Perelman’s Functionals on Cones

Abstract

In this paper, we are interested in conical structures of manifolds with respect to the Ricci flow. In a first part, we study Perelman’s $\lambda$ and $\nu$ functionals of cones and characterize their finiteness in terms of the $\lambda$-functional of the link. As an application, we characterize manifolds with conical singularities on which a $\lambda$-functional can be defined and get upper bounds on the $\nu$-functional of asymptotically conical manifolds. We then present an adaptation of the proof of Perelman’s pseudolocality theorem and prove that cones over some perturbations of the unit sphere can be smoothed out by type III immortal solutions of the Ricci flow.

Appendices

Appendix A: Making the Manifold Look Locally Conical

In this Appendix, we want to prove that given a flow on a manifold diffeomorphic to a sphere that ends at the unit sphere, it is possible to construct a manifold asymptotic to the cone over the initial manifold for which at each point, the quantities involved in the entropy are arbitrarily close to that over a cone.

We will consider a manifold:

$$\begin{aligned} (M,g) = ({\mathbb {R}}^+\times N,\mathrm{{d}}r^2+r^2g(r)). \end{aligned}$$

In the case when g(r) is constant, we have a cone and the expression from which we have been able to control the $\nu$-functional after a separation of variables is

$$\begin{aligned}&{\mathcal {W}}^{C(N)}(f,\mathrm{{d}}r^2+r^2g^N,\tau ) \\&\quad = \int _0^\infty \left[ {\mathcal {W}}^{N}\left( \tilde{f},g^N,\frac{\tau }{r^2}\right) -n(n-1)\frac{\tau }{r^2}\right] \left( \frac{e^{-a_r}}{(4\pi \tau )^{\frac{1}{2}}} \mathrm{{d}}r\right) \\&\qquad +\int _0^\infty \left( \int _N \left[ \tau (\partial _r f)^2+a_r-1\right] \left( \frac{e^{-\tilde{f}}}{\left( 4\pi \tau r^{-2}\right) ^{\frac{n}{2}}} \mathrm{{d}}v\right) \left( \frac{e^{-a_r}}{(4\pi \tau )^{\frac{1}{2}}} \mathrm{{d}}r\right) \right) . \end{aligned}$$

The goal is to have a similar expression up to a controlled error term for manifolds of the form $(M,g) = ({\mathbb {R}}^+\times N,\mathrm{{d}}r^2+r^2g(r))$ for an adapted choice of g(r), that is,

$$\begin{aligned}&{\mathcal {W}}^{M}(f,\mathrm{{d}}r^2+r^2 g(r),\tau ) \\&\quad = \int _0^\infty \left[ {\mathcal {W}}^{N}\left( \tilde{f},g(r),\frac{\tau }{r^2}\right) -n(n-1)\frac{\tau }{r^2}\right] \left( \frac{e^{-a_r}}{(4\pi \tau )^{\frac{1}{2}}} \mathrm{{d}}r\right) \\&\qquad +\int _0^\infty \left( \int _N \left[ \tau (\partial _r f)^2+a_r-1\right] \left( \frac{e^{-\tilde{f}}}{\left( 4\pi \tau r^{-2}\right) ^{\frac{n}{2}}} \mathrm{{d}}v\right) \left( \frac{e^{-a_r}}{(4\pi \tau )^{\frac{1}{2}}} \mathrm{{d}}r\right) \right) \\&\qquad + \text {Error term}. \end{aligned}$$

The idea is to consider the variations of g(r) small enough to consider it locally conical in the expression of the ${\mathcal {W}}$-functional. More precisely, we will prove the following proposition.

Proposition A.1

Consider a family of metrics $({\hat{g}})_t$ which is exponentially fast in the $C^2$-sense: for $k\in \{0,1,2\}$, there exist $c_k>0$ and $C_k>0$ such that

$$\begin{aligned} |\partial ^k_{t^k}{\hat{g}}|\leqslant C_ke^{-c_kt}. \end{aligned}$$

Then, defining $g(r) = {\hat{g}}\left( \frac{\delta }{r^2}\right)$, we have the following control:

$$\begin{aligned} {\mathcal {W}}^{M}(f,g,\tau )&= \int _0^\infty \left[ {\mathcal {W}}^{N}\left( \tilde{f},g(r),\frac{\tau }{r^2}\right) -n(n-1)\frac{\tau }{r^2}\right] \left( \frac{e^{-a_r}}{(4\pi \tau )^{\frac{1}{2}}} \mathrm{{d}}r\right) \\&\quad +\int _0^\infty \left( \int _N \left[ \tau (\partial _r f)^2+a_r-1\right] \left( \frac{e^{-\tilde{f}}}{\left( 4\pi \tau r^{-2}\right) ^{\frac{n}{2}}} \mathrm{{d}}v\right) \left( \frac{e^{-a_r}}{(4\pi \tau )^{\frac{1}{2}}} \mathrm{{d}}r\right) \right) \\&\quad + {\mathcal {O}}\left( \delta \int _0^{\infty }\frac{\tau }{r^2}\Big (\frac{e^{-a_r}}{(4\pi \tau )^{\frac{1}{2}}} \mathrm{{d}}r\Big )\right) .\\ \end{aligned}$$

Remark A.2

The integral $\int _0^{\infty }\frac{\tau }{r^2}\left( \frac{e^{-a_r}}{(4\pi \tau )^{\frac{1}{2}}} \mathrm{{d}}r\right)$ is controlled by the weighted Hardy inequality of [2].

