Ricci Solitons, Conical Singularities, and Nonuniqueness

Abstract

In dimension $n=3$, there is a complete theory of weak solutions of Ricci flow—the singular Ricci flows introduced by Kleiner and Lott (Acta Math 219(1):65–134, 2017, in: Chen, Lu, Lu, Zhang (eds) Geometric analysis. Progress in mathematics, vol 333. Birkhäuser, Cham, 2018)—which Bamler and Kleiner (Uniqueness and stability of Ricci flow through singularities, arXiv:1709.04122v1, 2017) proved are unique across singularities. In this paper, we show that uniqueness should not be expected to hold for Ricci flow weak solutions in dimensions $n\ge 5$. Specifically, for any integers $p_1,p_2\ge 2$ with $p_1+p_2\le 8$, and any $K\in {\mathbb N}$, we construct a complete shrinking soliton metric $g_K$ on $\mathcal S^{p_1}\times {\mathbb R}^{p_2+1}$ whose forward evolution $g_K(t)$ by Ricci flow starting at $t=-1$ forms a singularity at time $t=0$. As $t\nearrow 0$, the metric $g_K(t)$ converges to a conical metric on $\mathcal S^{p_1}\times \mathcal S^{p_2}\times (0,\infty )$. Moreover there exist at least K distinct, non-isometric, forward continuations by Ricci flow expanding solitons on $\mathcal S^{p_1}\times {\mathbb R}^{p_2+1}$, and also at least K non-isometric, forward continuations expanding solitons on ${\mathbb R}^{p_1+1}\times \mathcal S^{p_2}$. In short, there exist smooth complete initial metrics for Ricci flow whose forward evolutions after a first singularity forms are not unique, and whose topology may change at the singularity for some solutions but not for others.

Appendices

Appendix A. Doubly-Warped Product Geometries

1.1 The general case.

It is well known (see, e.g., [Pet16]) that all curvatures of a doubly-warped product metric

$$\begin{aligned} g = (\mathrm ds)^2 + \varphi _1^2\,g_{\mathcal S^{p_1}} + \varphi _2^2\,g_{\mathcal S^{p_2}} \end{aligned}$$

on $\mathbb R_+\times \mathcal S^{p_1}\times \mathcal S^{p_2}$ are convex linear combinations of the five sectional curvatures⁸

$$\begin{aligned} \kappa _{\alpha ,1}=\frac{1-\varphi _{\alpha ,s}^2}{\varphi _\alpha ^2},\quad \kappa _{\alpha ,2}=-\frac{\varphi _{\alpha ,ss}}{\varphi _\alpha },\quad \kappa _{12,1}=-\frac{\varphi _{1,s} \varphi _{2,s}}{\varphi _1\varphi _2}. \end{aligned}$$

We note that the functions $\kappa _{\alpha ,1}$ ($\alpha \in \{1,2\}$) are the sectional curvatures of orthonormal planes tangent to $\mathcal S^{p_\alpha }$. The functions $\kappa _{\alpha ,2}$ are the sectional curvatures of orthonormal planes spanned by $\frac{\partial }{\partial s}$ and vectors tangent to $\mathcal S^{p_\alpha }$. And $\kappa _{12,1}$ is the sectional curvature of a plane spanned by one vector tangent to $\mathcal S^{p_1}$ and one tangent to $\mathcal S^{p_2}$.

