On the P-Scalar Curvature

Abstract

We study the P-scalar curvature operator used by Perelman in the context of manifolds with density. We verify that the Gromov–Lawson surgery for positive scalar curvature extends naturally to positive P-scalar curvature. By studying the first variation of the P-scalar curvature operator, we obtain local perturbation results for the P-scalar curvature. As a corollary, we construct small perturbations (in the category of manifolds with density) of the round unit hemisphere $\mathbb {S}^n_+$ which are compactly supported away from the boundary and yield manifolds with density with non-constant P-scalar curvature at least $n(n-1)$.

Appendix

1.1 Product Formulas for the Gradient and Laplacian

Let D be the open disk $B_{r_0}(p)$ or the open tube $\mathcal {N}_{r_0}(\varSigma )$, and let $M\subset D\times \mathbb {R}$ be the submanifold of the product defined using the curve $\gamma$, as in Sect. 3. Let g be the metric on D, while $\tilde{g}$ is the metric induced by the product metric $\bar{g}=g+dt^2$ on M, and h is the metric induced (from g or $\tilde{g}$) on a slice $M\cap (D\times \{t\})$, which is diffeomorphic to $\mathbb {S}^{n-1}$ or $\varSigma \times \mathbb {S}^{q-1}$, respectively. We can view h as an r-dependent metric, or a t-dependent metric, via the relation $(r, t)\in \gamma$. We can write $\tilde{g}= h + ds^2$, as the s-direction along the appropriate $\gamma _\beta$ (see Sect. 3) is tangent to M; similarly, the metric on D can be written as $g=h+dr^2$, where $(r, t)\in \gamma$. For $\tilde{f}$ a smooth function on $D\times \mathbb {R}$, we can restrict $\tilde{f}$ to M (and compute with respect to $\tilde{g}$), or we can restrict to a slice $D\times \{ t\}$ (and compute with respect to g); we do not introduce new notation for these two restrictions of $\tilde{f}$.

We show details for the computation of $\varDelta _{\tilde{g}} \tilde{f}$ and $|\tilde{\nabla } \tilde{f}|_{\tilde{g}}^2$ on $(M,\tilde{g})$. We first compute the following Christoffel symbols:

$$\begin{aligned} \tilde{\varGamma }_{ss}^\ell= & {} \frac{1}{2} \tilde{g}^{\ell m} \left( \tilde{g}_{sm,s} + \tilde{g}_{sm,s} - \tilde{g}_{ss,m}\right) = 0\\ \tilde{\varGamma }_{ij}^s= & {} \frac{1}{2} \tilde{g}^{sm} \left( \tilde{g}_{im,j} + \tilde{g}_{jm,i} - \tilde{g}_{ij,m}\right) = -\frac{1}{2} \tilde{g}^{ss} \tilde{g}_{ij,s} = -\frac{1}{2} h_{ij,s}, \end{aligned}$$

and similarly $\tilde{\varGamma }^s_{sj}=0$.

We note $\varDelta _{\tilde{g}} \tilde{f} = \tilde{g}^{ij}\left( \tilde{f}_{,ij} - \tilde{\varGamma }_{ij}^k \tilde{f}_{,k}\right) = \varDelta _{h} \tilde{f} - h^{ij} \tilde{\varGamma }_{ij}^s \tilde{f}_{,s} + \tilde{f}_{,ss} - \tilde{\varGamma }_{ss}^k \tilde{f}_k$, so that

$$\begin{aligned} \varDelta _{\tilde{g}} \tilde{f} = \varDelta _{h} \tilde{f} + \frac{1}{2} h^{ij} h_{ij,s} \tilde{f}_{,s} + \tilde{f}_{,ss}. \end{aligned}$$

(27)

Thus we also have

$$\begin{aligned} \varDelta _{g} \tilde{f}= & {} \varDelta _{h} \tilde{f} + \tilde{f}_{,rr} + \frac{1}{2} h^{ij} h_{ij,r} \tilde{f}_{,r} . \end{aligned}$$

(28)