Some Formulas for Riemannian Foliations by Hypersurfaces

Now, we have the following formula for the curvature of $M =({\mathbb {R}}\times N, \mathrm{{d}}t^2 + g_t)$ with

$$\begin{aligned} \partial _t g_t = 2K(t). \end{aligned}$$

For the Ricci curvature, we have the components,

$$\begin{aligned} {{\,\mathrm{\mathrm{Ric}}\,}}_{00}= & {} -\,\partial _t K^i_i + g^{ij}K_{ij},\\ {{\,\mathrm{\mathrm{Ric}}\,}}_{0i}= & {} -\,D_i K^j_j + D_jK^j_i,\\ {{\,\mathrm{\mathrm{Ric}}\,}}_{ij}= & {} {{\,\mathrm{\mathrm{Ric}}\,}}^{n(t)}_{ij}-\partial _t K_{ij}+2 K_{il}K^l_j-K^l_lK_{ij}, \end{aligned}$$

which gives that the scalar curvature is

$$\begin{aligned} {{\,\mathrm{{ R}}\,}}^M = {{\,\mathrm{{ R}}\,}}^{N(t)}-\partial _t K^i_i + g^{ij}K_{ij} + \sum _{i,j}g^{ij}(-\partial _t K_{ij}+2 K_{il}K^l_j-K^l_lK_{ij}). \end{aligned}$$

(26)

Estimating the difference with the expression of ${\mathcal {W}}$for a cone

Let us define

$$\begin{aligned} {{\,\mathrm{{ R}}\,}}_\mathrm{{rest}} := {{\,\mathrm{{ R}}\,}}^M(r)-{{\,\mathrm{{ R}}\,}}^{C\big ({\hat{N}}\big (\frac{\delta }{r^2}\big )\big )}(r). \end{aligned}$$

Lemma A.3

If the convergence of the family of metric considered is exponentially fast in the $C^2$-sense, we have the following estimate for small $\delta$,

$$\begin{aligned} {{\,\mathrm{{ R}}\,}}_\mathrm{{rest}}(r)={\mathcal {O}}(\delta ) \end{aligned}$$

uniformly on the manifold.

Remark A.4

The convergence will always be exponentially fast for the constructions by renormalized Ricci flow that we will consider.

Proof

Thanks to (26), we have an explicit expression for the scalar curvature in our case depending on the first and second derivatives. Let us just note that this quantity is ${\mathcal {O}}\left( |\partial ^2_{r^2}g(r)|+\frac{1}{r}|\partial _r g(r)|+\frac{1}{r^2}|\partial _r g(r)|^2\right)$.

Now, if the convergence of ${\hat{g}}$ is exponentially fast in the $C^2$-sense:

$$\begin{aligned} |\partial ^k_{t^k}{\hat{g}}|\leqslant C_ke^{-c_kt}, \end{aligned}$$

so, since $g(r) = {\hat{g}}(\frac{\delta }{r^2})$, for some small $\delta$, by the ${\mathcal {O}}\Big (|\partial ^2_{r^2}g(r)|+\frac{1}{r}|\partial _r g(r)|+\frac{1}{r^2}|\partial _r g(r)|^2\Big )$ estimate, we have

$$\begin{aligned} {{\,\mathrm{{ R}}\,}}_\mathrm{{rest}}&= {\mathcal {O}}\left( |\partial ^2_{r^2}g(r)|+\frac{1}{r}|\partial _r g(r)|+\frac{1}{r^2}|\partial _r g(r)|^2\right) \\&={\mathcal {O}}\left( \left( (\theta ')^2|\partial ^2_{r^2}{\hat{g}}|+\theta ''|\partial _r{\hat{g}}|\right) +\left( \frac{1}{r}\theta '|\partial _r {\hat{g}}|\right) +\left( \frac{1}{r^2}(\theta ')^2|\partial _r {\hat{g}}|^2\right) \right) \\&={\mathcal {O}}\left( \left( \frac{\delta ^2}{r^6}e^{-\frac{\delta c_2}{r^2}}+\frac{\delta }{r^4}e^{-\frac{\delta c_1}{r^2}}\right) +\left( \frac{\delta }{r^4}e^{-\frac{\delta c_1}{r^2}}\right) +\left( \frac{\delta ^2}{r^8}e^{-\frac{2\delta c_1}{r^2}}\right) \right) \\&={\mathcal {O}}(\delta +\delta ^2). \end{aligned}$$

uniformly in r. $\square$

Proof of Proposition A.1

Since on M we have the following formulas coming from the foliations formulas,

$$\begin{aligned} {{\,\mathrm{{ R}}\,}}^M = \frac{{{\,\mathrm{{ R}}\,}}^{N(r)}-n(n-1)}{r^2}+{{\,\mathrm{{ R}}\,}}_\mathrm{{rest}} \end{aligned}$$

by definition of ${{\,\mathrm{{ R}}\,}}_{rest}$,

$$\begin{aligned} |\nabla ^M f|^2= & {} (\partial _r f)^2+\frac{|\nabla ^{N(r)} f|^2}{r^2},\\ \mathrm{{d}}v^M= & {} r^n\mathrm{{d}}v^{N(r)}\mathrm{{d}}r. \end{aligned}$$

This implies that ${\mathcal {W}}^M$ has the following expression:

$$\begin{aligned} {\mathcal {W}}^M(f,g,\tau )&=\int _0^\infty \int _{N(r)} \left[ \tau \left( (\partial _r f)^2+\frac{|\nabla ^{N(r)} f|^2+({{\,\mathrm{{ R}}\,}}^{N(r)}-n(n-1)+{{\,\mathrm{{ R}}\,}}_\mathrm{{rest}})}{r^2}\right) \right. \\&\quad \left. +\,\,f-(n+1) \right] \times \left( \frac{e^{-\tilde{f}}}{\left( 4\pi \tau r^{-2}\right) ^{\frac{n}{2}}} \mathrm{{d}}v\right) \left( \frac{e^{-a_r}}{(4\pi \tau )^{\frac{1}{2}}} \mathrm{{d}}r\right) . \end{aligned}$$

