An integral estimate in conformal geometry
$begingroup$
Let $g_0$ be a Riemannian metric of positive scalar curvature $R_0$ on a closed 4-manifold, and for some fixed constant $C$, define the set
begin{equation}
mathcal{S} = {uin C^infty(M): ||u||_{W^{1,12}(M)}leq C}.
end{equation}
Then consider the set of conformal metrics
begin{equation}
mathcal{T} = {g=e^{2u}g_0:uinmathcal{S},~R_ggeq C(g_0)>0}.
end{equation}
The motivation behind this is I have a set of smooth solutions $u$ satisfying some PDE, for which I have uniform $W^{1,12}$ estimates and know that the corresponding scalar curvatures are uniformly bounded (i.e. a bound depending only on $g_0$) away from zero.
I am trying to obtain the following estimate:
begin{equation}tag{1}
int R^p|nabla^2 u|^4|nabla u|^4 ,dv_gleq Cint R^{p+6},dv_g + C
end{equation}
where $C$ again depends only on $g_0$. When $p>0$ I have no problem: the conformal transformation law for the scalar curvature implies
begin{equation}
|R|leq CBig(|R_0| + |nabla u|^2 + |Delta u|Big)\
implies|R|^p leq CBig(|R_0|^p + |nabla u|^{2p} + |Delta u|^pBig)
end{equation}
for some uniform $C$, and we can plug this into $(1)$ and apply Holder etc. But when $p<0$, I am really stuck - one attempt is to use $R^{-1}leq C$ (which follows from $Rgeq C>0$), but then I'm not sure how to reintroduce to scalar curvature to get something like $(1)$. Any help would be much appreciated. Thanks!
dg.differential-geometry riemannian-geometry curvature conformal-geometry
asked Dec 13 '18 at 23:26
jl2jl2
1632
$endgroup$
migrated from math.stackexchange.com Dec 23 '18 at 0:32
This question came from our site for people studying math at any level and professionals in related fields.
add a comment |
$begingroup$
Let $g_0$ be a Riemannian metric of positive scalar curvature $R_0$ on a closed 4-manifold, and for some fixed constant $C$, define the set
begin{equation}
mathcal{S} = {uin C^infty(M): ||u||_{W^{1,12}(M)}leq C}.
end{equation}
Then consider the set of conformal metrics
begin{equation}
mathcal{T} = {g=e^{2u}g_0:uinmathcal{S},~R_ggeq C(g_0)>0}.
end{equation}
The motivation behind this is I have a set of smooth solutions $u$ satisfying some PDE, for which I have uniform $W^{1,12}$ estimates and know that the corresponding scalar curvatures are uniformly bounded (i.e. a bound depending only on $g_0$) away from zero.
I am trying to obtain the following estimate:
begin{equation}tag{1}
int R^p|nabla^2 u|^4|nabla u|^4 ,dv_gleq Cint R^{p+6},dv_g + C
end{equation}
where $C$ again depends only on $g_0$. When $p>0$ I have no problem: the conformal transformation law for the scalar curvature implies
begin{equation}
|R|leq CBig(|R_0| + |nabla u|^2 + |Delta u|Big)\
implies|R|^p leq CBig(|R_0|^p + |nabla u|^{2p} + |Delta u|^pBig)
end{equation}
for some uniform $C$, and we can plug this into $(1)$ and apply Holder etc. But when $p<0$, I am really stuck - one attempt is to use $R^{-1}leq C$ (which follows from $Rgeq C>0$), but then I'm not sure how to reintroduce to scalar curvature to get something like $(1)$. Any help would be much appreciated. Thanks!
dg.differential-geometry riemannian-geometry curvature conformal-geometry
asked Dec 13 '18 at 23:26
jl2jl2
1632
$endgroup$
migrated from math.stackexchange.com Dec 23 '18 at 0:32
This question came from our site for people studying math at any level and professionals in related fields.
1
$begingroup$
This is a bit specialistic, if you do not obtain answers consider flagging for migration to MathOverflow.
$endgroup$
– Giuseppe Negro
Dec 17 '18 at 10:34
add a comment |
$begingroup$
Let $g_0$ be a Riemannian metric of positive scalar curvature $R_0$ on a closed 4-manifold, and for some fixed constant $C$, define the set
begin{equation}
mathcal{S} = {uin C^infty(M): ||u||_{W^{1,12}(M)}leq C}.
end{equation}
Then consider the set of conformal metrics
begin{equation}
mathcal{T} = {g=e^{2u}g_0:uinmathcal{S},~R_ggeq C(g_0)>0}.
end{equation}
The motivation behind this is I have a set of smooth solutions $u$ satisfying some PDE, for which I have uniform $W^{1,12}$ estimates and know that the corresponding scalar curvatures are uniformly bounded (i.e. a bound depending only on $g_0$) away from zero.