Recall that $\phi$ is the angle between the unit normal $\nu$ to M and $\frac{\partial }{\partial t}$, so that $\frac{\partial }{\partial s} = -\cos \phi \frac{\partial }{\partial r} + \sin \phi \frac{\partial }{\partial t}$. Using (27) and (28), we have

$$\begin{aligned} \varDelta _{\tilde{g}} \tilde{f}= & {} \varDelta _{g} \tilde{f} - f_{,rr} - \frac{1}{2} h^{ij} h_{ij,r} \tilde{f}_{,r} + \frac{1}{2} h^{ij} h_{ij,s} \tilde{f}_{,s} + \tilde{f}_{,ss} \nonumber \\= & {} \varDelta _{g} \tilde{f} - \tilde{f}_{,rr} - \frac{1}{2} h^{ij} h_{ij,r} \tilde{f}_{,r} - \frac{1}{2} h^{ij} h_{ij,r} \cos \phi \left( -\cos \phi \tilde{f}_{,r} + \sin \phi \tilde{f}_{,t}\right) \nonumber \\&+\, \left( -\cos \phi \tilde{f}_{,r} + \sin \phi \tilde{f}_{,t}\right) _{,s} \nonumber \\= & {} \varDelta _{g} \tilde{f}- \tilde{f}_{,rr} - \frac{1}{2} h^{ij} h_{ij,r} \tilde{f}_{,r} - \frac{1}{2} h^{ij} h_{ij,r} \cos \phi \left( -\cos \phi \tilde{f}_{,r} + \sin \phi \tilde{f}_{,t}\right) \nonumber \\&+\, \sin \phi \frac{d\phi }{ds} \tilde{f}_{,r} - \cos \phi \left( -\cos \phi \tilde{f}_{,rr} + \sin \phi \tilde{f}_{,rt} \right) + \cos \phi \frac{d\phi }{ds} \tilde{f}_{,t} \nonumber \\&+\, \sin \phi \left( -\cos \phi \tilde{f}_{,tr} + \sin \phi \tilde{f}_{,tt}\right) \nonumber \\= & {} \varDelta _{ g} \tilde{f} + \sin ^2(\phi ) \left( \tilde{f}_{,tt} - \tilde{f}_{,rr}\right) + \tilde{f}_{,t}\left( -\frac{1}{2} h^{ij} h_{ij,r} \cos \phi \sin \phi +\, \cos \phi \frac{d\phi }{ds}\right) \nonumber \\&+\, \tilde{f}_{,r}\left( -\frac{1}{2} \sin ^2( \phi ) h^{ij} h_{ij,r}+ \sin \phi \frac{d\phi }{ds}\right) - 2\sin \phi \cos \phi \tilde{f}_{,tr}. \end{aligned}$$

(29)

We now compute $|\tilde{\nabla } \tilde{f}|_{\tilde{g}}^2$. Note $\tilde{\nabla }^\ell \tilde{f} = \tilde{g}^{\ell k} \tilde{f}_{,k}= h^{\ell k} \tilde{f}_{,k}+ \delta ^{\ell }_s \tilde{f}_{,s}$, while ${\nabla }^\ell \tilde{f} = {g}^{\ell k} \tilde{f}_k = h^{\ell k} \tilde{f}_k + \delta ^{\ell }_r \tilde{f}_{,r}$. Let Y be the vector field tangent to the level sets of r given in components by $Y^\ell = h^{\ell k} \tilde{f}_{,k}$ . Then $\tilde{\nabla } \tilde{f} = Y + \tilde{f}_{,s} \frac{\partial }{\partial s}$ and $\nabla \tilde{f} = Y + \tilde{f}_{,r}\frac{\partial }{\partial r}$ are $\tilde{g}$-orthogonal and g-orthogonal decompositions, respectively. Therefore,

$$\begin{aligned} |\tilde{\nabla } \tilde{f}|_{\tilde{g}}^2= & {} |Y|_{\tilde{g}}^2+ (\tilde{f}_{,s})^2= |Y|_h^2 + (\tilde{f}_{,s})^2= |{\nabla } \tilde{f}|_{ g}^2+(\tilde{f}_{,s})^2-(\tilde{f}_{,r})^2 \nonumber \\= & {} |{\nabla } \tilde{f}|_{ g}^2 + 2 \sin \phi \cos \phi \tilde{f}_{,r} \tilde{f}_{,t} + \sin ^2( \phi ) \left[ (\tilde{f}_{,t})^2 - (\tilde{f}_{,r})^2\right] . \end{aligned}$$

