Deformation of scalar curvature and volume

Abstract

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equal to a given constant. In this paper, we localize a condition satisfied by such stationary points to smooth bounded domains. The condition involves a generalization of the static equations, and we interpret solutions (and their boundary values) of this equation variationally. On domains carrying a metric that does not satisfy the condition, we establish a local deformation theorem that allows one to achieve simultaneously small prescribed changes of the scalar curvature and of the volume by a compactly supported variation of the metric. We apply this result to obtain a localized gluing theorem for constant scalar curvature metrics in which the total volume is preserved. Finally, we note that starting from a counterexample of Min-Oo’s conjecture such as that of Brendle–Marques–Neves, counterexamples of arbitrarily large volume and different topological types can be constructed.

Appendices

Appendix A. Schauder theory

Here we discuss interior Schauder estimates in weighted spaces, following Chruściel and Delay [4, Appendix B], for the particular example of the operator $u \mapsto P(u):= \rho ^{-1}L_g\rho L_g^* u$. Note that, in local coordinates, $P(u)$ has the form

$$\begin{aligned} (n-1)\Delta _g^2 u + \sum \limits _{|\beta |\le 3} b_{\beta } D^{\beta } u. \end{aligned}$$

Recall that the weight $\rho$ is a smooth function that behaves like $e^{- 1/d}$ near $\partial \Omega$. It is easy to check that $||b_{\beta }||_{C^{0, \alpha }_{\phi , \phi ^{4 - |\beta |} }} < \infty$. By appropriate scaling, one can obtain interior Schauder estimates on small balls near the boundary of $\Omega$ from interior Schauder estimates on balls of a fixed size for an operator whose coefficients are well controlled. The weighted Hölder norms defined in Sect. 3.1 are designed for this purpose.

For simplicity, we assume that we are working in $\mathbb R ^n$ with the standard metric, and that $x$ is close to $\partial \Omega$ so that $\phi (x)=d(x)^2$. For $z\in B(0,1)$, let $y= x+\phi (x) z$, and for any $f$, let $\tilde{f}(z)= f(x+\phi (x)z)=f(y)$. Then $D_z\tilde{u}|_z = \phi (x)D_y u|_{x+\phi (x) z}$. We compute that

$$\begin{aligned} (Pu) (x+\phi (x)z)&= (n-1)\Delta ^2_y u \big |_{x+\phi (x) z}+ \sum \limits _{|\beta |\le 3} b_{\beta } D_y^{\beta } u\big |_{x+\phi (x) z}\\&= \phi (x)^{-4} (n-1)\Delta ^2_z \tilde{u}\big |_z+ \sum \limits _{|\beta |\le 3} \phi (x)^{-|\beta |} \tilde{b}_{\beta }(z) D^{\beta }_z \tilde{u} \big |_z. \end{aligned}$$

We obtain that

$$\begin{aligned} \phi (x)^4 \widetilde{Pu}(z)= \left( (n-1) \Delta ^2_z+ \sum \limits _{|\beta |\le 3} \phi (x)^{4-|\beta |} \tilde{b}_{\beta }(z) D_z^{\beta }\right) \tilde{u} =: \tilde{P} \tilde{u} (z). \end{aligned}$$

We see that $\tilde{P}$ is uniformly elliptic on $B(0,1)$ and has coefficients that are bounded in $C^{0, \alpha }$ by bounds for $\Vert b_{\beta }\Vert _{C^{0, \alpha }_{\phi , \phi ^{4-|\beta |}}}$. The standard interior Schauder estimate gives

$$\begin{aligned} \Vert \tilde{u}\Vert _{C^{4, \alpha }(B(0, \frac{1}{4}))}&\le C \left(\Vert \tilde{P} \tilde{u} \Vert _{C^{0,\alpha }(B(0, \frac{1}{2}))} + \Vert \tilde{u} \Vert _{L^2(B(0, \frac{1}{2}))}\right) \\&\le C \left( \phi (x)^4 \Vert \widetilde{Pu} \Vert _{C^{0,\alpha }(B(0, \frac{1}{2}))} + \Vert \tilde{u} \Vert _{L^2(B(0, \frac{1}{2}))}\right). \end{aligned}$$

Scaling back, we see that

$$\begin{aligned}&\sum \limits _{j=0}^4 \phi (x)^j \Vert \nabla ^j_g u\Vert _{C^{0, \alpha }(B(x, \frac{\phi (x)}{4}))} + \phi (x)^{4+\alpha } [\nabla _g^4 u]_{0,\alpha ;B(x, \frac{\phi (x)}{4})} \\&\quad \le C \left( \phi (x)^4 \Vert Pu \Vert _{C^0(B(x, \frac{\phi (x)}{2}))}+\phi (x)^{4+\alpha }[Pu]_{0,\alpha ;B(x, \frac{\phi (x)}{2})} +\phi (x)^{-\frac{n}{2}} \Vert u \Vert _{L^2(B(x, \frac{\phi (x)}{2}))}\right)\!. \end{aligned}$$

