Pointwise lower scalar curvature bounds for $C^0$ metrics via regularizing Ricci flow

Abstract

In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from $C^0$ initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from $C^0$ initial data.

Iteration Scheme for the Ricci–DeTurck Flow

The aim of this section is essentially to show that closeness of two metrics is stable under the Ricci–DeTurck flow, by appealing to the Banach fixed point theorem. We first record the unweighted versions of several results that we have proven in Section 4; see also [KL12, Lemmata 2.2, 4.1, and 4.2].

Lemma A.1

We have, for every $0<\gamma < 1$ and every two $h', h''\in X^{\gamma }$,

$$\begin{aligned} ||Q^0[h'] - Q^0[h''] + \nabla ^*Q^1[h']- \nabla ^*Q^1[h'']||_{Y} \le c_1(||h'||_X + ||h''||_X)||h' - h''||_X, \end{aligned}$$

(A.1)

where $c_1 = c_1(n,\gamma , T\sup _{t\in [0,T]}|{\text {Rm}}|(\bar{g}_t))$. Moreover, if $(\partial _t + L)h\equiv 0$ with initial condition $h_0\in L^\infty (M)$, then

$$\begin{aligned} ||h||_X \le c_2||h_0||_{L^\infty (M)}, \end{aligned}$$

(A.2)

where $c_2$ is the constant from Lemma 4.5. Finally, if $(\partial _t + L)h = Q\in Y$ and $h_0 \equiv 0$, then

$$\begin{aligned} ||h||_X \le c_3||Q||_Y, \end{aligned}$$

(A.3)

where $c_3$ is the constant from Lemma 4.6.

Proof

The proofs of (A.2) and (A.3) follow similarly to the proofs of Lemmata 4.5 and 4.6, by omitting the weights. The proof of (A.1) is similar to the analysis of Terms I and II in the proof of Theorem 4.1 but we shall give more details here. The analysis is simplified by the fact that $h'$ and $h''$ are solutions with respect to the same background metric. First observe that if $||h'||_X, ||h''||_X< \gamma < 1$, then (2.14) and (2.16) imply

$$\begin{aligned}&|(\bar{g} + h')^{-1}\star (\bar{g} + h')^{-1} \\&\qquad - (\bar{g} + h'')^{-1}\star (\bar{g} + h'')^{-1}| \le |(\bar{g} + h')^{-1}\star (\bar{g} + h')^{-1} - (\bar{g} + h')^{-1}\star (\bar{g} + h'')^{-1}|\\&\qquad + |(\bar{g} + h')^{-1}\star (\bar{g} + h'')^{-1} - (\bar{g} + h'')^{-1}\star (\bar{g} + h'')^{-1}|\\&\quad \le |(\bar{g} + h')^{-1}||(\bar{g} + h')^{-1} - (\bar{g} + h'')^{-1}| + |(\bar{g} + h')^{-1} - (\bar{g} + h'')^{-1}||(\bar{g} + h'')^{-1}|\\&\quad \le |(\bar{g} + h')^{-1}|^2 |(\bar{g} + h'')^{-1}| |h' - h''| + |(\bar{g} + h'')^{-1}|^2 |(\bar{g} + h')^{-1}| |h' - h''|\\&\quad \le c(n,\gamma )|h' - h''|. \end{aligned}$$