(30)

1.2 Metric Estimates for Tubes

Lemma 4

Suppose $\varSigma \subset (X,g)$ is a closed submanifold of codimension $q\ge 3$, with normal bundle $N(\varSigma )$. Take $r_0>0$ so that for $0<r<r_0$, there is a regular tubular neighborhood $\mathcal {N}_r(\varSigma )= \{ \exp _p(v): p\in \varSigma ,\; v\in N_p\varSigma ,\; |v|_g< r\}$ about $\varSigma$ in M. Under a local trivialization of the normal bundle over $U\subset \varSigma$ using smooth orthonormal sections $\nu _1, \ldots , \nu _q$ of N(U), we have a diffeomorphism $\varPsi : \mathcal {N}_{r_0}(U) \rightarrow U \times \mathbb {D}^q(r_0)$. Using a covering of $\varSigma$ by such open sets U, the pullback metrics $\varPsi ^*(g|_{T\varSigma } \oplus g_{\mathrm {eucl}})$ patch together to give a metric $\mathring{g}$ on $\mathcal {N}_{r_0}(\varSigma )$. For $\epsilon >0$ sufficiently small, there is a smooth homotopy through metrics with positive scalar curvature on $\partial \mathcal {N}_{\epsilon }( \varSigma )$ from the metric $h_{\epsilon }$ induced from g to the metric $\mathring{h}_{\epsilon }$ induced from $\mathring{g}$.

Proof

We begin by setting up notation, and record the expansion of the metric $h_{\epsilon }$ on $\partial \mathcal {N}_{\epsilon }(\varSigma )$. We first choose adapted coordinates $(F=\varPsi ^{-1},(x^1, \ldots x^q, y^1, \ldots , y^\ell ))$ on $\mathcal {N}_{r_0}(U)$, where $(f,y=(y^1, \ldots , y^\ell ))$ are coordinates for an open set $U\subset \varSigma$, say $f(y)\in U$, and then $F(x,y)= \exp _{f(y)}(x^i \nu _i)$, and $\sum _{i=1}^q (x^i)^2 <r_0$. For instance, we could let f be given by normal coordinates at $p\in U$, which we assume here. If we let $z\mapsto \varTheta (z)\in \mathbb {S}^{q-1}\subset \mathbb {R}^q$ be a local parameterization of the (unit) sphere, then we can obtain local coordinates on $\partial \mathcal {N}_r(\varSigma )$ by $G_r(y,z)= F(r\varTheta (z), y)$.

We want to expand the metric $h_{\epsilon }$ on $\partial \mathcal {N}_{\epsilon }(\varSigma )$ for small $\epsilon$. Fix $p\in \varSigma$, and let f be normal coordinates on $\varSigma$ centered at $p=f(0)$. The coordinate frame induced by F evaluates at p to $\{E_1, \ldots , E_{n}\}$, where $E_k= F_*\Big |_{(0,0)}\left( \frac{\partial }{\partial x^k}\right) ,\;k\in \{ 1, \ldots , q\}$, and $E_{q+j}= F_*\Big |_{(0,0)}\left( \frac{\partial }{\partial y^j}\right) = f_*\Big |_{0}\left( \frac{\partial }{\partial y^j}\right) ,\;j\in \{ 1, \ldots \ell \}$. Note that $E_k= \nu _k$ at p, for $k=1, \ldots , q$. By use of normal coordinates at p, we see that $(E_1, \ldots , E_{n})$ is an orthonormal frame for $T_p X$. We let $Z_j\Big |_{G_{\epsilon }(y,z)}= (G_{\epsilon })_*\Big |_{(y,z)}\left( \frac{\partial }{\partial z^j}\right)$ for $j\in \{1, \ldots , q-1\}$, and $Y_{a}\Big |_{G_{\epsilon }(y,z)}=( G_{\epsilon })_*\Big |_{(y,z)}\left( \frac{\partial }{\partial y^a}\right)$ for $a\in \{1, \ldots , \ell \}$. In what follows below, we let $i, j \in \{ 1, \ldots , q-1\}$, while $a, b, c, d \in \{ 1, \ldots , \ell \}$.