We can multiply this inequality by $\varphi (x)$ where $\varphi =\phi ^r \rho ^s$ to obtain the following weighted estimate on $\Omega$:

$$\begin{aligned} \Vert u\Vert _{C^{4, \alpha }_{\phi , \varphi }}\le C (\Vert Pu\Vert _{C^{0,\alpha }_{\phi , \phi ^4 \varphi }}+ \Vert u\Vert _{L^2_{\phi ^{-n}\varphi ^2}}). \end{aligned}$$

(7.1)

This estimate is similar to that in [4, Appendix B]. Note that we impose slightly different conditions on the lower order coefficients here, and that we use a different convention for the weighted $L^2$ norms. As for higher regularity, we obtain similarly that

$$\begin{aligned} \Vert u\Vert _{C^{k+4, \alpha }_{\phi , \varphi }}\le C (\Vert Pu\Vert _{C^{k,\alpha }_{\phi , \phi ^4 \varphi }}+ \Vert u\Vert _{L^2_{\phi ^{-n}\varphi ^2}}) \end{aligned}$$

(7.2)

where the constant $C$ depends on the domain, the weight, and bounds for $\Vert b_{\beta }\Vert _{C^{k, \alpha }_{\phi , \phi ^{4-|\beta |}}}$.

Appendix B. Proof of Proposition 4.6

Here we sketch the proof of Proposition 4.6, which is similar to that of [13, Proposition 6] and [14, Proposition 3.6].

Recall that $[-\frac{T}{2}, \frac{T}{2}]\times \mathbb S ^{n-1} \cong C_T\subset \Sigma _T$. In the proof below, Hölder norms on $C_T$ or $Q:= [-1,1]\times \mathbb S ^{n-1} \subset C_T$ are indicated with an additional subscript.

Proof

We recall that on $\Sigma _T{\setminus } C_T$, $\mathcal{N }_T(\Psi _T)=0$, and that on $[-\frac{T}{2}+1,\frac{T}{2}-1]\times \mathbb S ^{n-1}$, $\Psi _T(s, \theta )=2 R^{\frac{n-2}{2}} e^{-\frac{(n-2)T}{4}} \cosh \big ( \tfrac{(n-2)s}{2}\big ).$ Let $\mathring{\gamma }= ds^2 + g_\mathbb{S ^{n-1}}$ be the standard cylindrical metric. Then $\Psi _T$ solves the equation $(-\Delta _{\mathring{\gamma }} + c_n R(\mathring{\gamma }))( \Psi _T) =0$ on $[-\frac{T}{2}+1,\frac{T}{2}-1]\times \mathbb S ^{n-1}$. Therefore, by (4.1) and Lemma 4.3, we have

$$\begin{aligned} \begin{aligned} \Vert \Delta _{\gamma _T}f-\Delta _{\mathring{\gamma }}f\Vert _{k-2, \alpha ,C_T}&\lesssim e^{-T} \cosh 2s \Vert f\Vert _{k, \alpha , C_T} \\ \Vert R(\gamma _T)-R(\mathring{\gamma })\Vert _{k-2, \alpha , C_T}&\lesssim e^{-T} \cosh 2s . \end{aligned} \end{aligned}$$

(8.1)

On $Q\cong [-1,1]\times \mathbb S ^{n-1}$, (8.1) implies $\Vert \mathcal{N }_T(\Psi _T)\Vert _{k-2, \alpha , Q}\lesssim e^{-\frac{(n+2)T}{4}}.$ This completes the estimate on $Q$.

We now consider $C_T{\setminus } Q \cong ([-\frac{T}{2}, -1]\times \mathbb S ^{n-1})\cup ([1, \frac{T}{2}]\times \mathbb S ^{n-1})$. The estimates on the two components are similar. We will do one of them.

Recall that on $[1, \frac{T}{2}]\times \mathbb S ^{n-1}$ we have that $\gamma _T= \Psi _2^{-{4/(n-2)}} \gamma _2$ and $\Psi _T(s, \theta )=\chi _{1,T}(s)\psi (R e^{-s- \frac{T}{2}})+ \chi _{2,T}(s) \psi ( R e^{s - \frac{T}{2}})$. Moreover, $\mathcal{N }_T(\Psi _2)=0$ in this region, so that

$$\begin{aligned} \mathcal{N }_T(\Psi _T)= (-\Delta _{\gamma _T} +c_n R(\gamma _T))(\chi _{1,T} \Psi _1) -c_n \sigma _n \Psi _T^{\frac{n+2}{n-2}} +c_n\sigma _n \Psi _2^{\frac{n+2}{n-2}}. \end{aligned}$$

