# Smoothing metrics on closed Riemannian manifolds through the Ricci flow

Yunyan Yang<EMAIL_ADDRESS>Department of Mathematics, Renmin
University of China, Beijing 100872, P. R. China

###### Abstract

Under the assumption of the uniform local Sobolev inequality, it is proved
that Riemannian metrics with an absolute Ricci curvature bound and a small
Riemannian curvature integral bound can be smoothed to having a sectional
curvature bound. This partly extends previous a priori estimates of Ye Li (J.
Geom. Anal. 17 (2007) 495-511; Advances in Mathematics 223 (2010) 1924-1957).

###### keywords:

Ricci flow; Smoothing; Moser iteration

###### MSC:

53C20; 53C21; 58J35

††journal: Ann Glob Anal Geom

##  1 Introduction

If a Riemannian manifold has bounded sectional curvature, then its geometric
structure is better understood than that with weaker curvature bounds, say
Ricci curvature bounds. Thus it is of significance to deform or smooth a
Riemannian metric with a Ricci curvature bound to a metric with a sectional
curvature bound. One way to do this is using the Ricci flow. In this regard we
refer the reader to the pioneer works [3, 6, 17, 18]. If the initial metric
has bounded curvatures, one can show the short time existence of the Ricci
flow and obtain the covariant derivatives bounds for the curvature tensors
along the Ricci flow [2, 14]. If the initial metric has bounded Ricci
curvature, under some additional assumption on conjugate radius, Dai, etc.
studied how to deform the metric on closed manifolds [6]. Also one can deform
a metric locally by using the local Ricci flow [11, 12, 13, 16, 18].
Throughout this paper, we use ${\rm Rm}(g)$ and ${\rm Ric}(g)$ to denote the
Riemannian curvature tensor and Ricci tensor with respect to the metric $g$
respectively. Our main result is the following:

Theorem 1.1. Assume $(M,g_{0})$ is a closed Riemannian manifold of dimension
$n$ ($n\geq 3$) and $|{\rm Ric}(g_{0})|\leq K$ for some constant $K$. Let
$B_{r}(x)$ be a geodesic ball centered at $x\in M$ with radius $r$. Suppose
there exists a constant $A_{0}>0$ such that for all $x\in M$ and some
$r\leq\min(\frac{1}{2}{\rm diam}(g_{0}),1)$

$\left(\int_{B_{r}(x)}|u|^{\frac{2n}{n-2}}dv_{g_{0}}\right)^{(n-2)/n}\leq
A_{0}\int_{B_{r}(x)}|\nabla_{g_{0}}u|^{2}dv_{g_{0}},\quad\forall u\in
C_{0}^{\infty}(B_{r}(x)).$ (1.1)

Then there exist constants $\epsilon$, $c_{1}$, $c_{2}$ depending only on $n$
and $K$ such that if

$\left(\int_{B_{r}(x)}|{\rm
Rm}(g_{0})|^{\frac{n}{2}}dv_{g_{0}}\right)^{{2}/{n}}\leq{\epsilon}{A_{0}^{-1}}\quad{\rm
for\,\,all}\,\,x\in M,$ (1.2)

then the Ricci flow

$\left\\{\begin{array}[]{lll}\displaystyle\frac{\partial g}{\partial
t}&=-2Ric(g),\\\\[5.16663pt] g(0)&=g_{0}\end{array}\right.$ (1.3)

has a unique smooth solution satisfying the following estimates

$\displaystyle|g(t)-g_{0}|_{g_{0}}$ $\displaystyle\leq$ $\displaystyle
c_{2}t^{\frac{2}{n+2}},$ (1.4) $\displaystyle|{\rm Rm}(g(t))|_{\infty}$
$\displaystyle\leq$ $\displaystyle c_{2}t^{-1},$ (1.5) $\displaystyle|{\rm
Ric(g(t))}|_{\infty}$ $\displaystyle\leq$ $\displaystyle
c_{2}t^{-\frac{n}{n+2}}$ (1.6)

for $0\leq t\leq T$ with $T\geq c_{1}\min(r^{2},K^{-1})$.