Then by [19, cf. Proposition 2.1], we have the following at $q=G_{\epsilon }( 0,z)$ where the partial derivatives are evaluated at z:

$$\begin{aligned} h_{\epsilon }(Y_a, Y_b)= & {} g(Y_a, Y_b) =\delta _{ab} + O(\epsilon )\\ \epsilon ^{-1}h_{\epsilon }(Y_a, Z_j)= & {} \epsilon ^{-1} g(Y_a, Z_j)= O(\epsilon ^2)\\ \epsilon ^{-2} h_{\epsilon }(Z_i, Z_j)= & {} \epsilon ^{-2} g(Z_i, Z_j)= \sum _{k=1}^q \frac{\partial \varTheta ^k}{\partial z^i} \frac{\partial \varTheta ^k}{\partial z^j} +O(\epsilon ^2). \end{aligned}$$

These give the components of the metric $h_{\epsilon }$ in the given chart.

We compare this to the metric $\mathring{h}_{\epsilon }$ on $\partial \mathcal {N}_{\epsilon }(\varSigma )$ induced locally by the product metric on $\varSigma \times \mathbb {S}^{q-1}$, where the spherical factor has the round metric $\epsilon ^2 \mathring{\sigma }$ of radius $\epsilon$. The metric $\mathring{h}_{\epsilon }$ is given at $q=G_{\epsilon }(0,z)$ by

$$\begin{aligned} \mathring{h}_{\epsilon }(Y_a, Y_b)= & {} \delta _{ab} \\ \epsilon ^{-1}\mathring{h}_{\epsilon }(Y_a, Z_j)= & {} 0\\ \epsilon ^{-2} \mathring{h}_{\epsilon }(Z_i, Z_j)= & {} \mathring{g}\left( \frac{\partial \varTheta ^k}{\partial z^i}E_k, \frac{\partial \varTheta ^{\ell }}{\partial z^j}E_{\ell }\right) = \sum _{k=1}^q \frac{\partial \varTheta ^k}{\partial z^i} \frac{\partial \varTheta ^k}{\partial z^j} . \end{aligned}$$

This implies that the difference is given by

$$\begin{aligned} h_{\epsilon }-\mathring{h}_{\epsilon }=A_{ab}(y,z) dy^a dy^b + 2B_{cj}(y,z) dy^c dz^j + C_{ij} (y,z) dz^i dz^j, \end{aligned}$$

where $A_{ab}=A_{ba},\;C_{ij}=C_{ji}$, and where for one-forms $\theta$ and $\eta$, we let $\theta \eta = \frac{1}{2} (\theta \otimes \eta + \eta \otimes \theta )$. From the above, we have $A_{ab}=O(\epsilon ),\;B_{cj}= O(\epsilon ^3)$ and $C_{ij}=O(\epsilon ^4)$; partial derivatives in $y^a$ or $z^i$ of these component functions have the same error estimates. (We can cover $\varSigma$ by finitely many such charts, and obtain uniform error estimates.) This implies that the raised metrics (each respectively raised with respect to itself) are related by

$$\begin{aligned} h_{\epsilon }^{\sharp }(y,z)= \mathring{h}_{\epsilon }^{\sharp }(y,z)+ \widehat{A}^{ab}(y,z) \partial _{y^a} \partial _{y^b} + 2\widehat{B}^{cj}(y,z) \partial _{y^c} \partial _{z^j} + \widehat{C}^{ij}(y,z) \partial _{z^i} \partial _{z^j}, \end{aligned}$$