Thus we have

$$\begin{aligned}&|Q^0[h'] - Q^0[h'']|\\&\quad \le |(\bar{g} + h')^{-1}\star (\bar{g} + h')^{-1} \star \nabla h' \star \nabla h' - (\bar{g} + h'')^{-1}\star (\bar{g} + h'')^{-1} \star \nabla h'' \star \nabla h''|\\&\qquad + |[(\bar{g} + h')^{-1} - \bar{g}]\star {\text {Rm}}^{\bar{g}}\star h' \star h' - [(\bar{g} + h'')^{-1} - \bar{g}]\star {\text {Rm}}^{\bar{g}}\star h'' \star h''|\\&\quad \le |(\bar{g} + h')^{-1}\star (\bar{g} + h')^{-1} \star \nabla h' \star \nabla h' - (\bar{g} + h')^{-1}\star (\bar{g} + h')^{-1} \star \nabla h' \star \nabla h''|\\&\qquad + |(\bar{g} + h')^{-1}*(\bar{g} + h')^{-1} \star \nabla h' \star \nabla h'' - (\bar{g} + h'')^{-1}\star (\bar{g} + h'')^{-1} \star \nabla h' \star \nabla h''|\\&\qquad + |(\bar{g} + h'')^{-1}\star (\bar{g} + h'')^{-1} \star \nabla h' \star \nabla h'' - (\bar{g} + h'')^{-1}\star (\bar{g} + h'')^{-1} \star \nabla h'' \star \nabla h''|\\&\qquad + |[(\bar{g} + h')^{-1} - \bar{g}]\star {\text {Rm}}^{\bar{g}}\star h' \star h' - [(\bar{g} + h')^{-1} - \bar{g}]\star {\text {Rm}}^{\bar{g}}\star h' \star h''|\\&\qquad +|[(\bar{g} + h')^{-1} - \bar{g}]\star {\text {Rm}}^{\bar{g}}\star h' \star h'' - [(\bar{g} + h'')^{-1} - \bar{g}]\star {\text {Rm}}^{\bar{g}}\star h' \star h''|\\&\qquad + |[(\bar{g} + h'')^{-1} - \bar{g}]\star {\text {Rm}}^{\bar{g}}\star h' \star h'' - [(\bar{g} + h'')^{-1} - \bar{g}]\star {\text {Rm}}^{\bar{g}}\star h'' \star h''|\\&\quad \le c(n, \gamma )|\nabla h'||\nabla (h' - h'')| + c(n)|h' - h''||\nabla h'||\nabla h''| + c(n,\gamma )|\nabla (h' - h'')||\nabla h''|\\&\qquad + c(n,\gamma )|{\text {Rm}}^{\bar{g}}||h'||h' - h''| + c(n)|h' - h''||{\text {Rm}}^{\bar{g}}||h'||h''| \\&\qquad + c(n,\gamma )|{\text {Rm}}^{\bar{g}}||h'-h''||h''|, \end{aligned}$$

appealing to (2.14). We also have

$$\begin{aligned} |Q^1[h'] - Q^1[h'']|&\le |[(\bar{g} + h')^{-1} - \bar{g}^{-1}]\star h' \star \nabla h' - [(\bar{g} + h'')^{-1} - \bar{g}^{-1}]\star h'' \star \nabla h''|\\&\le |[(\bar{g} + h')^{-1} - \bar{g}^{-1}]\star h' \star \nabla h' - [(\bar{g} + h')^{-1} - \bar{g}^{-1}]\star h' \star \nabla h''|\\&\quad + |[(\bar{g} + h')^{-1} - \bar{g}^{-1}]\star h' \star \nabla h'' - [(\bar{g} + h')^{-1} - \bar{g}^{-1}]\star h'' \star \nabla h''|\\&\quad + |[(\bar{g} + h')^{-1} - \bar{g}^{-1}]\star h'' \star \nabla h'' - [(\bar{g} + h'')^{-1} - \bar{g}^{-1}]\star h'' \star \nabla h''|\\&\le c(n,\gamma )|h'||\nabla (h' - h'')| + c(n,\gamma )|h' - h''||\nabla h''| \\&\quad + c(n,\gamma )|h''||\nabla h''||h' - h''|. \end{aligned}$$

Then the estimate (A.1) follows from the definitions of the X and Y norms, much as in the proof of Lemma 4.4. $\square$

