points, then one can show that $\int{v}$ at time $t$ is bounded away from zero for (small) time intervals $\bar{t}-t$ of the order of $\vert Rm\vert^{-1}(\bar{x},\bar{t}).$ By monotonicity we conclude that $\int{v}$ is bounded away from zero at $t=0.$ In fact, using (9.1) and an appropriate cut-off function, we can show that at $t=0$ already the integral of $v$ over $B(x_0,r)$ is bounded away from zero, whereas the integral of $u$ over this ball is close to $1,$ where $r$ can be made as small as we like compared to $r_0.$ Now using the control over the scalar curvature and isoperimetric constant in $B(x_0r_0),$ we can obtain a contradiction to the logarithmic Sobolev inequality.

Now let us go into details. By scaling assume that $r_0=1.$ We may also assume that $\alpha$ is small, say $\alpha<\frac{1}{100n}.$ From now on we fix $\alpha$ and denote by $M_{\alpha}$ the set of pairs $(x,t),$ such that $\vert Rm\vert(x,t)\ge\alpha t^{-1}.$

Claim 1.For any $A>0,$ if $g_{ij}(t)$ solves the Ricci flow equation on $0\le t\le \epsilon^2, A\epsilon<\frac{1}{100n},$ and $\vert Rm\vert(x,t)>\alpha t^{-1}+\epsilon^{-2}$ for some $(x,t),$ satisfying $0\le t\le \epsilon^2, d(x,t)<\epsilon,$ then one can find $(\bar{x},\bar{t})\in M_{\alpha },$ with $0<\bar{t}\le \epsilon ^2, d(\bar{x},\bar{t})<(2A+1)\epsilon ,$ such that

$$\vert Rm\vert(x,t)\le 4\vert Rm\vert(\bar{x},\bar{t}),$$ (10.1)

whenever

$$(x,t)\in M_{\alpha }, 0<t\le \bar{t}, d(x,t)\le d(\bar{x},\bar{t})+A\vert Rm\vert^{-\frac{1}{2}}(\bar{x},\bar{t})$$ (10.2)

Proof of Claim 1. We construct $(\bar{x},\bar{t})$ as a limit of a (finite) sequence $(x_k,t_k),$ defined in the following way. Let $(x_1,t_1)$ be an arbitrary point, satisfying $0<t_1\le \epsilon ^2, d(x_1,t_1)<\epsilon , \vert Rm\vert(x_1,t_1)\ge \alpha t^{-1}+\epsilon ^{-2}.$ Now if $(x_k,t_k)$ is already constructed, and if it can not be taken for $(\bar{x},\bar{t}),$ because there is some $(x,t)$ satisfying (10.2), but not (10.1), then take any such $(x,t)$ for $(x_{k+1},t_{k+1}).$ Clearly, the sequence, constructed in such a way, satisfies $\vert Rm\vert(x_k,t_k)\ge 4^{k-1}\vert Rm\vert(x_1,t_1)\ge 4^{k-1}\epsilon ^{-2},$ and therefore, $d(x_k,t_k)\le (2A+1)\epsilon .$ Since the solution is smooth, the sequence is finite, and its last element fits.

Claim 2. For $(\bar{x},\bar{t}),$ constructed above, (10.1) holds whenever

$$\bar{t}-\frac{1}{2}\alpha Q^{-1}\le t\le \bar{t}, _{\bar{t}}(x,\bar{x})\le \frac{1}{10}AQ^{-\frac{1}{2}},$$ (10.3)

where $Q=\vert Rm\vert(\bar{x},\bar{t}).$

Proof of Claim 2. We only need to show that if $(x,t)$ satisfies (10.3), then it must satisfy (10.1) or (10.2). Since $(\bar{x},\bar{t})\in M_{\alpha },$ we have $Q\ge \alpha \bar{t}^{-1},$ so $\bar{t}-\frac{1}{2}\alpha Q^{-1}\ge \frac{1}{2}\bar{t}.$ Hence, if $(x,t)$ does not satisfy (10.1), it definitely belongs to $M_{\alpha }.$ Now by the triangle inequality, $d(x,\bar{t})\le d(\bar{x},\bar{t})+\frac{1}{10}AQ^{-\frac{1}{2}}.$ On the other hand, using lemma 8.3(b) we see that, as $t$ decreases from $\bar{t}$ to $\bar{t}-\frac{1}{2}\alpha Q^{-1},$ the point $x$ can not escape from the ball of radius $d(\bar{x},\bar{t})+AQ^{-\frac{1}{2}}$ centered at $x_0.$