where $\widehat{A}^{ab}=\widehat{A}^{ba},\;\widehat{C}^{ij}=\widehat{C}^{ji}$ and for vectors X and $Y,\;X Y= \frac{1}{2} (X\otimes Y+ Y\otimes X)$. Here we have $\widehat{A}^{ab}=O(\epsilon ),\;\widehat{B}^{cj}=O(\epsilon )$ and $\widehat{C}^{ij}=O(1)$, with the same estimates holding for the $y^a$ and $z^i$ partial derivatives, i.e., for any multi-indices $\alpha$ and $\beta$, there is a constant $C=C_{\alpha \beta }$ so that for all $a,\;b,\;|\partial ^{\alpha }_y \partial ^{\beta }_z \widehat{A}^{ab}|\le C \epsilon$, and analogously for the other components.

We wish to compute the scalar curvature, for which we record the following estimates for the Christoffel symbols $\varGamma ^{\mu }_{\kappa \lambda }$ of $h_{\epsilon }$. For the summation convention, we let $a, b, c, d \in \{ 1, \ldots , \ell \}$ correspond to directions along $\varSigma$, whereas $s, t, u,v \in \{ \ell +1, \ldots , \ell +q-1=n-1\}$ correspond to spherical directions (so $s=j+\ell$ corresponds to $z^j$), and Greek letters denote elements of $\{ 1, \ldots , n-1\}$. We let $\mathring{\varGamma }^{\mu }_{\kappa \lambda }$ be the Christoffel symbols for $\mathring{g}$, so that $\mathring{\varGamma }^{a}_{bc}$ are the Christoffel symbols for the metric h on $\varSigma$ in the coordinates y, whereas $\mathring{\varGamma }^{s}_{tu}$ are the Christoffel symbols of the round metric $\mathring{\sigma }$ in the coordinates z, where we note that the Christoffel symbols are invariant under constant metric rescaling. In the equations below, $\Lambda =O(\epsilon ^m)$ for a function $\Lambda$ on (an open subset of) $\varSigma \times \mathbb {S}^{q-1}\times (0, r_0)$ pulled back from $\mathcal {N}_{r_0} (\varSigma ) \setminus \varSigma$ means there exist $C>0$ so that $|\Lambda |\le C \epsilon ^m$, and for any index a or $j,\;|\partial _{y^a} \Lambda |\le C\epsilon ^m,\;|\partial _{z^j} \Lambda | \le C \epsilon ^m$ (and likewise for higher derivatives).

Lemma 5

For $a, b, c \in \{ 1, \ldots , \ell \}$ and $s, t, u \in \{ \ell +1, \ldots , n-1\}$,

$$\begin{aligned} \varGamma ^{s}_{tu}= \mathring{\varGamma }^{s}_{tu} + O(\epsilon ^2),&\qquad&\varGamma ^a_{bc}=\mathring{\varGamma }^a_{bc}+O(\epsilon )\\ \varGamma ^{a}_{tu}= O(\epsilon ^3),&\varGamma ^a_{tb} = O(\epsilon )\\ \varGamma ^s_{ta}= O(\epsilon ),&\varGamma ^{s}_{bc}= O(\epsilon ^{-1}). \end{aligned}$$

Proof

The proof is a simple calculation, using the above metric estimates. We pay particular attention to terms involving $(\mathring{h}_{\epsilon })^{st}=\epsilon ^{-2} \mathring{\sigma }^{st}$. We have

$$\begin{aligned} \varGamma ^{s}_{tu}= & {} \frac{1}{2} (h_{\epsilon })^{sv} \left( (h_{\epsilon })_{vu,t} + (h_{\epsilon })_{tv,u}-(h_{\epsilon })_{tu,v}\right) + \frac{1}{2} (h_{\epsilon })^{sa}((h_{\epsilon })_{au,t}+(h_{\epsilon })_{ta,u}-(h_{\epsilon })_{tu,a})\\= & {} \mathring{\varGamma }^s_{tu}+ \frac{1}{2} (\mathring{h}_{\epsilon })^{sv} \left( C_{vu,t} + C_{tv,u}-C_{tu,v}\right) +\frac{1}{2} \widehat{C}^{sv} \left( (h_{\epsilon })_{vu,t} + (h_{\epsilon })_{tv,u}-(h_{\epsilon })_{tu,v}\right) \\&+\, \frac{1}{2} \widehat{B}^{sa} \left( (h_{\epsilon })_{au,t}+(h_{\epsilon })_{ta,u}-(h_{\epsilon })_{tu,a}\right) \\= & {} \mathring{\varGamma }^s_{tu} + O(\epsilon ^2). \end{aligned}$$