I am trying to obtain the following estimate:
begin{equation}tag{1}
int R^p|nabla^2 u|^4|nabla u|^4 ,dv_gleq Cint R^{p+6},dv_g + C
end{equation}
where $C$ again depends only on $g_0$. When $p>0$ I have no problem: the conformal transformation law for the scalar curvature implies
begin{equation}
|R|leq CBig(|R_0| + |nabla u|^2 + |Delta u|Big)\
implies|R|^p leq CBig(|R_0|^p + |nabla u|^{2p} + |Delta u|^pBig)
end{equation}
for some uniform $C$, and we can plug this into $(1)$ and apply Holder etc. But when $p<0$, I am really stuck - one attempt is to use $R^{-1}leq C$ (which follows from $Rgeq C>0$), but then I'm not sure how to reintroduce to scalar curvature to get something like $(1)$. Any help would be much appreciated. Thanks!
dg.differential-geometry riemannian-geometry curvature conformal-geometry
asked Dec 13 '18 at 23:26
jl2jl2
1632
$endgroup$
Let $g_0$ be a Riemannian metric of positive scalar curvature $R_0$ on a closed 4-manifold, and for some fixed constant $C$, define the set
begin{equation}
mathcal{S} = {uin C^infty(M): ||u||_{W^{1,12}(M)}leq C}.
end{equation}
Then consider the set of conformal metrics
begin{equation}
mathcal{T} = {g=e^{2u}g_0:uinmathcal{S},~R_ggeq C(g_0)>0}.
end{equation}
The motivation behind this is I have a set of smooth solutions $u$ satisfying some PDE, for which I have uniform $W^{1,12}$ estimates and know that the corresponding scalar curvatures are uniformly bounded (i.e. a bound depending only on $g_0$) away from zero.
I am trying to obtain the following estimate:
begin{equation}tag{1}
int R^p|nabla^2 u|^4|nabla u|^4 ,dv_gleq Cint R^{p+6},dv_g + C
end{equation}
where $C$ again depends only on $g_0$. When $p>0$ I have no problem: the conformal transformation law for the scalar curvature implies
begin{equation}
|R|leq CBig(|R_0| + |nabla u|^2 + |Delta u|Big)\
implies|R|^p leq CBig(|R_0|^p + |nabla u|^{2p} + |Delta u|^pBig)
end{equation}
for some uniform $C$, and we can plug this into $(1)$ and apply Holder etc. But when $p<0$, I am really stuck - one attempt is to use $R^{-1}leq C$ (which follows from $Rgeq C>0$), but then I'm not sure how to reintroduce to scalar curvature to get something like $(1)$. Any help would be much appreciated. Thanks!
dg.differential-geometry riemannian-geometry curvature conformal-geometry
dg.differential-geometry riemannian-geometry curvature conformal-geometry
asked Dec 13 '18 at 23:26
jl2jl2
1632
asked Dec 13 '18 at 23:26
jl2jl2
1632
asked Dec 13 '18 at 23:26
jl2jl2
1632
asked Dec 13 '18 at 23:26
jl2jl2
1632
asked Dec 13 '18 at 23:26
jl2jl2
1632
1632
migrated from math.stackexchange.com Dec 23 '18 at 0:32
This question came from our site for people studying math at any level and professionals in related fields.
migrated from math.stackexchange.com Dec 23 '18 at 0:32
This question came from our site for people studying math at any level and professionals in related fields.
1
$begingroup$
This is a bit specialistic, if you do not obtain answers consider flagging for migration to MathOverflow.
$endgroup$
– Giuseppe Negro
Dec 17 '18 at 10:34
add a comment |
1
$begingroup$
This is a bit specialistic, if you do not obtain answers consider flagging for migration to MathOverflow.
$endgroup$
– Giuseppe Negro
Dec 17 '18 at 10:34
1
1
$begingroup$
This is a bit specialistic, if you do not obtain answers consider flagging for migration to MathOverflow.
$endgroup$
– Giuseppe Negro
Dec 17 '18 at 10:34
$begingroup$
This is a bit specialistic, if you do not obtain answers consider flagging for migration to MathOverflow.
$endgroup$
– Giuseppe Negro
Dec 17 '18 at 10:34
add a comment |
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$begingroup$
This is a bit specialistic, if you do not obtain answers consider flagging for migration to MathOverflow.
$endgroup$
– Giuseppe Negro
Dec 17 '18 at 10:34