When $(M,g_{0})$ is a complete noncompact Riemannian manifold, similar results
were obtained by Ye Li [13] and G. Xu [16]. The assumptions of [13] is much
weaker than (1.1) and (1.2) in case $n=4$. It comes from Cheeger and Tian’s
work [5] concerning the collapsing Einstein 4-manifolds. Here Theorem 1.1 is
just the beginning of extending the results [5, 12, 13], which may depend on
the Gauss-Bonnet-Chern formula, to general dimensional case.

For the proof of Theorem 1.1, we follow the lines of [6, 7, 13, 18]. Let’s
roughly describe the idea. First it is well known [10, 8] that the Ricci flow
(1.3) has a unique smooth solution $g(t)$ for a very short time interval.
Using Moser’s iteration and Gromov’s covering argument, we derive a priori
estimates on ${\rm Rm}(g(t))$ and ${\rm Ric}(g(t))$. Let $[0,T_{\rm max}]$ be
a maximum time interval on which $g(t)$ exists. Then based on those a priori
estimates, $T_{\rm max}$ has the desired lower bound.

Such kind of results are very useful when considering the relation between
curvature and topology [1, 6, 12]. Using Theorem 1.1, we can easily generalize
Gromov’s almost flat manifold theorem [9]. Particularly one has the following:

Theorem 1.2. There exist constants $\epsilon$ and $\delta$ depending only on
$n$ and $K$ such that if a closed Riemannian manifold $(M,g_{0})$ satisfies
$|{\rm Ric(g_{0})}|\leq K$, ${\rm diam}(g_{0})\leq\delta$, (1.1) and (1.2)
hold for all $x\in M$, then the universal covering space of $(M,g_{0})$ is
$\mathbb{R}^{n}$. If all the above hypothesis on $(M,g_{0})$ are satisfied and
moreover the fundamental group $\pi(g_{0})$ is commutative, then $(M,g_{0})$
is diffeomorphic to a torus.

Before ending this introduction, we would like to mention [15] for local
regularity estimates for Riemannian curvatures. The remaining part of the
paper is organized as follows. In Sect. 2, we derive two weak maximum
principles by using the Moser’s iteration. In Sect. 3, we estimate the time
interval on which the solution of Ricci flow exists, and prove Theorem 1.1.
Finally Theorem 1.2 is proved in Sect. 4.

##  2 Weak maximum principles

In this section, following the lines of [13, 18], we give two maximum
principles via the Moser’s iteration. Throughout this section the manifolds
need not to be compact. Suppose $(M,g(t))$ are complete Riemannian manifolds
for $0\leq t\leq T$. Let $\nabla_{g(t)}$ denote the covariant differentiation
with respect to $g(t)$ and $-\Delta_{g(t)}$ be the corresponding Laplace-
Beltrami operator, which will be also denoted by $\nabla$ and $-\Delta$ for
simplicity, the reader can easily recognize it from the context. Let $A$ be a
constant such that for all $t\in[0,T]$,

$\left(\int_{B_{r}(x)}|u|^{\frac{2n}{n-2}}dv_{t}\right)^{(n-2)/n}\leq
A\int_{B_{r}(x)}|\nabla u|^{2}dv_{t},\quad\forall u\in
C_{0}^{\infty}(B_{r}(x)),$ (2.1)

where $dv_{t}=dv_{g(t)}$. Assume that for all $0\leq t\leq T$,

$\frac{1}{2}g_{0}\leq g(t)\leq 2g_{0}\quad{\rm on}\quad M.$ (2.2)

Here and in the sequel, all geodesic balls are defined with respect to
$g_{0}$.

Firstly we have the following maximum principle:

Theorem 2.1. Let $(M,g(t))$ be complete Riemannian manifolds and (2.1), (2.2)
are satisfied for $0\leq t\leq T$. Let $f(x,t)$ be such that

$\frac{\partial f}{\partial t}\leq\Delta f+uf\quad{\rm on}\quad
B_{r}(x)\times[0,T]$ (2.3)

with $f\geq 0$, $u\geq 0$,

$\frac{\partial}{\partial t}dv_{t}\leq cudv_{t},$ (2.4)

for some constant $c$ depending only on $n$ and for some $q>n$

$\left(\int_{B_{r}(x)}u^{\frac{q}{2}}dv_{t}\right)^{\frac{2}{q}}\leq\mu
t^{-\frac{q-n}{q}},$ (2.5)

where $\mu>0$ is a constant. Then for any $p>1$, $t\in[0,T]$, we have

$f(x,t)\leq
CA^{\frac{n}{2p}}\left(\frac{1+A^{\frac{n}{q-n}}\mu^{\frac{q}{q-n}}}{t}+\frac{1}{r^{2}}\right)^{\frac{n+2}{2p}}\left(\int_{0}^{T}\int_{B_{r}(x)}f^{p}dv_{t}\right)^{\frac{1}{p}},$
(2.6)

where $C$ is a constant depending only on $n$, $q$ and $p$.