Now, defining the same separation of variables as in the cone case and using the last lemma stating that $R_\mathrm{{rest}} = {\mathcal {O}}(\delta )$, we get

$$\begin{aligned} {\mathcal {W}}^{M}(f,g,\tau )&= \int _0^\infty [\tilde{{\mathcal {W}}}^{N(r)}(\tilde{f},g(r),\tau r^{-2})]\left( \frac{e^{-a_r}}{(4\pi \tau )^{\frac{1}{2}}}\mathrm{{d}}r\right) \\&\quad +\int _0^\infty \int _N [\tau (\partial _r f)^2+a_r-1]\left( \frac{e^{-\tilde{f}}}{(4\pi \tau r^{-2})^{\frac{n}{2}}}\mathrm{{d}}v\right) \left( \frac{e^{-a_r}}{(4\pi \tau )^{-\frac{1}{2}}}\mathrm{{d}}r\right) \\&\quad + {\mathcal {O}}\left( \delta \int _0^{\infty }\frac{\tau }{r^2}\Big (\frac{e^{-a_r}}{(4\pi \tau )^{\frac{1}{2}}} \mathrm{{d}}r\Big )\right) .\\ \end{aligned}$$

$\square$

Appendix B: Lower Bounds on the ${\mathcal {W}}$-Functional of Perturbation of the Sphere

In this section, we see under which conditions, we can control the ${\mathcal {W}}$-functional of perturbations of the unit sphere at all scales.

We will obtain a precise enough control for our purpose by assuming a lower bound on the scalar curvature and the $C^0$-closeness. Recall that the ${\mathcal {W}}$-functional has the following expression,

$$\begin{aligned} {\mathcal {W}}\left( f,g^N,\tau \right) = \tau {\mathcal {F}}\left( f+\frac{n}{2}\log (4\pi \tau ),\;g\right) +\mathcal {N}\left( f+\frac{n}{2}\log (4\pi \tau ),\;g\right) -\frac{n}{2}\log (4\pi \tau ) - n. \end{aligned}$$

(27)

Proposition B.1

Let $(\mathbb {S}^n, g^N)$ satisfy the two following properties:

• $C^0$-closeness,

  $$\begin{aligned} \beta _1^2 g^{\mathbb {S}^n}\leqslant g^N \leqslant \beta _2^2g^{\mathbb {S}^n}. \end{aligned}$$

• Lower bound on the scalar curvature,

  $$\begin{aligned} {{\,\mathrm{{ R}}\,}}^N\geqslant \frac{n(n-1)}{\beta _2^2}. \end{aligned}$$

Then, we have the following lower bounds on the different components of the ${\mathcal {W}}$-functional decomposed as in (27):

• For ${\mathcal {F}}$,

  $$\begin{aligned} {\mathcal {F}}\left( f+\frac{n}{2}\log (4\pi \tau ),\;g^N\right) \geqslant \frac{\beta _1^{n}}{\beta _2^{n+4}}{\mathcal {F}}\left( f +\delta +\frac{n}{2}\log (4\pi \tau ),\;g^{\mathbb {S}^n}\right) . \end{aligned}$$

• For $\mathcal {N}$,

  $$\begin{aligned} \mathcal {N}\left( f+\frac{n}{2}\log (4\pi \tau ),\;g^N\right)\geqslant & {} \frac{\beta _2^{n}}{\beta _1^n}\mathcal {N}\left( f+\delta +\frac{n}{2}\log (4\pi \tau )\;,\;g^{\mathbb {S}^n}\right) \\&+\left( \frac{\beta _1^{n}}{\beta _2^n}-\frac{\beta _2^{n}}{\beta _1^n}\right) \frac{vol(\mathbb {S}^n)}{e}-n\log \beta _2, \end{aligned}$$

where $\delta$ is defined as the real number ensuring that $\int _{\mathbb {S}^n}\frac{e^{-f-\delta }}{(4\pi \tau )^{\frac{n}{2}}}\mathrm{{d}}v^{\mathbb {S}^n} = 1$. Note that it depends on f, but that there are bounds on it that only depend on the $C^0$-closeness.

Proof

Throughout the proof, we will write $\theta :=\frac{\mathrm{{d}}v^N}{\mathrm{{d}}v^{\mathbb {S}^n}}$. By the $C^{0}$-closeness, we have $\beta _1^n\leqslant \theta \leqslant \beta _2^n$, and this implies that $\delta$ defined in the statement of Proposition B.1 satisfies

$$\begin{aligned} -\,n\log \beta _2\leqslant \delta \leqslant -\,n\log \beta _1. \end{aligned}$$

Let us start by proving the estimate for ${\mathcal {F}}$. We have a lower bound on the scalar curvature and the volume form is controlled; we only have to control $g^N\left( \nabla ^N f,\nabla ^N f\right)$ thanks to $g^{\mathbb {S}^n}\left( \nabla ^{\mathbb {S}^n} f,\nabla ^{\mathbb {S}^n} f\right)$. By definition, we have, for any v in $T_x\mathbb {S}^n$ for some $x\in \mathbb {S}^n$,

$$\begin{aligned} \mathrm{{d}}f(v)&= g^N\left( \nabla ^N f,v\right) \\&= g^{\mathbb {S}^n}\left( \nabla ^{\mathbb {S}^n} f,v\right) . \end{aligned}$$