It follows easily that the Ricci tensor of such a metric is given by

$$\begin{aligned} {{\,\mathrm{Rc}\,}}&=\big \{p_1\kappa _{1,2} +p_2\kappa _{2,2}\big \}(\mathrm ds)^2\\&\quad +\big \{\kappa _{1,2}+ (p_1-1)\kappa _{1,1}+p_2\kappa _{12,1}\big \}\varphi _1^2g_{\mathcal S^{p_1}}\\&\quad +\big \{\kappa _{2,2}+(p_2-1)\kappa _{2,1}+p_1\kappa _{12,1}\big \}\varphi _2^2g_{\mathcal S^{p_2}}. \end{aligned}$$

Now let $\mathfrak X = f(s)\,\frac{\partial }{\partial s}$ denote the gradient of a potential function F(s). Applying the general formula

$$\begin{aligned} (\mathcal L_{\mathfrak X}g)(V_1,V_2) = \mathfrak X\big \{g(V_1,V_2)\big \}+g(\nabla _{V_1}\mathfrak X,V_2)+g(V_1,\nabla _{V_2}\mathfrak X) \end{aligned}$$

for the Lie derivative of a covariant 2-tensor to this special case, one sees that

$$\begin{aligned} \mathcal L_{\mathfrak X} g = 2f_s(\mathrm d s)^2+2f\varphi _1\varphi _{1,s}\,g_{\mathcal S^{p_1}}+2f\varphi _2\varphi _{2,s}\,g_{\mathcal S^{p_2}}. \end{aligned}$$

In the main body of this work, we apply the equations above to the soliton condition

$$\begin{aligned} -2{{\,\mathrm{Rc}\,}}[g] = 2\lambda g + \mathcal L_{\mathfrak X}g, \end{aligned}$$

were $\lambda \in \{-1,0,+1\}$ controls the rescaling of the soliton.⁹

1.2 Cones.

In the case of a cone metric,

$$\begin{aligned} \varphi _1= s\,\sqrt{\frac{p_1-1}{{{\bar{x}}}_1}} \qquad \text{ and }\qquad \varphi _2= s\, \sqrt{\frac{p_2-1}{{{\bar{x}}}_2}} \end{aligned}$$

with asymptotic apertures ${{\bar{x}}}_1,\,{{\bar{x}}}_2$, the sectional curvature $\kappa _{12,1}$ is given by

$$\begin{aligned} \kappa _{12,1} = -\frac{1}{s^2}. \end{aligned}$$

In particular, the norm of the curvature tensor of the cone becomes unbounded as $s\searrow 0$.

Appendix B. A Representation as a Mechanical System on ${\mathbb R}^3$

As a curiosity, we observe that our assumption that $p_\alpha \ge 2$ allows us to define

$$\begin{aligned} u_\alpha = \log \varphi _\alpha -\tfrac{1}{2} \ln (p_\alpha -1),\quad \alpha \in \{1,2\}, \qquad \text { and }\qquad v = f - p_1\frac{\mathrm du_1}{\mathrm ds} - p_2\frac{\mathrm du_2}{\mathrm ds}. \end{aligned}$$

In these variables, the differential equations (8) become

$$\begin{aligned} {\ddot{u}}_\alpha&= v \dot{u}_\alpha + e^{-2u_\alpha }+\lambda ,\qquad \alpha \in \{1,2\}, \end{aligned}$$

(102a)

$$\begin{aligned} \dot{v}&= p_1\dot{u}_1^2 + p_2\dot{u}_2^2 - \lambda . \end{aligned}$$

(102b)

These equations can be interpreted as a mechanical system in which s is “time” and where (in this section only) we write s-derivatives as fluxions. In this interpretation, $u_1$ and $u_2$ are the coordinates of two unit-mass particles on the real line that are each subject to a force field given by $F(u) = e^{-2u}+\lambda$, and whose motion is subject to friction with friction coefficient v. The only unusual aspect of this system from the point of view of mechanics is that the friction coefficient v can be either positive or negative, and that it is itself a function of time that satisfies an ode.