Proof. Let $\eta$ be a nonnegative Lipschitz function supported in $B_{r}(x)$.
We first consider the case $p\geq 2$. By the partial differential inequality
(2.3) and (2.4), we have

$\displaystyle\frac{1}{p}\frac{\partial}{\partial t}\int\eta^{2}f^{p}dv_{t}$
$\displaystyle\leq$ $\displaystyle\int\eta^{2}f^{p-1}\Delta fdv_{t}+C_{1}\int
uf^{p}\eta^{2}dv_{t},$

where $C_{1}$ is a constant depending only on $n$. Integration by parts
implies

$\displaystyle\int\eta^{2}f^{p-1}\Delta fdv_{t}$ $\displaystyle=$
$\displaystyle-2\int\eta f^{p-1}\nabla\eta\nabla
fdv_{t}-(p-1)\int\eta^{2}f^{p-2}|\nabla f|^{2}dv_{t}$ $\displaystyle=$
$\displaystyle-\frac{4}{p}\int\left(f^{\frac{p}{2}}\nabla\eta\nabla(\eta
f^{\frac{p}{2}})-|\nabla\eta|^{2}f^{p}\right)dv_{t}-\frac{4(p-1)}{p^{2}}$
$\displaystyle\times\int\left(|\nabla(\eta
f^{\frac{p}{2}})|^{2}+|\nabla\eta|^{2}f^{p}-2f^{\frac{p}{2}}\nabla\eta\nabla(\eta
f^{\frac{p}{2}})\right)dv_{t}$ $\displaystyle=$
$\displaystyle-\frac{4(p-1)}{p^{2}}\int|\nabla(\eta
f^{\frac{p}{2}})|^{2}dv_{t}+\frac{4}{p^{2}}\int|\nabla\eta|^{2}f^{p}dv_{t}$
$\displaystyle+\frac{4p-8}{p^{2}}\int f^{\frac{p}{2}}\nabla\eta\nabla(\eta
f^{\frac{p}{2}})dv_{t}$ $\displaystyle\leq$
$\displaystyle-\frac{2}{p}\int|\nabla(\eta
f^{\frac{p}{2}})|^{2}dv_{t}+\frac{2}{p}\int|\nabla\eta|^{2}f^{p}dv_{t}.$

Here we have used the elementary inequality $2ab\leq a^{2}+b^{2}$. By the
Hölder inequality, we have

$\displaystyle\int uf^{p}\eta^{2}dv_{t}$ $\displaystyle\leq$
$\displaystyle\left(\int
u^{\frac{q}{2}}dv_{t}\right)^{\frac{2}{q}}\left(\int(\eta^{2}f^{p})^{\alpha
q_{1}}dv_{t}\right)^{\frac{1}{q_{1}}}\left(\int(\eta^{2}f^{p})^{(1-\alpha)q_{2}}dv_{t}\right)^{\frac{1}{q_{2}}},$

where $\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{2}{q}=1$ and $0<\alpha<1$. Let
$\alpha q_{1}=\frac{n}{n-2}$ and $(1-\alpha)q_{2}=1$. This implies
$q_{1}=\frac{q}{n-2}$, $q_{2}=\frac{q}{q-n}$ and $\alpha=\frac{n}{q}$. Using
the Sobolev inequality (2.1) and the Young inequality, we obtain