We can decompose $\nabla ^N f =: \alpha \nabla ^{\mathbb {S}^n} f +p^N(\nabla ^{N}f)$ where $p^N$ is the projection on the orthogonal of $\nabla ^{\mathbb {S}^n} f$ for $g^N$ (note that $\alpha$ depends on the point at which we look at the tangent space and the directions of $\nabla ^{\mathbb {S}^n} f$ and $\nabla ^N f$). Let us find bounds on $\alpha$ which will give us a lower bound on $g^N\left( \nabla ^N f,\nabla ^N f\right)$. By definition, we have

$$\begin{aligned} \mathrm{{d}}f(\nabla ^{\mathbb {S}^n} f)&= g^N\left( \nabla ^N f,\nabla ^{\mathbb {S}^n} f\right) \\&= \alpha g^N\left( \nabla ^{\mathbb {S}^n} f,\nabla ^{\mathbb {S}^n} f\right) , \end{aligned}$$

but we also have: $\mathrm{{d}}f(\nabla ^{\mathbb {S}^n} f) = g^{\mathbb {S}^n}\left( \nabla ^{\mathbb {S}^n} f,\nabla ^{\mathbb {S}^n} f\right)$. Now, since $\beta _1^2g^{\mathbb {S}^n}\leqslant g^N\leqslant \beta _2^2g^{\mathbb {S}^n}$, we have

$$\begin{aligned} \frac{1}{\beta _2^2}\leqslant \alpha \leqslant \frac{1}{\beta _1^2}. \end{aligned}$$

We obtain the following bound for the ${\mathcal {F}}$-functional,

$$\begin{aligned}&{\mathcal {F}}\left( f+\frac{n}{2}\log (4\pi \tau )\;,\;g^N\right) = \int _{\mathbb {S}^n} \left( g^N\left( \nabla ^N f,\nabla ^N f\right) +{{\,\mathrm{{ R}}\,}}^N\right) \frac{e^{-f}}{(4\pi \tau )^{\frac{n}{2}}}\;\theta \mathrm{{d}}v^{\mathbb {S}^n} \\&\quad \geqslant \int _{\mathbb {S}^n} \left( \alpha ^2g^N\left( \nabla ^{\mathbb {S}^n} f,\nabla ^{\mathbb {S}^n} f\right) +g^N(p^N(\nabla ^N f)\;,\;p^N(\nabla ^N f))\right. \\&\quad \left. +\,\,\frac{n(n-1)}{\beta _2^2}\right) \frac{e^{-f-\delta }}{(4\pi \tau )^{\frac{n}{2}}}\; e^{\delta }\beta _1^n \mathrm{{d}}v^{\mathbb {S}^n}. \end{aligned}$$

We can finally now use our bound on $\alpha$ and $\delta$ and conclude that

$$\begin{aligned}&{\mathcal {F}}\left( f+\frac{n}{2}\log (4\pi \tau )\;,\;g^N\right) \geqslant \int _{\mathbb {S}^n} \left( \frac{1}{\beta _2^4}g^N\left( \nabla ^{\mathbb {S}^n} f+\delta ,\nabla ^{\mathbb {S}^n} f+\delta \right) \right. \\&\quad \left. +\,\,\frac{n(n-1)}{\beta _2^2}\right) \frac{e^{-f-\delta }}{(4\pi \tau )^{\frac{n}{2}}}\;\frac{\beta _1^{n}}{\beta _2^n}\mathrm{{d}}v^{\mathbb {S}^n}\\&\quad \geqslant \frac{\beta _1^{n}}{\beta _2^{n+4}}{\mathcal {F}}\left( f+\delta +\frac{n}{2}\log (4\pi \tau )\;,\;g^{\mathbb {S}^n}\right) , \end{aligned}$$

which is the stated inequality.

Let us now take care of the $\mathcal {N}$ functional, and denote $v = \frac{e^{-f}}{(4\pi \tau )^{\frac{n}{2}}}$ and $w = \frac{e^{-f-\delta }}{(4\pi \tau )^{\frac{n}{2}}}$. They satisfy

$$\begin{aligned} \int _{\mathbb {S}^n}v\; \theta \mathrm{{d}}v^{\mathbb {S}^n} =1 \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {S}^n}w \;\mathrm{{d}}v^{\mathbb {S}^n} =1. \end{aligned}$$

The expressions to compare are

$$\begin{aligned} \mathcal {N}\left( f+\frac{n}{2}\log (4\pi \tau )\;,\;g^N\right) = -\int _{\mathbb {S}^n}v\log v\; \theta \mathrm{{d}}v^{\mathbb {S}^n} \end{aligned}$$

and

$$\begin{aligned} \mathcal {N}\left( f+\delta +\frac{n}{2}\log (4\pi \tau )\;,\;g^{\mathbb {S}^n}\right) = -\int _{\mathbb {S}^n}w\log w \;\mathrm{{d}}v^{\mathbb {S}^n}. \end{aligned}$$

Let us start the comparison of both expressions by separating the positive and negative part of the integrand, recall that $v = e^\delta w$,

$$\begin{aligned}&\mathcal {N}\left( f+\frac{n}{2}\log (4\pi \tau )\;,\;g^N\right) = -\int _{\mathbb {S}^n}v\log v\; \theta \mathrm{{d}}v^{\mathbb {S}^n}\\&\quad =-e^\delta \int _{\mathbb {S}^n}w\log w\; \theta \mathrm{{d}}v^{\mathbb {S}^n} + \delta \\&\quad =-e^\delta \int _{\{w\geqslant 1\}}w\log w\; \theta \mathrm{{d}}v^{\mathbb {S}^n}-e^\delta \int _{\{w < 1\}}w\log w \; \theta \mathrm{{d}}v^{\mathbb {S}^n} + \delta .\\ \end{aligned}$$

Now, we have bounds on $\delta$ and $\theta$ coming from the $C^0$-closeness to the unit sphere, namely,

$$\begin{aligned} -\,n\log \beta _2\leqslant \delta \leqslant -\,n\log \beta _1 \end{aligned}$$

and

$$\begin{aligned} \beta _1^n\leqslant \theta \leqslant \beta _2^n. \end{aligned}$$