The derivation of the equations for $u_1, u_2, v$ from (8) is a simple calculus exercise. Even though it appears simpler than the original equations (8), we will not use the mechanical system (102) in this paper. It does however make several cameo appearances. For example, the Ivey invariant for the stationary soliton flow can be interpreted as the energy dissipation in the mechanical system. Indeed, if $\lambda =0$, then any solution of (102) satisfies

$$\begin{aligned} \frac{\mathrm d}{\mathrm ds}\left\{ \frac{p_1}{2} \bigl (\dot{u}_1^2 + e^{-2u_1} \bigr ) + \frac{p_2}{2} \bigl (\dot{u}_2^2 + e^{-2u_1} \bigr ) \right\} = v\dot{v} = \frac{\mathrm d\frac{1}{2} v^2}{\mathrm ds}, \end{aligned}$$

which implies that the quantity $I = p_1(\dot{u}_1^2+e^{-2u_1}) + p_2(\dot{u}_2^2+e^{-2u_2}) - v^2$ is preserved along solutions of (102). Similarly, the non-obvious Lyapunov function W in Gastel and Kronz’ construction of the Böhm soliton (see Section 7.1) can be interpreted as Kinetic$+$Potential Energy for a renormalized version of the mechanical system (102).

Appendix C. An Estimate for Orbits Near a Hyperbolic Fixed Point

1.1 A model nonlinear system.

Consider a system

$$\begin{aligned} x_-' = -A_- x_- + B_-(x)x, \qquad x_+' = A_+ x_+ + B_+(x)x, \end{aligned}$$

(103)

where $x = (x_-, x_+) \in {\mathbb R}^{k_-}\times {\mathbb R}^{k_+}$, where $A_- : {\mathbb R}^{k_-} \rightarrow {\mathbb R}^{k_-}$, $A_+ : {\mathbb R}^{k_+}\rightarrow {\mathbb R}^{k_+}$ are constant linear maps, and where $B_\pm$ are smooth functions on some neighborhood of the origin in ${\mathbb R}^{k_-+k_+}$ such that $B_-(x)$ is a linear map from ${\mathbb R}^{k_-+k_+}$ to ${\mathbb R}^{k_-}$ and $B_+(x)$ is a linear map from ${\mathbb R}^{k_-+k_+}$ to ${\mathbb R}^{k_+}$. We assume furthermore that $B_\pm (0) = 0$.

The origin (0, 0) is a fixed point for our system (103). The linearization of this system at the origin has the matrix

$$\begin{aligned} \begin{bmatrix} -A_- &{} 0 \\ 0 &{} A_+ \end{bmatrix}. \end{aligned}$$

We make one more assumption, namely that the eigenvalues of both $A_\pm$ all have strictly positive real parts.

1.2 An analysis lemma.

There is a constant $C\in {\mathbb R}$ that only depends on the matrices $A_\pm$ and the nonlinear functions $B_\pm$, such that for all $T>0$ and for any solution $x:[0, T]\rightarrow {\mathbb R}^{k_-+k_+}$ of (103) such that $\sup _{0\le t\le T} \Vert x(t)\Vert$ is sufficiently small, one has

$$\begin{aligned} \Vert x(t)\Vert \le C\bigl ( e^{-\epsilon t} + e^{-\epsilon (T-t)} \bigr )\sup _{0\le t\le T} \Vert x(t)\Vert \end{aligned}$$

and

$$\begin{aligned} \int _0^T \Vert x(t)\Vert \, {\mathrm d}t \le C \sup _{0\le t\le T} \Vert x(t)\Vert . \end{aligned}$$

Proof

Briefly, we use a Gronwall-type argument to establish an exponential upper bound for $\Vert x(t)\Vert$ in the interval [0, T], and then integrate this upper bound to get the claimed estimate.