$\displaystyle\int uf^{p}\eta^{2}dv_{t}$ $\displaystyle\leq$ $\displaystyle\mu
t^{-\frac{q-n}{q}}\left(\int(\eta^{2}f^{p})^{\frac{n}{n-2}}dv_{t}\right)^{\frac{n-2}{q}}\left(\int\eta^{2}f^{p}dv_{t}\right)^{\frac{q-n}{q}}$
$\displaystyle\leq$ $\displaystyle\mu
t^{-\frac{q-n}{q}}\left(A\int|\nabla(\eta
f^{\frac{p}{2}})|^{2}dv_{t}\right)^{\frac{n}{q}}\left(\int\eta^{2}f^{p}dv_{t}\right)^{\frac{q-n}{q}}$
$\displaystyle\leq$ $\displaystyle\frac{1}{pC_{1}}\int|\nabla(\eta
f^{\frac{p}{2}})|^{2}dv_{t}+C_{2}p^{\frac{n}{q-n}}\mu^{\frac{q}{q-n}}A^{\frac{n}{q-n}}t^{-1}\int\eta^{2}f^{p}dv_{t}$

for some constant $C_{2}$ depending only on $n$ and $q$. Combining all the
above estimates one has

$\displaystyle\frac{\partial}{\partial
t}\int\eta^{2}f^{p}dv_{t}+\int|\nabla(\eta f^{\frac{p}{2}})|^{2}dv_{t}\leq
2\int|\nabla\eta|^{2}f^{p}dv_{t}$ (2.7)
$\displaystyle\quad\quad\quad\quad\quad\quad+C_{1}C_{2}p^{\frac{q}{q-n}}\mu^{\frac{q}{q-n}}A^{\frac{n}{q-n}}t^{-1}\int\eta^{2}f^{p}dv_{t}.{}$

For $0<\tau<\tau^{\prime}<T$, let

$\psi(t)=\left\\{\begin{array}[]{lll}0,&0\leq t\leq\tau\\\\[5.16663pt]
\frac{t-\tau}{\tau^{\prime}-\tau},&\tau\leq t\leq\tau^{\prime}\\\\[5.16663pt]
1,&\tau^{\prime}\leq t\leq T.\end{array}\right.$

Multiplying (2.7) by $\psi$, we have

$\displaystyle\frac{\partial}{\partial
t}\left(\psi\int\eta^{2}f^{p}dv_{t}\right)+\psi\int|\nabla(\eta
f^{\frac{p}{2}})|^{2}dv_{t}\leq 2\psi\int|\nabla\eta|^{2}f^{p}dv_{t}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\left(C_{1}C_{2}p^{\frac{q}{q-n}}\mu^{\frac{q}{q-n}}A^{\frac{n}{q-n}}t^{-1}\psi+\psi^{\prime}\right)\int\eta^{2}f^{p}dv_{t}.$
(2.8)

Assume $\tau<\tau^{\prime}<t\leq T$. Since on the time interval
$[\tau,\tau^{\prime}]$

$0\leq\frac{\psi(t)}{t}=\frac{1}{\tau^{\prime}-\tau}-\frac{\tau}{\tau^{\prime}-\tau}\frac{1}{t}\leq\frac{1}{\tau^{\prime}-\tau}\left(1-\frac{\tau}{\tau^{\prime}}\right)=\frac{1}{\tau^{\prime}},$

and on the time interval $[\tau^{\prime},T]$

$\frac{1}{T}\leq\frac{\psi(t)}{t}\leq\frac{1}{\tau^{\prime}},$

we have

$\int_{\tau}^{t}\frac{\psi(t)}{t}\left(\int\eta^{2}f^{p}dv_{t}\right)dt\leq\frac{1}{\tau^{\prime}}\int_{\tau}^{t}\int\eta^{2}f^{p}dv_{t}dt.$
(2.9)

Notice that $0\leq\psi\leq 1$ and
$0\leq\psi^{\prime}\leq\frac{1}{\tau^{\prime}-\tau}$. Integrating the
differential inequality (2.8) from $\tau$ to $t$, we obtain by using (2.9)

$\displaystyle\int\eta^{2}f^{p}dv_{t}+\int_{\tau^{\prime}}^{t}\int|\nabla(\eta
f^{\frac{p}{2}})|^{2}dv_{t}dt\leq
2\int_{\tau}^{t}\int|\nabla\eta|^{2}f^{p}dv_{t}dt$
$\displaystyle\quad\quad\quad\quad+\left(\frac{C_{1}C_{2}p^{\frac{q}{q-n}}\mu^{\frac{q}{q-n}}A^{\frac{n}{q-n}}}{\tau^{\prime}}+\frac{1}{\tau^{\prime}-\tau}\right)\int_{\tau}^{T}\int\eta^{2}f^{p}dv_{t}dt.$