Continuing the proof of the theorem, and arguing by contradiction, take sequences $\epsilon \to 0,\delta \to 0$ and solutions $g_{ij}(t),$ violating the statement; by reducing $\epsilon ,$ we'll assume that

$$\vert Rm\vert(x,t)\le \alpha t^{-1}+2\epsilon ^{-2}\ \ 0\le t\le \epsilon ^2 \ d(x,t)\le \epsilon$$ (10.4)

Take $A=\frac{1}{100n\epsilon }\to\infty ,$ construct $(\bar{x},\bar{t}),$ and consider solutions $u=(4\pi(\bar{t}-t))^{-\frac{n}{2}}e^{-f}$ of the conjugate heat equation, starting from $\delta$-functions at $(\bar{x},\bar{t}),$ and corresponding nonpositive functions $v.$

Claim 3.As $\epsilon ,\delta \to 0,$ one can find times $\tilde{t}\in[\bar{t}-\frac{1}{2}\alpha Q^{-1},\bar{t}],$ such that the integral $\int_B{v}$ stays bounded away from zero, where $B$ is the ball at time $\tilde{t}$ of radius $\sqrt{\bar{t}-\tilde{t}}$ centered at $\bar{x}.$

Proof of Claim 3(sketch). The statement is invariant under scaling, so we can try to take a limit of scalings of $g_{ij}(t)$ at points $(\bar{x},\bar{t})$ with factors $Q.$ If the injectivity radii of the scaled metrics at $(\bar{x},\bar{t})$ are bounded away from zero, then a smooth limit exists, it is complete and has $\vert Rm\vert(\bar{x},\bar{t})=1$ and $\vert Rm\vert(x,t)\le 4$ when $\bar{t} -\frac{1}{2}\alpha \le t\le \bar{t}.$ It is not hard to show that the fundamental solutions $u$ of the conjugate heat equation converge to such a solution on the limit manifold. But on the limit manifold, $\int_B{v}$ can not be zero for $\tilde{t}=\bar{t}-\frac{1}{2}\alpha ,$ since the evolution equation (9.1) would imply in this case that the limit is a gradient shrinking soliton, and this is incompatible with $\vert Rm\vert(\bar{x},\bar{t})=1.$

If the injectivity radii of the scaled metrics tend to zero, then we can change the scaling factor, to make the scaled metrics converge to a flat manifold with finite injectivity radius; in this case it is not hard to choose $\tilde{t}$ in such a way that $\int_B{v}\to -\infty.$

The positive lower bound for $-\int_B{v}$ will be denoted by $\beta.$

Our next goal is to construct an appropriate cut-off function. We choose it in the form $h(y,t)=\phi(\frac{\tilde{d}(y,t)}{10A\epsilon }),$ where $\tilde{d}(y,t)=d(y,t)+200n\sqrt{t},$ and $\phi$ is a smooth function of one variable, equal one on $(-\infty,1]$ and decreasing to zero on $[1,2].$ Clearly, $h$ vanishes at $t=0$ outside $B(x_0,20A\epsilon );$ on the other hand, it is equal to one near $(\bar{x},\bar{t}).$

Now $\Box h=\frac{1}{10A\epsilon }(d_t-\triangle d+\frac{100n}{\sqrt{t}})\phi '-\frac{1}{(10A\epsilon )^2}\phi ''.$ Note that $d_t-\triangle t+\frac{100n}{\sqrt{t}}\ge 0$ on the set where $\phi '\neq 0 \ \ -$ this follows from the lemma 8.3(a) and our assumption (10.4). We may also choose $\phi$ so that $\phi ''\ge -10\phi, (\phi ')^2\le 10\phi.$ Now we can compute $(\int_M{hu})_t=\int_M{(\Box h)u}\le \frac{1}{(A\epsilon )^2},$ so $\int_M{hu}\mid_{t=0}\ge \int_M{hu}\mid_{t=\bar{t}}-\frac{\bar{t}}{(A\epsilon )^2}\ge 1-A^{-2}.$ Also, by (9.1), $(\int_M{-hv})_t\le \int_M{-(\Box h)v}\le \frac{1}{(A\epsilon )^2}\int_M{-hv},$ so by Claim 3, $-\int_M{hv}\mid_{t=0}\ge \beta$   exp$(-\frac{\bar{t}}{(A\epsilon )^2})\ge \beta (1-A^{-2}).$