Similarly, we see

$$\begin{aligned} \varGamma ^{a}_{bc}= & {} \frac{1}{2} (h_{\epsilon })^{as} \left( (h_{\epsilon })_{sc,b} + (h_{\epsilon })_{bs,c}-(h_{\epsilon })_{bc,s}\right) + \frac{1}{2} (h_{\epsilon })^{ad}((h_{\epsilon })_{dc,b}+(h_{\epsilon })_{bd,c}-(h_{\epsilon })_{bc,d})\\= & {} \frac{1}{2} \widehat{B}^{as} \left( (h_{\epsilon })_{sc,b} + (h_{\epsilon })_{bs,c}-(h_{\epsilon })_{bc,s}\right) + \mathring{\varGamma }^a_{bc} + \frac{1}{2} (\mathring{h}_{\epsilon })^{ad}\left( A_{dc,b} + A_{bd,c}-A_{bc,d}\right) \\&+\, \frac{1}{2} \widehat{A}^{ad} \left( (h_{\epsilon })_{dc,b}+(h_{\epsilon })_{bd,c}-(h_{\epsilon })_{bc,d}\right) \\= & {} \mathring{\varGamma }^a_{bc} + O(\epsilon ). \end{aligned}$$

We compute similarly with the other terms:

$$\begin{aligned} \varGamma ^{a}_{tu}= & {} \frac{1}{2} (h_{\epsilon })^{ab} \left( (h_{\epsilon })_{bu,t} + (h_{\epsilon })_{tb,u}-(h_{\epsilon })_{tu,b}\right) + \frac{1}{2} (h_{\epsilon })^{as}\left( (h_{\epsilon })_{su,t}+(h_{\epsilon })_{ts,u}-(h_{\epsilon })_{tu,s}\right) \\= & {} \frac{1}{2} (h_{\epsilon })^{ab} \left( B_{bu,t}+B_{tb,u}-C_{tu,b}\right) + \frac{1}{2} \hat{B}^{as} \left( (h_{\epsilon })_{su,t}+(h_{\epsilon })_{ts,u}-(h_{\epsilon })_{tu,s}\right) \\= & {} O(\epsilon ^3) \\ \varGamma ^{a}_{tb}= & {} \frac{1}{2} (h_{\epsilon })^{ac} \left( (h_{\epsilon })_{cb,t}+(h_{\epsilon })_{tc,b}-(h_{\epsilon })_{tb,c}\right) + \frac{1}{2} (h_{\epsilon })^{as} \left( (h_{\epsilon })_{sb,t}+(h_{\epsilon })_{ts,b}-(h_{\epsilon })_{tb,s} \right) \\= & {} \frac{1}{2} (h_{\epsilon })^{ac} \left( A_{cb,t}+B_{tc,b}-B_{tb,c}\right) + \frac{1}{2} \widehat{B}^{as} \left( B_{sb,t}+C_{ts,b}-B_{tb,s} \right) \\= & {} O(\epsilon )\\ \varGamma ^{s}_{ta}= & {} \frac{1}{2} (h_{\epsilon })^{sb} \left( (h_{\epsilon })_{ba,t}+(h_{\epsilon })_{tb,a}-(h_{\epsilon })_{ta,b}\right) + \frac{1}{2} (h_{\epsilon })^{su}\left( (h_{\epsilon })_{ua,t}+(h_{\epsilon })_{tu,a}-(h_{\epsilon })_{ta,u}\right) \\= & {} \frac{1}{2} \widehat{B}^{sb} \left( A_{ba,t}+ B_{tb,a}-B_{ta.b}\right) + \frac{1}{2} \left( (\mathring{h}_{\epsilon })^{su}+ \widehat{C}^{su} \right) \left( B_{ua,t}+C_{tu,a}-B_{ta,u}\right) \\= & {} O(\epsilon )\\ \varGamma ^{s}_{bc}= & {} \frac{1}{2} (h_{\epsilon })^{sa} \left( (h_{\epsilon })_{ac,b}+(h_{\epsilon })_{ba,c}-(h_{\epsilon })_{bc,a}\right) + \frac{1}{2} (h_{\epsilon })^{su}\left( (h_{\epsilon })_{uc,b}+(h_{\epsilon })_{bu,c}-(h_{\epsilon })_{bc,u}\right) \\= & {} \frac{1}{2} \widehat{B}^{sa} \left( (h_{\epsilon })_{ac,b}+(h_{\epsilon })_{ba,c}-(h_{\epsilon })_{bc,a}\right) \\&+\,\frac{1}{2} (\mathring{h}^{su}+ \widehat{C}^{su} ) \left( (\mathring{h}_{\epsilon })_{uc,b}+B_{uc,b}+(\mathring{h}_{\epsilon })_{bu,c}+B_{bu,c}-(\mathring{h}_{\epsilon })_{bc,u}-A_{bc,u}\right) \\= & {} O(\epsilon )+ \frac{1}{2} \left( \epsilon ^{-2}\mathring{\sigma }^{su}+ \widehat{C}^{su} \right) \left( B_{uc,b}+B_{bu,c}-A_{bc,u}\right) \\= & {} O(\epsilon ^{-1}). \end{aligned}$$