Therefore, by the separation of the positive and negative parts, we get

$$\begin{aligned}&\mathcal {N}\left( f+\frac{n}{2}\log (4\pi \tau )\;,\;g^N\right) = -e^\delta \int _{\{w\geqslant 1\}}w\log w\; \theta \mathrm{{d}}v^{\mathbb {S}^n}-e^\delta \int _{\{w< 1\}}w\log w\; \theta \mathrm{{d}}v^{\mathbb {S}^n} + \delta \\&\quad \geqslant -\beta _2^{2n}\int _{\{w\geqslant 1\}}w\log w \; \mathrm{{d}}v^{\mathbb {S}^n}-\beta _1^{2n}\int _{\{w < 1\}}w\log w\; \mathrm{{d}}v^{\mathbb {S}^n}+n\log \beta _1. \end{aligned}$$

We can now use the fact that for any positive real number x, $x\log x\geqslant -\frac{1}{e}$:

$$\begin{aligned}&\mathcal {N}\left( f+\frac{n}{2}\log (4\pi \tau )\;,\;g^N\right) \geqslant -\frac{\beta _2^{n}}{\beta _1^n}\int _{\{w\geqslant 1\}}w\log w \; \mathrm{{d}}v^{\mathbb {S}^n} -\frac{\beta _1^{n}}{\beta _2^n}\int _{\{w< 1\}}w\log w \; \mathrm{{d}}v^{\mathbb {S}^n}+n\log \beta _1\\&\quad = -\frac{\beta _2^{n}}{\beta _1^n}\int _{\mathbb {S}^n}w\log w \;\mathrm{{d}}v^{\mathbb {S}^n}+\left( \frac{\beta _1^{n}}{\beta _2^n}-\frac{\beta _2^{n}}{\beta _1^n}\right) \int _{\{w < 1\}}w\log w\; \mathrm{{d}}v^{\mathbb {S}^n}+n\log \beta _1 \\&\quad \geqslant -\frac{\beta _2^{n}}{\beta _1^n}\int _{\mathbb {S}^n}w\log w\; \mathrm{{d}}v^{\mathbb {S}^n}+\left( \frac{\beta _1^{n}}{\beta _2^n}-\frac{\beta _2^{n}}{\beta _1^n}\right) \frac{vol(\mathbb {S}^n)}{e}+n\log \beta _1, \end{aligned}$$

which is exactly what we stated. $\square$

In particular, such manifolds satisfy the lower bound $L(\epsilon _1,\; \epsilon _2,\;\epsilon _3)$ defined at the end of section 3 with

$$\begin{aligned} \left\{ \begin{array}{lll} \epsilon _1 = 1-\frac{\beta _1^{n}}{\beta _2^{n+4}},\\ \epsilon _2 = \frac{\beta _2^{n}}{\beta _1^n}-1,\\ \epsilon _3 = \left( \frac{\beta _1^{n}}{\beta _2^n}-\frac{\beta _2^{n}}{\beta _1^n}\right) \frac{vol(\mathbb {S}^n)}{e}+n\log \beta _1. \end{array} \right. \end{aligned}$$

Remark B.2

For a sphere of radius $\beta >0$, we have the following bounds,

• If $\beta \geqslant 1$, then

  $$\begin{aligned} {\mathcal {W}}^{\beta \mathbb {S}^n}(f,\beta ^2g^{\mathbb {S}^n},\tau )\geqslant & {} \frac{\tau }{\beta ^2}{\mathcal {F}}^{\mathbb {S}^n}\left( f+C,g^{\mathbb {S}^n}\right) +\mathcal {N}^{\mathbb {S}^n}\left( f+C,g^{\mathbb {S}^n}\right) \\&+\frac{n}{2}\log {4\pi \tau }-n, \end{aligned}$$

  that is, $1-\epsilon _1 = \frac{1}{\beta ^2}$, $\epsilon _2 = 0$, and $\epsilon _3 = 0$.

• If $\beta \leqslant 1$, then

  $$\begin{aligned} {\mathcal {W}}^{\beta \mathbb {S}^n}\left( f,\beta ^2g^{\mathbb {S}^n},\tau \right)\geqslant & {} \tau {\mathcal {F}}^{\mathbb {S}^n}(f+C,g^{\mathbb {S}^n})+\mathcal {N}^{\mathbb {S}^n}(f+C,g^{\mathbb {S}^n})\\&+\frac{n}{2}\log {4\pi \tau }-n+n\log \beta , \end{aligned}$$

  that is, $\epsilon _1 = 0$, $\epsilon _2 = 0$, and $\epsilon _3 = -n\log \beta$.

And in dimension 3, thanks to the explicit $\eta _3$ given in the last section, we can smooth out the cones over a sphere of radius $\beta$ if $\beta \in [0.77,1.05]$ by type III Ricci flows.

Here we give the proofs of the remarks given along the paper about positively curved Einstein manifolds, with a particular interest in the round sphere.

Appendix C: The Behavior of the $\mu$-Functional of Positively Curved Einstein Manifolds

Positively curved Einstein manifolds have the following particularity: the minimizing function of ${\mathcal {W}}^N(.,g,\tau )$ is constant if $\tau \geqslant T_N$.