Applying this estimate and the Sobolev inequality we derive

$\displaystyle\quad\quad\int_{\tau^{\prime}}^{T}\int
f^{p(1+\frac{2}{n})}\eta^{2+\frac{1}{n}}dv_{t}dt$ $\displaystyle\leq$
$\displaystyle\int_{\tau^{\prime}}^{T}\left(\int\eta^{2}f^{p}dv_{t}\right)^{\frac{2}{n}}\left(\int
f^{\frac{pn}{n-2}}\eta^{\frac{2n}{n-2}}dv_{t}\right)^{\frac{n-2}{n}}dt$
$\displaystyle\leq$ $\displaystyle A\left(\sup_{\tau^{\prime}\leq t\leq
T}\int\eta^{2}f^{p}\right)^{\frac{2}{n}}\int_{\tau^{\prime}}^{T}\int|\nabla(\eta
f^{\frac{p}{2}})|^{2}dv_{t}dt$ $\displaystyle\leq$ $\displaystyle
A\left[2\int_{\tau}^{t}\int|\nabla\eta|^{2}f^{p}dv_{t}dt+\left(\frac{C_{1}C_{2}p^{\frac{q}{q-n}}\mu^{\frac{q}{q-n}}A^{\frac{n}{q-n}}}{\tau^{\prime}}\right.\right.$
$\displaystyle\quad\quad\quad\left.\left.+\frac{1}{\tau^{\prime}-\tau}\right)\int_{\tau}^{T}\int\eta^{2}f^{p}dv_{t}dt\right]^{1+\frac{2}{n}}.$

For $p\geq p_{0}\geq 2$ and $0\leq\tau\leq T$, we set

$H(p,\tau,r)=\int_{\tau}^{T}\int_{B_{r}(x)}f^{p}dv_{t}dt,$

where $B_{r}(x)$ is the geodesic ball centered at $x$ with radius $r$ measured
in $g(0)$. Choosing a suitable cut-off function $\eta$ and noting that
$|\nabla\eta|_{t}\leq 2|\nabla\eta|_{0}$, we obtain from (2)

$\displaystyle H\left(p\left(1+\frac{2}{n}\right),\tau^{\prime},r\right)$
(2.11) $\displaystyle\leq
AC_{3}\left(\frac{p^{\frac{q}{q-n}}\mu^{\frac{q}{q-n}}A^{\frac{n}{q-n}}}{\tau^{\prime}}+\frac{1}{\tau^{\prime}-\tau}+\frac{1}{(r^{\prime}-r)^{2}}\right)^{1+\frac{2}{n}}H(p,\tau,r^{\prime})^{1+\frac{2}{n}},$

where $0<r<r^{\prime}$, $C_{3}$ is a constant depending only on $n$ and $q$.
Set

$\nu=1+\frac{2}{n},\quad
p_{k}=p_{0}\nu^{k},\quad\tau_{k}=(1-\nu^{-\frac{qk}{q-n}})t,\quad\quad
r_{k}=(1+\nu^{-\frac{qk}{q-n}})r/2.$

Then the inequality (2.11) gives

$H(p_{k+1},\tau_{k+1},r_{k+1})\leq
AC_{3}\left(\frac{1+p_{0}^{\frac{q}{q-n}}\mu^{\frac{q}{q-n}}A^{\frac{n}{q-n}}}{t}+\frac{1}{r^{2}}\right)^{\nu}\eta^{k\nu}H(p_{k},\tau_{k},r_{k})^{\nu},$

where $\eta=\nu^{\frac{2q}{q-n}}$. It follows that

$\displaystyle H(p_{k+1},\tau_{k+1},r_{k+1})^{\frac{1}{p_{k+1}}}$
$\displaystyle\leq(AC_{3})^{\frac{1}{p_{k+1}}}\left(\frac{1+p_{0}^{\frac{q}{q-n}}\mu^{\frac{q}{q-n}}A^{\frac{n}{q-n}}}{t}+\frac{1}{r^{2}}\right)^{\frac{1}{p_{k}}}\eta^{\frac{k}{p_{k}}}H(p_{k},\tau_{k},r_{k})^{\frac{1}{p_{k}}}.$

