In this note, we talk about the following interesting result obtained by X. Xu in Nagoya Mathematical Journal in 1995, here.
To state his result, let us fix 
a compact manifold of dimension 
. We shall denote by 
the scalar curvature computed with respect to the metric 
. It is clear that under the conformal change 
, 
and are related by the following rule
Furthermore, we may also assume that is a negative constant. We are now able to state his result.
Theorem. If 
and 
, then there exists a constant 
such that 
where 
.
Proof. Using the conformal change rule shown above, we obtain
![\begin{array}{lcl}{C_0} &\ge& \displaystyle\int_M {|\text{Scal}_g{|^4}d{v_g}} \\ &=& \displaystyle\int_M {{{[{u^{ - 3}}(u \text{Scal}_{g_0} - 6{\Delta _{{g_0}}}u)]}^4}({u^4}d{v_{{g_0}}})} \\ &=& \displaystyle\int_M {{{[{u^{ - 1}}\text{Scal}_{g_0} - 6{u^{ - 2}}{\Delta _{{g_0}}}u]}^4}d{v_{{g_0}}}} \\ &=& \displaystyle\int_M {{u^{ - 4}}|\text{Scal}_{g_0}|^4 d{v_{{g_0}}}} + {\rm{1296}}\int_M {{u^{ - 8}}{{({\Delta _{{g_0}}}u)}^4}d{v_{{g_0}}}} + \\&& \displaystyle + {\rm{216}}\int_M {{u^{ - 6}}|\text{Scal}_{g_0}|^2{{({\Delta _{{g_0}}}u)}^2}d{v_{{g_0}}}} - 24\int_M {{u^{ - 5}}|\text{Scal}_{g_0}|^3{\Delta _{{g_0}}}ud{v_{{g_0}}}} \\ &&\displaystyle - 864\int_M {{u^{ - 7}} \text{Scal}_{g_0} {{({\Delta _{{g_0}}}u)}^3}d{v_{{g_0}}}} \\ &=& \displaystyle |\text{Scal}_{g_0}|^4\int_M {{u^{ - 4}}d{v_{{g_0}}}} - 24|\text{Scal}_{g_0}|^3 \int_M {{u^{ - 5}}{\Delta _{{g_0}}}ud{v_{{g_0}}}} \\ &&\displaystyle +432\int_M {{u^{ - 8}}{{({\Delta _{{g_0}}}u)}^4}d{v_{{g_0}}}} + 216\int_M {{u^{ - 8}}{{({\Delta _{{g_0}}}u)}^2}{{[\text{Scal}_{g_0} - 2{\Delta _{{g_0}}}u]}^2}d{v_{{g_0}}}} .\end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blcl%7D%7BC_0%7D+%26%5Cge%26+%5Cdisplaystyle%5Cint_M+%7B%7C%5Ctext%7BScal%7D_g%7B%7C%5E4%7Dd%7Bv_g%7D%7D+%5C%5C+%26%3D%26+%5Cdisplaystyle%5Cint_M+%7B%7B%7B%5B%7Bu%5E%7B+-+3%7D%7D%28u+%5Ctext%7BScal%7D_%7Bg_0%7D+-+6%7B%5CDelta+_%7B%7Bg_0%7D%7D%7Du%29%5D%7D%5E4%7D%28%7Bu%5E4%7Dd%7Bv_%7B%7Bg_0%7D%7D%7D%29%7D+%5C%5C+%26%3D%26+%5Cdisplaystyle%5Cint_M+%7B%7B%7B%5B%7Bu%5E%7B+-+1%7D%7D%5Ctext%7BScal%7D_%7Bg_0%7D+-+6%7Bu%5E%7B+-+2%7D%7D%7B%5CDelta+_%7B%7Bg_0%7D%7D%7Du%5D%7D%5E4%7Dd%7Bv_%7B%7Bg_0%7D%7D%7D%7D+%5C%5C+%26%3D%26+%5Cdisplaystyle%5Cint_M+%7B%7Bu%5E%7B+-+4%7D%7D%7C%5Ctext%7BScal%7D_%7Bg_0%7D%7C%5E4+d%7Bv_%7B%7Bg_0%7D%7D%7D%7D+%2