We write the last two terms using $\Psi _T^{\frac{n+2}{n-2}}- \Psi _2^{\frac{n+2}{n-2}}= \Psi _2^{\frac{n+2}{n-2}} \left( \Big ( 1+ \chi _{1,T} \frac{ \Psi _1}{\Psi _2}\Big )^{\frac{n+2}{n-2}} - 1\right).$ Since also $\frac{\Psi _1}{\Psi _2}= e^{-s(n-2)}$ and $\Psi _2^{\frac{n+2}{n-2}}= (Re^{s-\frac{T}{2}})^{\frac{n+2}{2}}$ in this region, we obtain that

$$\begin{aligned} \left| \Psi _T^{\frac{n+2}{n-2}}- \Psi _2^{\frac{n+2}{n-2}} \right|\lesssim \Psi _2^{\frac{n+2}{n-2}}\cdot \frac{\Psi _1}{\Psi _2}&\lesssim e^{-s\frac{n-6}{2}} e^{-\frac{(n+2)T}{4}}. \end{aligned}$$

(8.2)

On $[1, \frac{T}{2}-1]\times \mathbb S ^{n-1}$, (8.2) shows

$$\begin{aligned} \Big | \Psi _T^{\frac{n+2}{n-2}}&- \Psi _2^{\frac{n+2}{n-2}} \Big | \lesssim {\left\{ \begin{array}{ll} e^{-\frac{(n+2)T}{4}}&\quad \text{ for}\; n > 6 \\ e^{-\frac{(n-2)T}{2}}&\quad \text{ for}\; n \le 6 \end{array}\right.} \end{aligned}$$

while on $[\frac{T}{2}-1, \frac{T}{2}]\times \mathbb S ^{n-1}$, (8.2) gives that

$$\begin{aligned} \left| \Psi _T^{\frac{n+2}{n-2}}- \Psi _2^{\frac{n+2}{n-2}} \right|\lesssim e^{-\frac{(n-2)T}{2}}. \end{aligned}$$

The required Hölder bounds of $\Vert \Psi _T^{\frac{n+2}{n-2}}- \Psi _2^{\frac{n+2}{n-2}} \Vert _{k-2,\alpha ,C_T}$ follow analogously.

It remains to estimate $\Vert (-\Delta _{\gamma _T} + c_n R(\gamma _T))(\chi _{1,T}\Psi _1)\Vert _{k-2,\alpha ,C_T}$. We first estimate on $[1, \frac{T}{2}-1]\times \mathbb S ^{n-1}$, where $\chi _{1,T}=1$. Using this along with (8.1) and the fact that $\Psi _1$ is in the kernel of the conformal Laplacian on the cylinder, we obtain

$$\begin{aligned}&\Big |(-\Delta _{\gamma _T} + c_n R(\gamma _T))(\chi _{1,T}\Psi _1) \Big | \\&\quad = \Big | (-\Delta _{\gamma _T} + c_n R(\gamma _T))(\Psi _1)- (-\Delta _{\mathring{\gamma }} + c_n R(\mathring{\gamma }))(\Psi _1) \Big | \\&\quad \lesssim e^{-T} \cosh (2s)\Vert \Psi _1\Vert _{C^2(C_T)}\\&\quad \lesssim e^{s(2-\frac{n-2}{2})}e^{-T}e^{-\frac{(n-2)T}{4}}\\&\quad \lesssim {\left\{ \begin{array}{ll} e^{-T}e^{-\frac{(n-2)T}{4}} = e^{ - \frac{(n+2)}{4} T }&\quad \text{ for}\; n > 6 \\ e^{(1-\frac{n-2}{4})T} e^{-T}e^{-\frac{(n-2)T}{4}}= e^{-\frac{(n-2)T}{2}}&\quad \text{ for}\; n \le 6 . \end{array}\right.} \end{aligned}$$

For $s\in [\frac{T}{2}-1, \frac{T}{2}]$, using that $\Psi _1= (Re^{-s})^{\frac{n-2}{2}}e^{-\frac{(n-2)T}{4}}$, we have

$$\begin{aligned}&\Big | (-\Delta _{\gamma _T} + c_n R(\gamma _T))(\chi _{1,T}\Psi _1) \Big | \\&\quad \lesssim e^{-T} \cosh (2s) \Vert \Psi _1\Vert _{C^2(C_T)} + \Big | (-\Delta _{\mathring{\gamma }} + c_n R(\mathring{\gamma }))(\chi _{1,T}\Psi _1) \Big | \lesssim e^{-\frac{(n-2)T}{2}}. \end{aligned}$$

This proves the desired $C^0$ bound in (4.3) and (4.4). The estimate of the derivatives and the Hölder bound follow from similar reasoning.