There is a $\delta > 0$ such that all eigenvalues $\mu$ of $A_+$ and $A_-$ satisfy ${\mathrm {Re}}\mu \ge \delta$. Furthermore, there is a constant $C_A>0$ such that

$$\begin{aligned} \Vert e^{-tA_\pm }\Vert \le C_A e^{-\delta t} \end{aligned}$$

holds for all $t\ge 0$. Applying the variation of constants formula to the system (103), we find that on the interval [0, T], both $x_+$ and $x_-$ are given by

$$\begin{aligned} x_-(t)&= e^{-t A_-}x_-(0) + \int _0^t e^{-(t-s)A_-} B_-(x(s))x(s) \,{\mathrm d}s, \\ x_+(t)&= e^{-(T-t)A_+} x_+(T) - \int _t^T e^{-(s-t)A_+} B_+(x(s))x(s)\, {\mathrm d}s. \end{aligned}$$

Since $B_\pm (0) = 0$, there is a constant $C_B>0$ such that $\Vert B_\pm (x)\Vert \le C_B\Vert x\Vert$ holds for all sufficiently small x. Thus we get

$$\begin{aligned} \Vert x_-(t)\Vert&\le C_A e^{-\delta t}\Vert x_-(0)\Vert + C_AC_B \int _0^t e^{-\delta (t-s)} \Vert x(s)\Vert ^2 \,{\mathrm d}s, \\ \Vert x_+(t)\Vert&\le C_A e^{-\delta (T-t)} \Vert x_+(T) \Vert + C_AC_B \int _t^T e^{-\delta (s-t)} \Vert x(s)\Vert ^2 \,{\mathrm d}s. \end{aligned}$$

For K to be fixed below, we may assume that $\Vert x(s)\Vert \le K$ for all $s\in [0,T]$, whence we get

$$\begin{aligned} \Vert x_-(t)\Vert&\le C_AKe^{-\delta t} + C_AC_BK \int _0^t e^{-\delta (t-s)} \Vert x(s)\Vert \,{\mathrm d}s, \\ \Vert x_+(t)\Vert&\le C_AKe^{-\delta (T-t)} + C_AC_BK \int _t^T e^{-\delta (s-t)} \Vert x(s)\Vert \,{\mathrm d}s. \end{aligned}$$

After adding these two inequalities, we find there exists $C_0=C_0(C_A,C_B)$ such that that for all $t\in [0, T]$, one has

$$\begin{aligned} \Vert x(t)\Vert&\le \Vert x_-(t)\Vert + \Vert x_+(t)\Vert \nonumber \\&\le C_0K \bigl (e^{-\delta t} + e^{-\delta (T-t)}\bigr ) + C_0K \int _0^T e^{-\delta |s-t|} \Vert x(s)\Vert \,{\mathrm d}s. \end{aligned}$$

(104)

If we define

$$\begin{aligned} \rho (t) = \frac{1}{2\delta } \int _0^T e^{-\delta |s-t|} \Vert x(s)\Vert \,{\mathrm d}s, \end{aligned}$$

then since $G(t) = (2\delta )^{-1}e^{-\delta |t|}$ is the Green’s function for $-\frac{{\mathrm d}^2}{{\mathrm d}t^2} + \delta ^2$, the quantity $\rho$ satisfies

$$\begin{aligned} -\rho ''(t) + \delta ^2 \rho (t) = \Vert x(t)\Vert \end{aligned}$$

for all $t\in (0, T)$. We can therefore rewrite the integral inequality (104) as

$$\begin{aligned} -\rho ''(t) + \bigl (\delta ^2 - 2\delta C_0K \bigr )\rho (t) \le C_0 \bigl (e^{-\delta t} + e^{-\delta (T-t)}\bigr ), \qquad (0<t<T). \end{aligned}$$

(105)

Moreover, we have the boundary conditions

$$\begin{aligned} \rho (0) = \frac{1}{2\delta }\int _0^Te^{-\delta t}\Vert x(t)\Vert \,{\mathrm d}t \le \frac{K}{2\delta ^2}(1-e^{-\delta T})\quad \text{ and }\quad \rho (T) \le \frac{K}{2\delta ^2}(1-e^{-\delta T}). \nonumber \\ \end{aligned}$$

(106)