Lemma A.2

Suppose $\bar{g}(t)$ is a smooth Ricci flow background defined for $t\in [0,T]$. Suppose $g_0', g_0''\in L^{\infty }(M)$ are two initial metrics such that $||g_0' - \bar{g}(0)||_{L^\infty (M)}, ||g_0''-\bar{g}(0)||_{L^\infty (M)} < \varepsilon$, where $\varepsilon$ is given by Lemma 3.3 and reduced, if necessary, so that $2c_1c_3C\varepsilon < 1$, where C is the constant from Lemma 3.3 and $c_1$ and $c_3$ are as in (A.1) and (A.3) respectively. Let $h'(t)$ and $h''(t)$ be solutions to the integral equation (3.1) in $X^\gamma$ for some $0<\gamma < 1$, given by Lemma 3.3, starting from $g_0'$ and $g_0''$ respectively. Then

$$\begin{aligned} ||h' - h''||_{X} \le c||g_0' - g_0''||_{L^\infty (M)}, \end{aligned}$$

(A.4)

where $c = c(n, \gamma , T\sup _{[0,T]}|{\text {Rm}}|(\bar{g}_t))$.

Proof

Observe that $||h'||_{X}, ||h''||_{X} \le C\varepsilon$, by Lemma 3.3. Moreover, the proof of Lemma 3.3 implies that $F[\cdot , h_0']$ and $F[\cdot , h_0'']$ are contraction mappings $X^{C\varepsilon }\rightarrow X^{C\varepsilon }$. For $i\in \mathbb {N}$ let $h_i' = F^i[0, h_0']$, i.e. $F[\cdot , h_0']$ applied to the 0-tensor i times, and let $h_i'' = F^i[0,h_0'']$, so that $h_i'\rightarrow h'$ in X as $i\rightarrow \infty$, and similarly for $h''$, by the Banach fixed point theorem. We show

$$\begin{aligned} ||h_i' - h_i''||_{X} \le c_2\sum _{k=0}^{i-1}(2c_1c_3C\varepsilon )^{k}||h_0' - h_0''||_{L^\infty (M)}. \end{aligned}$$

(A.5)

To prove (A.5) we induct. We have

$$\begin{aligned} ||h_1' - h_1''||_{X} = \left| \left| \int _M\bar{K}(x,y) (h_0' - h_0'')(y)dy\right| \right| _{X} \le c_2||h_0' - h_0''||_{L^\infty (M)}, \end{aligned}$$

by (A.2). Moreover, supposing (A.5) holds for $i-1$, we appeal to (A.3), (A.2), and (A.1) to find

$$\begin{aligned} ||h_i' - h_i''||_X&\le ||F[h_{i-1}', h_0'] - F[h_{i-1}'', h_0']||_X + ||F[h_{i-1}'', h_0'] - F[h_{i-1}'', h_0'']||_X\\&= \left| \left| \int _0^t\int _M\bar{K}_{t-s}(x,y)(Q[h_{i-1}'] - Q[h_{i-1}''])(y,s)dyds\right| \right| _X \\&\quad + \left| \left| \int _M\bar{K}(x,y)(h_0' - h_0'')(y)dy\right| \right| _X\\&\le c_3||Q[h_{i-1}'] - Q[h_{i-1}'']||_Y + c_2||h_0' - h_0''||_{L^\infty (M)}\\&\le c_1c_3(||h_{i-1}'||_{X} + ||h_{i-1}''||_{X})||h_{i-1}' - h_{i-1}''||_X + c_2||h_0' - h_0''||_{L^\infty (M)}\\&\le 2c_1c_3C\varepsilon c_2\sum _{k=0}^{i-2}(2c_1c_3C\varepsilon )^{k}||h_0' - h_0''||_{L^\infty (M)} + c_2||h_0' - h_0''||_{L^\infty (M)} . \end{aligned}$$

Taking limits, we obtain (A.4), with $c = c_2\sum _{k=0}^{\infty }(2c_1c_3C\varepsilon )^k$. $\square$