Lemma C.1

Let us consider (N, g) an Einstein manifold such that,

$$\begin{aligned}{{\,\mathrm{\mathrm{Ric}}\,}}= \frac{1}{2T_N}g,\end{aligned}$$

(of shrinking time $T_N$), and $\phi _N$ its constant potential function such that,

$$\begin{aligned}e^{-\phi _N} vol(N) = 1,\end{aligned}$$

and let us denote $\phi ^\tau$ a function such that $\phi ^\tau -\frac{n}{2}\log (4\pi \tau )$ is a minimizer for ${\mathcal {W}}^N$ at $\tau$,

$$\begin{aligned} {\mathcal {W}}^N\left( \phi ^\tau -\frac{n}{2}\log (4\pi \tau ),g^N,\tau \right) = \mu ^N(g^N,\tau ). \end{aligned}$$

Then, for all $\tau \geqslant T_N$,

$$\begin{aligned} \phi ^\tau = \phi ^{T_N}. \end{aligned}$$

Remark C.2

In particular, it is also a minimizer of the ${\mathcal {F}}^N$-functional (which is consistent with the fact that when $\tau \rightarrow \infty$, $\phi ^\tau$ gets closer and closer to a minimizer of the ${\mathcal {F}}$ functional).

Remark C.3

This is not true for $\tau <T_N$. For example, $\phi ^{T_{\mathbb {S}^n}}$ is a constant function, but for small $\tau$, $\phi ^{\tau }$ looks like a Gaussian at small scale. Another way to see this is to note that the value of ${\mathcal {W}}^{\mathbb {S}^n}$ with a constant function tends to $+\infty$ when $\tau \rightarrow 0$.

Proof

Let us also consider N a positively curved Einstein manifold of shrinking time $T_N$, that is,

$$\begin{aligned} g^N(t) = \left( 1-\frac{t}{T_N}\right) g^N(0), \end{aligned}$$

and choose $\tau _0 \geqslant T_N$. There exists $0\leqslant t_0<T_N$ such that

$$\begin{aligned} \tau _0 = \frac{T_N}{1-\frac{t_0}{T_N}}. \end{aligned}$$

Let us compute ${\mathcal {W}}^N\left( g^N(0),\left[ f^{T_N}+\frac{n}{2}\log \frac{\tau _0}{T_N}\right] ,\tau _0\right)$. Any positively curved Einstein metric g has a shrinking soliton structure, ${{\,\mathrm{\mathrm{Ric}}\,}}(g) + {\mathcal {L}}_{\nabla \phi ^{T_N}}g = \frac{1}{2T_N}g$, with $\phi ^{T_N} = \phi _N$.

Thus, $\phi ^{T_N}$ is also a minimizer of ${\mathcal {F}}^N$, then we have the following equality,

$$\begin{aligned}&\mu ^N(g^N,\tau _0) \leqslant {\mathcal {W}}^N\left( g^N,\left[ \phi ^{T_N}-\frac{n}{2}\log (4\pi \tau _0)\right] ,\tau _0\right) \\&\quad = {\mathcal {W}}^N\left( \left[ \phi ^{T_N}-\frac{n}{2}\log (4\pi T_N)\right] ,g^N,T_N\right) \\&\qquad + \lambda ^N(\tau _0-T_N)-\frac{n}{2}\log \frac{\tau _0}{T_N}\\&\quad =\mu (g^N,T_N)+ \lambda ^N(\tau _0-T_N)-\frac{n}{2}\log \frac{\tau _0}{T_N}. \end{aligned}$$

Then, there is equality in the inequality and

$$\begin{aligned} \mu ^N(\tau ) \geqslant \mu ^N(T_N)+ \lambda ^N(\tau -T_N)-\frac{n}{2}\log \frac{\tau }{T_N}, \end{aligned}$$

so for all $T_N<\tau <\tau _0$, $\phi ^\tau$ minimizes ${\mathcal {W}}^N(.,g^N,\tau )$, that is,

$$\begin{aligned} \phi ^{\tau _0} = \phi ^{T_N}. \end{aligned}$$

$\square$

As a consequence, we have the following lemma.

Corollary C.4

If N is a positively curved Einstein manifold, such that

$$\begin{aligned} {{\,\mathrm{\mathrm{Ric}}\,}}= \frac{1}{2T_N}g, \end{aligned}$$

(of shrinking time $T_N$).

Then, for all $\tau \geqslant T_N$, the minimizing function for ${\mathcal {W}}^N(.,g^N,\tau )$ is constant and

$$\begin{aligned} \mu ^N(\tau ) = \mu ^N(T_N)+ \lambda ^N(\tau -T_N)-\frac{n}{2}\log \frac{\tau }{T_N}, \end{aligned}$$

(28)

that is,

if $\tau \geqslant T_N$, then

$$\begin{aligned} \mu ^N(\tau ,g^N) = \tau \lambda ^N+\log (vol(N))-\frac{n}{2}\log (4\pi \tau )-n, \end{aligned}$$

(29)

and if $\tau \leqslant T_N$, then

$$\begin{aligned} 0\geqslant \mu ^N(\tau ,g^N) \geqslant \mu ^N(T_N,g^N) = \frac{n}{2}+\log (vol(N))-\frac{n}{2}\log (4\pi T_N)-n. \end{aligned}$$

(30)

Remark D.5

In particular, the $\mu$-functional of a manifold with ${{\,\mathrm{\mathrm{Ric}}\,}}= (n-1)g$ for large $\tau$ only depends on the dimension and the volume of the manifold.

Appendix D: Smoothing Rotationally Symmetric Cones by Manifolds of High Entropy

In this Appendix, we provide an explicit computation of the smoothing process of the last section of the paper applied to spheres. That will illustrate the last step of the construction of a renormalization of the Ricci flow preserving a $P(\beta _1',\beta _2')$ property, where a ball centered at the “tip” of the manifold is arbitrarily close to that of a cone over a sphere.

Lemma D.1

For all n, there exist $\beta _1(n)$ and $\beta _2(n)$, $0<\beta _1<1<\beta _2$ such that

For all $\beta \in (\beta _1,\beta _2)$, the cone $C(\beta \mathbb {S}^n)$ satisfies

$$\begin{aligned} \nu ^{C(\beta \mathbb {S}^n)}> -\,\eta _n, \end{aligned}$$

where $\eta _n$ is the number defined in the global pseudolocality, Theorem 3.2.