From now on we"ll work at $t=0$ only. Let $\tilde{u}=hu$ and correspondingly $\tilde{f}=f-$log$h.$ Then

$$\beta (1-A^{-2})\le -\int_M{hv}=\int_M{[(-2\triangle f+\vert\nabla f\vert^2-R)\bar{t}-f+n]hu}$$

$$=\int_M{[-\bar{t}\vert\nabla \tilde{f}\vert^2-\tilde{f}+n]\tilde{u}}+ \int_M{[\bar{t}(\vert\nabla h\vert^2/h-Rh)-h\mbox{log}h]u}$$

$$\le\int_M{[-\bar{t}\vert\nabla\tilde{f}\vert^2-\tilde{f}-n]\tilde{u}}+A^{-2}+100\epsilon ^2$$

( Note that $\int_M{-uh \log h}$ does not exceed the integral of $u$ over
$B(x_0,20A\epsilon )\backslash B(x_0,10A\epsilon ),$ and $\int_{B(x_0,10A\epsilon )}{u}\ge \int_M{\bar{h}u}\ge 1-A^{-2},$
where $\bar{h}=\phi(\frac{\tilde{d}}{5A\epsilon }))$

Now scaling the metric by the factor $\frac{1}{2}\bar{t}^{-1}$ and sending $\epsilon ,\delta$ to zero, we get a sequence of metric balls with radii going to infinity, and a sequence of compactly supported nonnegative functions $u=(2\pi)^{-\frac{n}{2}}e^{-f}$ with $\int{u}\to 1$ and $\int{[-\frac{1}{2}\vert\nabla f\vert^2-f+n]u}$ bounded away from zero by a positive constant. We also have isoperimetric inequalities with the constants tending to the euclidean one. This set up is in conflict with the Gaussian logarithmic Sobolev inequality, as can be seen by using spherical symmetrization.

10.2 Corollary(from the proof) Under the same assumptions, we also have at time $t, 0<t\le (\epsilon r_0)^2,$ an estimate $Vol B(x,\sqrt{t})\ge c\sqrt{t}^n$ for $x\in B(x_0,\epsilon r_0),$ where $c=c(n)$ is a universal constant.

10.3 Theorem. There exist $\epsilon ,\delta > 0$ with the following property. Suppose $g_{ij}(t)$ is a smooth solution to the Ricci flow on $[0,(\epsilon r_0)^2],$ and assume that at $t=0$ we have $\vert Rm\vert(x)\le r_0^{-2}$ in $B(x_0,r_0),$ and $VolB(x_0,r_0)\ge (1-\delta )\omega_n r_0^n,$ where $\omega_n$ is the volume of the unit ball in $\mathbb{R}^n.$ Then the estimate $\vert Rm\vert(x,t)\le (\epsilon r_0)^{-2}$ holds whenever $0\le t\le (\epsilon r_0)^2,$   dist$_t(x,x_0)<\epsilon r_0.$

The proof is a slight modification of the proof of theorem 10.1, and is left to the reader. A natural question is whether the assumption on the volume of the ball is superfluous.

10.4 Corollary(from 8.2, 10.1, 10.2) There exist $\epsilon ,\delta > 0$ and for any $A>0$ there exists $\kappa(A)>0$ with the following property. If $g_{ij}(t)$ is a smooth solution to the Ricci flow on $[0,(\epsilon r_0)^2],$ such that at $t=0$ we have $R(x)\ge -r_0^{-2}, Vol(\partial\Omega)^n\ge (1-\delta )c_nVol(\Omega)^{n-1}$ for any $x,\Omega\subset B(x_0,r_0),$ and $(x,t)$ satisfies $A^{-1}(\epsilon r_0)^2\le t\le (\epsilon r_0)^2,$   dist$_t(x,x_0)\le Ar_0,$ then $g_{ij}(t)$ can not be $\kappa$-collapsed at $(x,t)$ on the scales less than $\sqrt{t}.$

10.5 Remark. It is straightforward to get from 10.1 a version of the Cheeger diffeo finiteness theorem for manifolds, satisfying our assumptions on scalar curvature and isoperimetric constant on each ball of some fixed radius $r_0>0.$ In particular, these assumptions are satisfied (for some controllably smaller $r_0$), if we assume a lower bound for Ric and an almost euclidean lower bound for the volume of the balls of radius $r_0.$ (this follows from the Levy-Gromov isoperimetric inequality); thus we get one of the results of Cheeger and Colding [Ch-Co] under somewhat weaker assumptions.

10.6* Our pseudolocality theorem is similar in some respect to the results of Ecker-Huisken [E-Hu] on the mean curvature flow.