For any constant M, the function ${{\bar{\rho }}}(t) = M\bigl (e^{-\epsilon t}+e^{-\epsilon (T-t)}\bigr )$ satisfies ${{\bar{\rho }}}\,'' = \epsilon ^2{{\bar{\rho }}}$, so that

$$\begin{aligned} -{{\bar{\rho }}}\,''(t) +\bigl (\delta ^2 - 2\delta C_0K\bigr ){{\bar{\rho }}} = \bigl (\delta ^2 - 2\delta C_0K - \epsilon ^2\bigr ) {{\bar{\rho }}}. \end{aligned}$$

If we now choose K small enough that $2C_0K < \frac{1}{2} \delta$ and then choose $\epsilon = \frac{1}{2} \delta$, we have

$$\begin{aligned} \delta ^2 - 2\delta CK - \epsilon ^2 > \tfrac{1}{2} \delta ^2 - \epsilon ^2 = \tfrac{1}{4} \delta ^2. \end{aligned}$$

So the function ${{\bar{\rho }}}$ becomes a supersolution of the boundary-value problem (105, 106) provided that $M=C_1K$, where $C_1$ is a constant that depends on $\delta ,\epsilon$, and $C_0$. By the maximum principle, we conclude that $\rho \le {{\bar{\rho }}}$, and thus that

$$\begin{aligned} \rho (t) \le C_1 K \bigl ( e^{-\epsilon t} + e^{-\epsilon (T-t)} \bigr ). \end{aligned}$$

Applying this to (104) and using the fact that $\epsilon < \delta$, we find

$$\begin{aligned} e^{-\delta t} + e^{-\delta (T-t)} \le e^{-\epsilon t} + e^{-\epsilon (T-t)}. \end{aligned}$$

Since we may assume that $K\le 1$, it follows that

$$\begin{aligned} \Vert x(t)\Vert \le (C_2K + C_2K^2) \bigl ( e^{-\epsilon t} + e^{-\epsilon (T-t)} \bigr ) \le C_3K \bigl ( e^{-\epsilon t} + e^{-\epsilon (T-t)} \bigr ). \end{aligned}$$

We complete the proof by integrating over [0, T], obtaining

$$\begin{aligned} \int _0^T \Vert x(t)\Vert {\mathrm d}t \le \frac{2C_3}{\epsilon } K = C_4 K, \end{aligned}$$

where the constant C only depends on $\delta$, $C_A$, and $C_B$, but not on T. $\square$

Appendix D. Asymptotics of $\chi (s)$ as $s\rightarrow \infty$

Asymptotic expansions for the solutions $\chi$ of (39) are well documented and can be derived in a number of ways. Here we indicate one possible real-variable approach.

If $\chi :{\mathbb R}\rightarrow {\mathbb R}$ is a solution of (39), namely

$$\begin{aligned} \chi _{ss} + \left( \frac{n}{s}+\lambda s\right) \chi _s + \frac{2(n-1)}{s^2} \chi = 0, \end{aligned}$$

then the function

$$\begin{aligned} Z(s) = \frac{\chi _s(s)}{s\chi (s)} \end{aligned}$$

satisfies

$$\begin{aligned} \mathcal G(s,Z) {\mathop {=}\limits ^{{\mathrm{def}}}} \frac{1}{s}\frac{{\mathrm d}Z}{{\mathrm d}s} = -\frac{2(n-1)}{s^4} - \frac{n+1}{s^2}Z - \lambda Z - Z^2. \end{aligned}$$

(107)

As $s\rightarrow \infty$, this equation becomes

$$\begin{aligned} \frac{{\mathrm d}Z}{{\mathrm d}(s^2/2)} = - Z(Z+\lambda ) + {{\mathcal {O}}}(s^{-2}), \end{aligned}$$

which has two constant (approximate) solutions, $Z=0$ and $Z=-\lambda$.