Moreover, there exists $M_\beta$ a manifold smoothing it out such that

$$\begin{aligned} \nu ^{M_\beta }>-\,\eta _n. \end{aligned}$$

Since the ${\mathcal {W}}$-functional is continuous on the space of metrics with the $C^2$-topology, we have the following corollary:

Corollary D.2

For all $\beta >0$ such that $\beta _1<\beta <\beta _2$ metric $\mathrm{{d}}r^2 + r^2g(r)$ whose $\nu$-functional is larger than $-\,\eta _n$, and which is smoothly asymptotic to the metric $\mathrm{{d}}r^2+r^2 \beta ^2 g^{\mathbb {S}^n}$ when $r\rightarrow 0$, there exists a smoothing that preserves the inequality $\nu > -\,\eta _n$.

Proof

The first statement is a direct consequence of the lower bounds found on the $\nu$-functional of cones thanks to some closeness to the unit sphere, see Proposition B.1. We will now focus on the construction of the manifold $M_\beta$.

We will smoothen the cone by the euclidean space around its tip by warped product: $(M,g) = ({\mathbb {R}}^+\times \mathbb {S}^n,\mathrm{{d}}r^2+h^2g^{\mathbb {S}^n})$, where we define $h: [0,\infty ) \rightarrow {\mathbb {R}}^+$ by

• 1. $h(r) = \frac{r}{\beta }$ for $0\leqslant r\leqslant 1$,

• 2. $h(r) = \frac{r}{\beta }+ \frac{(r-1)^2}{2\beta A}$ for $1\leqslant r\leqslant b = 1+A(\beta -1)$,

• 3. $h(r) = r-b+h(b)$ for $r\geqslant b$.

This gives

• 1. $h'(r) = \frac{1}{\beta }$ for $0\leqslant r\leqslant 1$,

• 2. $h'(r) = \frac{1}{\beta }+ \frac{r-1}{\beta A}$ for $1\leqslant r\leqslant b = 1+A(\beta -1)$,

• 3. $h'(r) = 1$ for $r\geqslant b$

and

• 1. $h''(r) = 0$ for $0\leqslant r\leqslant 1$,

• 2. $h''(r) = \frac{1}{\beta A}$ for $1\leqslant r\leqslant b = 1+A(\beta -1)$,

• 3. $h''(r) = 0$ for $r\geqslant b$.

Let us now consider $(M,g) = ({\mathbb {R}}^+\times \beta \mathbb {S}^n,\mathrm{{d}}r^2+h^2g_{\beta \mathbb {S}^n})$ and $f: M\rightarrow {\mathbb {R}}$ such that $\int _M \frac{e^{-f}}{(4\pi \tau )^{\frac{n+1}{2}}}=1$, and denote $\beta \mathbb {S}^n = N$.

Note 4

Here, we note w such that $w^2 = e^{-f}$, it satisfies $\int _M w^2\mathrm{{d}}v = (4\pi \tau )^{\frac{n+1}{2}}$.

Using the following general formula for the entropy of a warped product,

$$\begin{aligned}&{\mathcal {W}}^M(f,\mathrm{{d}}r^2+h^2g_N,\tau ) \\&\quad = \int _0^{+\infty }h^n\int _N\left[ \tau \left( 4(\partial _r w)^2 +\frac{4|\nabla ^Nw|^2+({{\,\mathrm{{ R}}\,}}^N-n(n-1)(h')^2-2n h h'')w^2}{h^2}\right) \right. \\&\qquad \left. - \,w^2 \log (w^2)-(n+1) w^2 \right] (4\pi \tau )^{-\frac{n+1}{2}}\mathrm{{d}}v \mathrm{{d}}r, \end{aligned}$$

we get

$$\begin{aligned}&{\mathcal {W}}^M(f,\mathrm{{d}}r^2+h^2g_N,\tau ) \\&\quad = \int _0^{1}\left( \frac{r}{\beta }\right) ^n\int _N \left[ \tau \left( 4(\partial _r w)^2 +\frac{4|\nabla ^Nw|^2}{(\frac{r}{\beta })^2}\right) \right. \\&\qquad \left. - \,\,w^2 \log (w^2)-(n+1) w^2 \right] (4\pi \tau )^{-\frac{n+1}{2}}\mathrm{{d}}v \mathrm{{d}}r \\&\qquad +\int _1^{b}h^n\int _N \left[ \tau \left( 4(\partial _r w)^2 {+}\frac{4|\nabla ^Nw|^2+(n(n-1)(\beta ^{-2}-(h')^2)-2n h h'')w^2}{h^2}\right) \right. \\&\qquad \left. -\,\, w^2 \log (w^2)-(n+1) w^2 \right] (4\pi \tau )^{-\frac{n+1}{2}}\mathrm{{d}}v \mathrm{{d}}r\\&\qquad +\int _{h(b)}^{+\infty }r^n\int _N\left[ \tau \left( 4(\partial _r w)^2 +\frac{4|\nabla ^Nw|^2+n(n-1)(\beta ^{-2}-1)w^2}{r^2}\right) \right. \\&\qquad \left. - \,\,w^2 \log (w^2)-(n+1) w^2 \right] (4\pi \tau )^{-\frac{n+1}{2}}\mathrm{{d}}v \mathrm{{d}}r,\\&\quad =(1)+(2)+(3), \end{aligned}$$

where we have made the last term explicit to show that it is corresponding to what we would get with C(N).

In the second term, for A big enough, the term involving $h''$ is like $A^{-1}$ times the term in $h'$. We will assume that A is big enough to forget about $h''$ (we only have strict inequalities in the end).