B+%7B%5Crm%7B1296%7D%7D%5Cint_M+%7B%7Bu%5E%7B+-+8%7D%7D%7B%7B%28%7B%5CDelta+_%7B%7Bg_0%7D%7D%7Du%29%7D%5E4%7Dd%7Bv_%7B%7Bg_0%7D%7D%7D%7D+%2B+%5C%5C%26%26+%5Cdisplaystyle+%2B+%7B%5Crm%7B216%7D%7D%5Cint_M+%7B%7Bu%5E%7B+-+6%7D%7D%7C%5Ctext%7BScal%7D_%7Bg_0%7D%7C%5E2%7B%7B%28%7B%5CDelta+_%7B%7Bg_0%7D%7D%7Du%29%7D%5E2%7Dd%7Bv_%7B%7Bg_0%7D%7D%7D%7D+-+24%5Cint_M+%7B%7Bu%5E%7B+-+5%7D%7D%7C%5Ctext%7BScal%7D_%7Bg_0%7D%7C%5E3%7B%5CDelta+_%7B%7Bg_0%7D%7D%7Dud%7Bv_%7B%7Bg_0%7D%7D%7D%7D+%5C%5C+%26%26%5Cdisplaystyle+-+864%5Cint_M+%7B%7Bu%5E%7B+-+7%7D%7D+%5Ctext%7BScal%7D_%7Bg_0%7D+%7B%7B%28%7B%5CDelta+_%7B%7Bg_0%7D%7D%7Du%29%7D%5E3%7Dd%7Bv_%7B%7Bg_0%7D%7D%7D%7D+%5C%5C+%26%3D%26+%5Cdisplaystyle+%7C%5Ctext%7BScal%7D_%7Bg_0%7D%7C%5E4%5Cint_M+%7B%7Bu%5E%7B+-+4%7D%7Dd%7Bv_%7B%7Bg_0%7D%7D%7D%7D+-+24%7C%5Ctext%7BScal%7D_%7Bg_0%7D%7C%5E3+%5Cint_M+%7B%7Bu%5E%7B+-+5%7D%7D%7B%5CDelta+_%7B%7Bg_0%7D%7D%7Dud%7Bv_%7B%7Bg_0%7D%7D%7D%7D+%5C%5C+%26%26%5Cdisplaystyle+%2B432%5Cint_M+%7B%7Bu%5E%7B+-+8%7D%7D%7B%7B%28%7B%5CDelta+_%7B%7Bg_0%7D%7D%7Du%29%7D%5E4%7Dd%7Bv_%7B%7Bg_0%7D%7D%7D%7D+%2B+216%5Cint_M+%7B%7Bu%5E%7B+-+8%7D%7D%7B%7B%28%7B%5CDelta+_%7B%7Bg_0%7D%7D%7Du%29%7D%5E2%7D%7B%7B%5B%5Ctext%7BScal%7D_%7Bg_0%7D+-+2%7B%5CDelta+_%7B%7Bg_0%7D%7D%7Du%5D%7D%5E2%7Dd%7Bv_%7B%7Bg_0%7D%7D%7D%7D+.%5Cend%7Barray%7D&bg=ffffff&fg=333333&s=0&c=20201002)
Since 
, 
. Also, using integration by parts,
Therefore, each term on the right hand side of the preceding inequality is positive, hence we obtain
and
Now from the identity
it is easy to see that for any point 
d{v_{{g_0}}}}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cfrac%7B1%7D%7B%7Bu%28p%29%7D%7D+-+%5Cfrac%7B1%7D%7B%7B%5Ctext%7Bvol%7D%28M%2C%7Bg_0%7D%29%7D%7D%5Cint_M+%7B%5Cfrac%7B1%7D%7Bu%7Dd%7Bv_%7B%7Bg_0%7D%7D%7D%7D+%3D+-+%5Cint_M+%7BG%28p%2Cq%29%5Cleft%5B+-+%5Cfrac%7B%7B%7B%5CDelta+_%7B%7Bg_0%7D%7D%7Du%7D%7D%7B%7B%7Bu%5E2%7D%7D%7D+%2B+%5Cfrac%7B%7B2%7C%5Cnabla+u%7C%5E2%7D%7D%7B%7B%7Bu%5E3%7D%7D%7D+%5Cright%5D%28q%29d%7Bv_%7B%7Bg_0%7D%7D%7D%7D+&bg=ffffff&fg=333333&s=0&c=20201002)
where 
is the Green function on 
which can be assumed to be positive everywhere on . The well known fact is that the Green function on a -dimensional manifold is 
integrable for 
. Thus using the Holder inequality, it is clear that 
is bounded from below.
Ngô Quốc Anh 
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