Hence we obtain for any fixed $k$

$\displaystyle H(p_{k+1},\tau_{k+1},r_{k+1})^{\frac{1}{p_{k+1}}}$
$\displaystyle\leq$
$\displaystyle(AC_{3})^{\sum_{j=0}^{k}\frac{1}{p_{j+1}}}\left(\frac{1+p_{0}^{\frac{q}{q-n}}\mu^{\frac{q}{q-n}}A^{\frac{n}{q-n}}}{t}+\frac{1}{r^{2}}\right)^{\sum_{j=0}^{k}\frac{1}{p_{j}}}$
$\displaystyle\eta^{\sum_{j=0}^{k}\frac{j}{p_{j}}}H(p_{0},\tau_{0},r_{0})^{\frac{1}{p_{0}}}.$

Passing to the limit $k\rightarrow\infty$, one concludes

$\displaystyle
f(x,t)\leq(CA)^{\frac{n}{2p_{0}}}\left(\frac{1+(p_{0}\mu)^{\frac{q}{q-n}}A^{\frac{n}{q-n}}}{t}+\frac{1}{r^{2}}\right)^{\frac{n+2}{2p_{0}}}\left(\int_{0}^{T}\int
f^{p_{0}}dv_{t}dt\right)^{\frac{1}{p_{0}}}.$

This proves (2.6) in the case $p\geq 2$.

Assuming $f$ satisfies (2.3) and $f\geq 0$. We define a sequence of functions

$f_{j}=f+1/j,\quad j\in\mathbb{N}.$

Then $f_{j}$ also satisfies (2.3) and $f_{j}^{p/2}$ is Lipschitz continuous
for $1<p<2$. The same argument as the case $p\geq 2$ also yields

$\displaystyle
f_{j}(x,t)\leq(CA)^{\frac{n}{2p_{0}}}\left(\frac{1+(p_{0}\mu)^{\frac{q}{q-n}}A^{\frac{n}{q-n}}}{t}+\frac{1}{r^{2}}\right)^{\frac{n+2}{2p_{0}}}\left(\int_{0}^{T}\int
f_{j}^{p_{0}}dv_{t}dt\right)^{\frac{1}{p_{0}}}$

for some constant $C$ depending only on $n$ and $q$, where $1<p_{0}<2$.
Passing to the limit $j\rightarrow\infty$, we can see that (2.6) holds when
$1<p<2$. $\hfill\Box$

To proceed we need the following covering lemma belonging to M. Gromov.

Lemma 2.2 ([4], Proposition 3.11). Let $(M,g)$ be a complete Riemannian
manifold, the Ricci curvature of $M$ satisfy ${\rm Ric}(g)\geq(n-1)H$. Then
given $r,\epsilon>0$ and $p\in M$, there exists a covering,
$B_{r}(p)\subset\cup_{i=1}^{N}B_{\epsilon}(p_{i})$, ($p_{i}$ in $B_{r}(p)$)
with $N\leq N_{1}(n,Hr^{2},r/\epsilon)$. Moreover, the multiplicity of this
covering is at most $N_{2}(n,Hr^{2})$.

For any complete Riemannian manifold $(M,g_{0})$ of dimension $n$ with $|{\rm
Ric}(g_{0})|\leq K$, it follows from Lemma 2.2 that there exists an absolute
constant $N$ depending only on $K$ and $n$ such that

$B_{2r}(x)\subset\cup_{i=1}^{N}B_{r}(y_{i}),\quad y_{i}\in
B_{\frac{3}{2}r}(x).$ (2.12)

Suppose (2.1) and (2.2) hold for all $x\in M$ and $0\leq t\leq T$,
$g(0)=g_{0}$. Let $f(x,t)$ and $u(x,t)$ be two nonnegative functions
satisfying

$\frac{\partial f}{\partial t}\leq\Delta f+C_{0}f^{2},\quad\frac{\partial
u}{\partial t}\leq\Delta u+C_{0}fu$

on $M\times[0,T]$. Assume that there hold on $M\times[0,T]$

$u\leq c(n)f,\quad\frac{\partial}{\partial t}dv_{t}\leq c(n)fdv_{t}.$

Define

$e_{0}(t)=\sup_{x\in M,\,0\leq\tau\leq
t}\left(\int_{B_{r/2}(x)}f^{\frac{n}{2}}dv_{\tau}\right)^{{2}/{n}}.$ (2.13)

Then we have the following proposition of $f$ and $u$.