Let us consider v a function on C(N) such that

$$\begin{aligned} \int _{C(N)}v^2(4\pi \tau )^{-\frac{n+1}{2}} = 1. \end{aligned}$$

We are going to define $w=\phi (v)$ a function on M such that

$$\begin{aligned} \int _M w^2(4\pi \tau )^{-\frac{n+1}{2}} = 1, \end{aligned}$$

where $\phi$ is a natural one-to-one correspondence between functions on C(N) and M. We will then prove that the entropy of w on M is controlled by the entropy of v on C(n).

If we define

$$\begin{aligned} w(r,.) = \sqrt{h'(r)}v(h(r),.), \end{aligned}$$

we have

$$\begin{aligned} \int _M w^2(4\pi \tau )^{-\frac{n+1}{2}} = 1. \end{aligned}$$

And by the change of variable $\rho = h(r)$, we obtain

$$\begin{aligned}&{\mathcal {W}}^M(f,\mathrm{{d}}r^2+h^2g_N,\tau ) \\&\quad = \int _0^{\frac{1}{\beta }}(\rho )^n\int _N\left[ \tau \left( 4(\partial _\rho v)^2 +\frac{4|\nabla ^Nv|^2}{\rho ^2}\right) \right. \\&\qquad \left. -\,\, v^2 \left( \log (v^2)+n\log \beta \right) -(n+1) v^2\right] (4\pi \tau )^{-\frac{n+1}{2}}\mathrm{{d}}v \mathrm{{d}}\rho \\&\qquad +\int _{\frac{1}{\beta }}^{h(b)}\rho ^n\int _N\left[ \tau \left( 4(\partial _\rho v)^2 \right. \right. \\&\qquad \left. \left. +\,\,\frac{4|\nabla ^Nv|^2+n(n-1)\left[ \beta ^{-2}-\left( (h'\circ h^{-1}(\rho )\left( 1+{\mathcal {O}}(\frac{1}{A})\right) v^2+\frac{1}{2}\log h'\circ h^{-1}\right) \right] }{\rho ^2}\right) \right. \\&\qquad \left. -\,\, v^2 \log (v^2)-(n+1) v^2\right] (4\pi \tau )^{-\frac{n+1}{2}}\mathrm{{d}}v \mathrm{{d}}\rho \\&\qquad +\int _{h(b)}^{+\infty }\rho ^n\int _N\left[ \tau \left( 4(\partial _\rho v)^2 +\frac{4|\nabla ^Nv|^2+n(n-1)(\beta ^{-2}-1)v^2}{\rho ^2}\right) \right. \\&\qquad \left. - \,\,v^2 \log (v^2)-(n+1) v^2 \right] (4\pi \tau )^{-\frac{n+1}{2}}\mathrm{{d}}v \mathrm{{d}}\rho \\&\quad =(1)+(2)+(3). \end{aligned}$$

Now, what we would have with the cone is

$$\begin{aligned}&{\mathcal {W}}^{C(N)}(v,\mathrm{{d}}r^2+r^2g_N,\tau ) \\&\quad =\int _0^{\frac{1}{\beta }}(\rho )^n\int _N\left[ \tau \left( 4(\partial _\rho v)^2 +\frac{4|\nabla ^Nv|^2+n(n-1)(\beta ^{-2}-1)v^2}{\rho ^2}\right) \right. \\&\qquad \left. -\,\, v^2 \log (v^2)-(n+1) v^2 \right] (4\pi \tau )^{-\frac{n+1}{2}}\mathrm{{d}}v \mathrm{{d}}\rho \\&\qquad +\int _{\frac{1}{\beta }}^{h(b)}\rho ^n\int _N\left[ \tau \left( 4(\partial _\rho v)^2 +\frac{4|\nabla ^Nv|^2+n(n-1)(\beta ^{-2}-1)v^2}{\rho ^2}\right) \right. \\&\qquad \left. - \,\,v^2 \log (v^2)-(n+1) v^2 \right] (4\pi \tau )^{-\frac{n+1}{2}}\mathrm{{d}}v \mathrm{{d}}\rho \\&\qquad +\int _{h(b)}^{+\infty }\rho ^n\int _N\left[ \tau \left( 4(\partial _\rho v)^2 +\frac{4|\nabla ^Nv|^2+n(n-1)(\beta ^{-2}-1)v^2}{\rho ^2}\right) \right. \\&\qquad \left. - \,\,v^2 \log (v^2)-(n+1) v^2 \right] (4\pi \tau )^{-\frac{n+1}{2}}\mathrm{{d}}v \mathrm{{d}}\rho \\&\quad =(1')+(2')+(3'). \end{aligned}$$

If we assume $\beta >1$, by taking the difference of these two expressions, we get

$$\begin{aligned} {\mathcal {W}}^{M}(w,\mathrm{{d}}r^2 + r^2g_N,\tau )-{\mathcal {W}}^{C(\beta \mathbb {S}^n)}(v,\mathrm{{d}}r^2 + h^2g_N,\tau ) >-(n\log \beta +{\mathcal {O}}(A)). \end{aligned}$$

In the case of a $\beta <1$, it is more convenient to compare to the Euclidean space. The computation is exactly the same, and we get

$$\begin{aligned} {\mathcal {W}}^{M}(w,\mathrm{{d}}r^2 + r^2g_N,\tau )-{\mathcal {W}}^{{\mathbb {R}}^n}(v,\mathrm{{d}}r^2 + h^2g_N,\tau ) >(n\log \beta +{\mathcal {O}}(A)). \end{aligned}$$

Remark D.3

It is crucial that the right term does not depend on $\tau$.

In both cases, if $\beta$ is close enough to 1 and A large enough in our parametrization, we have $\nu ^M>\eta _n$. $\square$