Direct substitution reveals that¹⁰$Z_4(s) {\mathop {=}\limits ^{{\mathrm{def}}}} 2(n-1)/s^4$ satisfies

$$\begin{aligned} \frac{1}{s}\frac{{\mathrm d}Z_4}{{\mathrm d}s} - \mathcal G(s, Z_4(s)) = {{\mathcal {O}}}(s^{-6}), \qquad (s\rightarrow \infty ), \end{aligned}$$

while

$$\begin{aligned} Z_4^{\pm }(s) {\mathop {=}\limits ^{{\mathrm{def}}}} \frac{2(n-1)\pm 1}{s^4} \end{aligned}$$

satisfies

$$\begin{aligned} \frac{1}{s}\frac{{\mathrm d}Z_4^\pm }{{\mathrm d}s} - \mathcal G(s, Z_4^\pm (s)) = \pm \frac{\lambda }{s^4} + {{\mathcal {O}}}(s^{-6}), \qquad (s\rightarrow \infty ). \end{aligned}$$

If $\lambda >0$, this implies that $Z_4^-(s) < Z_4^+(s)$ are lower and upper barriers for the ode (107), and therefore that there is a solution Z(s) with $Z_4^-(s) \le Z(s) \le Z_4^+(s)$ for large s. In fact, if $s_0\gg 1$, then any solution Z(s) of (107) that satisfies $Z_4^-(s_0) \le Z(s_0) \le Z_4^+(s_0)$ will continue to satisfy $Z_4^-(s) \le Z(s) \le Z_4^+(s)$ for all $s\ge s_0$.

If $\lambda <0$, then $Z_4^-(s)$ is an upper barrier, and $Z_4^+(s)$ is a lower barrier. Since $Z_4^-(s) < Z_4^+(s)$ for all $s\ge s_0$ if $s_0$ is large enough, we can apply a Ważewski argument and conclude that there exists at least one $Z^*\in \big (Z_4^-(s_0) , Z_4^+(s_0)\big )$ such that the solution of (107) with $Z(s_0) = Z^*$ satisfies $Z_4^-(s) \le Z(s) \le Z_4^+(s)$ for all $s\ge s_0$.

In either case, the conclusion is that there exists a solution Z(s) of (107) with

$$\begin{aligned} Z(s) = \bigl (2(n-1)+{{\mathcal {O}}}(1)\bigr ) s^{-4}, \qquad (s\rightarrow \infty ). \end{aligned}$$

By repeating this argument, one finds that for any $m\in {\mathbb N}$, there exists a solution that satisfies the expansion

$$\begin{aligned} Z(s) = \frac{A_4}{s^4} + \frac{A_6}{s^6} + \frac{A_8}{s^8} + \cdots + \frac{A_{2m}}{s^{2m}} + {{\mathcal {O}}}\bigl (s^{-2m-2}\bigr ), \qquad (s\rightarrow \infty ). \end{aligned}$$

The coefficients $A_{2j}$ can be computed inductively by substituting the formal expansion; one finds for example that $A_4 = 2\lambda (n-1)$.

Integration then shows that $\chi$ satisfies

$$\begin{aligned} \chi (s) = e^ {\int Z(s) s{\mathrm d}s} = e^{C - \lambda \frac{n-1}{s^2} + {{\mathcal {O}}}(s^{-4})} = e^C \Bigl \{1 - \lambda \frac{n-1}{s^2} + {{\mathcal {O}}}(s^{-4})\Bigr \}, \qquad (s\rightarrow \infty ). \end{aligned}$$

As we noted above, there exists another solution ${{\tilde{Z}}}$ such that ${{\tilde{Z}}}(s)= -\lambda + o(1)$ for large s. Similar reasoning then leads to an expansion of the form

$$\begin{aligned} {{\tilde{Z}}}(s) = -\lambda - \frac{n+1}{s^2} + \frac{B_4}{s^4} + \cdots , \end{aligned}$$

which after integration leads to

$$\begin{aligned} \chi (s) = e^{-\lambda s^2/2} s^{-n-1} \bigl \{1 + {{\mathcal {O}}}(s^{-2})\bigr \}, \qquad (s\rightarrow \infty ). \end{aligned}$$