$\square$

Using the above lemma, a straightforward expansion of the scalar curvature in terms of Christoffel symbols, paying special attention to terms involving $\varGamma ^s_{bc}=O(\epsilon ^{-1})$ and $(\mathring{h}_{\epsilon })^{st}= \epsilon ^{-2} \mathring{\sigma }^{st}$, now shows that the scalar curvature satisfies the estimate

$$\begin{aligned} R(h_\epsilon )= R(\mathring{h}_{\epsilon })+ O(\epsilon ^{-1}) = R(\mathring{h}_{\epsilon })(1+ O(\epsilon ))\ge C \epsilon ^{-2}. \end{aligned}$$

For $\epsilon$ sufficiently small, then, not only is $R(h_{\epsilon })$ positive, but the above argument applies to the metric $h_{\epsilon }+t (\mathring{h}_{\epsilon }- h_{\epsilon })= \mathring{h}_{\epsilon }+ (1-t)(h_{\epsilon }- \mathring{h}_{\epsilon })$, from we can conclude $R(h_{\epsilon }+t (\mathring{h}_{\epsilon }- h_{\epsilon }))>0$ for all $t\in [0,1]$. Thus we can isotope $h_{\epsilon }$ to $\mathring{h}_{\epsilon }$ through metrics of positive scalar curvature.

We have the following as an immediate corollary of the proof.

Corollary 5

Let $\mathcal {N}_{r_0}(\varSigma )\subset (X,g)$ be a regular tubular neighborhood of a closed, embedded submanifold $\varSigma$ of codimension $q\ge 3$. For $0<\epsilon <r_0$, let $g_\epsilon$ be the metric induced on $\partial \mathcal {N}_{\epsilon }(\varSigma )$ from g, and let $\mathring{h}_{\epsilon }$ be the metric on $\partial \mathcal {N}_{\epsilon } (\varSigma )$ obtained by pullback of $\gamma \oplus \epsilon ^2 \mathring{\sigma }$ via the exponential map and a covering of local trivializations of the normal bundle as above, where $\mathring{\sigma }$ is the standard unit round metric on $\mathbb {S}^{q-1}$, and $\gamma$ is any metric on $\varSigma$. By choosing $\epsilon >0$ sufficiently small, one can isotope $g_{\epsilon }$ to $\mathring{h}_{\epsilon }$ through metrics of positive scalar curvature.