Proposition 2.3. Let $f$ and $u$ be as above, $A$ be given by (2.1) and
$e_{0}(t)$ be defined by (2.13). Suppose there holds for all $x\in M$

$\left(\int_{B_{r/2}(x)}f_{0}^{\frac{n}{2}}dv_{0}\right)^{\frac{2}{n}}\leq(2N^{1+\frac{2}{n}}n(C_{0}+c(n))A)^{-1},$

where $N=N(n,K)$ is given by (2.12), $f_{0}(x)=f(x,0)$ and
$dv_{0}=dv_{g_{0}}$. Then there exist two constants $C_{1}$ and $C_{2}$
depending only on $n$ and $C_{0}$ such that if $0<t<\min(T,C_{2}N^{-1}r^{2})$,
then $f(x,t)\leq C_{1}t^{-1}$ and

$\displaystyle u(x,t)\leq
C_{1}A^{\frac{n}{n+2}}t^{-\frac{n}{n+2}}\left[\left(\int_{B_{r}(x)}u_{0}^{\frac{n+2}{2}}dv_{0}\right)^{\frac{2}{n+2}}+{r^{-\frac{4}{n+2}}}e_{0}(t)\right].$

Proof. Let $[0,T^{\prime}]\subset[0,T]$ be the maximal interval such that

$e_{0}(T^{\prime})=\sup_{x\in M,\,0\leq t\leq
T^{\prime}}\left(\int_{B_{r/2}(x)}f^{\frac{n}{2}}dv_{t}\right)^{\frac{2}{n}}\leq((C_{0}+c(n))nNA)^{-1}.$
(2.14)

For any cut-off function $\phi$ supported in $B_{r}(x)$, using the same method
of deriving (2.7), we calculate when $p\leq n$ and $m\leq n$,

$\displaystyle\frac{1}{p}\frac{\partial}{\partial t}\int\phi^{m+2}f^{p}dv_{t}$
$\displaystyle\leq$ $\displaystyle\int\phi^{m+2}f^{p-1}(\Delta
f+C_{0}f^{2})dv_{t}+\frac{c(n)}{p}\int\phi^{m+2}f^{p+1}dv_{t}$
$\displaystyle\leq$ $\displaystyle-\int\nabla(\phi^{m+2}f^{p-1})\nabla
fdv_{t}+\left(C_{0}+\frac{c(n)}{p}\right)$
$\displaystyle\quad\times\left(\int_{B_{2r}(x)}f^{\frac{n}{2}}dv_{t}\right)^{\frac{2}{n}}\left(\int(\phi^{m+2}f^{p})^{\frac{n}{n-2}}dv_{t}\right)^{\frac{n-2}{n}}$
$\displaystyle\leq$
$\displaystyle-\frac{2}{p}\int|\nabla(\phi^{\frac{m}{2}+1}f^{\frac{p}{2}})|^{2}dv_{t}+\frac{2}{p}\int|\nabla\phi^{\frac{m}{2}+1}|^{2}f^{p}dv_{t}$
$\displaystyle+\left(C_{0}+\frac{c(n)}{p}\right)Ne_{0}A\int|\nabla(\phi^{\frac{m}{2}+1}f^{\frac{p}{2}})|^{2}dv_{t}.$
$\displaystyle\leq$
$\displaystyle-\frac{1}{p}\int|\nabla(\phi^{\frac{m}{2}+1}f^{\frac{p}{2}})|^{2}dv_{t}+\frac{(m+2)^{2}}{2p}|\nabla\phi|_{\infty}^{2}\int\phi^{m}f^{p}dv_{t}.$

Here in the second and third inequalities we used (2.12) and the Sobolev
inequality. Hence

$\frac{\partial}{\partial
t}\int\phi^{m+2}f^{p}dv_{t}+\int|\nabla(\phi^{\frac{m}{2}+1}f^{\frac{p}{2}})|^{2}dv_{t}\leq\frac{(m+2)^{2}}{2}|\nabla\phi|_{\infty}^{2}\int\phi^{m}f^{p}dv_{t}.$
(2.15)

Take $\phi$ supported in $B_{r}(x)$, which is 1 on $B_{r/2}(x)$ and
$|\nabla_{g_{0}}\phi|_{\infty}^{2}\leq 5/r^{2}$. Since
$\frac{1}{2}g_{ij}(0)\leq g_{ij}(t)\leq 2g_{ij}(0)$, we have
$|\nabla_{g(t)}\phi|_{\infty}^{2}\leq 10/r^{2}$. Taking $p=\frac{n}{2}$ in
(2.15) and integrating it from $0$ to $t$, we obtain by using (2.12) again

$\displaystyle\int_{B_{r/2}(x)}f^{\frac{n}{2}}dv_{t}{}$ $\displaystyle\leq$
$\displaystyle\int_{B_{r}(x)}f_{0}^{\frac{n}{2}}dv_{0}+\frac{2(m+2)^{2}}{r^{2}}\int_{0}^{t}\int\phi^{m}f^{\frac{n}{2}}dv_{t}dt$
(2.16) $\displaystyle\leq$ $\displaystyle
N\left(2N^{1+\frac{2}{n}}n(C_{0}+c(n))A\right)^{-\frac{n}{2}}+{2(m+2)^{2}}{r^{-2}}N(e_{0}(t))^{\frac{n}{2}}t.$

Noting that $x$ is arbitrary, one concludes

$\left(1-{2(m+2)^{2}}{r^{-2}}Nt\right)(e_{0}(t))^{\frac{n}{2}}\leq
N\left(2N^{1+\frac{2}{n}}n(C_{0}+c(n))A\right)^{-\frac{n}{2}}.$

If $T^{\prime}<\frac{r^{2}}{8(m+2)^{2}N}$, then for all $t\in[0,T^{\prime}]$

$e_{0}(t)<\left(\frac{4}{3}\right)^{{2}/{n}}\left(2Nn(C_{0}+c(n))A\right)^{-1}.$

This contradicts the maximality of $[0,T^{\prime}]$. We can therefore assume
that $T^{\prime}\geq\min(C_{2}N^{-1}r^{2},T)$.

It follows from (2.15) that

$\displaystyle\frac{\partial}{\partial
t}\left(t\int\phi^{m+2}f^{p}dv_{t}\right)$ $\displaystyle=$ $\displaystyle
t\frac{\partial}{\partial
t}\int\phi^{m+2}f^{p}dv_{t}+\int\phi^{m+2}f^{p}dv_{t}$ $\displaystyle\leq$
$\displaystyle\left(\frac{(m+2)^{2}}{2}|\nabla\phi|_{\infty}^{2}t+1\right)\int\phi^{m}f^{p}dv_{t}.$

When $0\leq t\leq\min(C_{2}N^{-1}r^{2},T)$, integrating the above inequality
from $0$ to $t$, we have

$\displaystyle\int\phi^{m+2}f^{p}dv_{t}$ $\displaystyle\leq$
$\displaystyle\left(\frac{2(m+2)^{2}}{r^{2}}+\frac{1}{t}\right)\int_{0}^{t}\int\phi^{m}f^{p}dv_{t}dt$
(2.17) $\displaystyle\leq$ $\displaystyle
c\,t^{-1}\int_{0}^{t}\int\phi^{m}f^{p}dv_{t}dt$

for some constant $c$ depending only on $n$. Moreover, integrating (2.15) from
$0$ to $t$, we derive

$\int_{0}^{t}\int|\nabla(\phi^{\frac{m}{2}+1}f^{\frac{p}{2}})|^{2}dv_{t}dt\leq\int\phi^{m+2}f_{0}^{p}dv_{0}+\frac{2(m+2)^{2}}{r^{2}}\int_{0}^{t}\int\phi^{m}f^{p}dv_{t}dt.$
(2.18)

Noting that $\frac{1}{r^{2}}\leq\frac{C_{2}}{Nt}$ and $m\leq n$, we calculate
by using (2.17) and (2.18)

$\displaystyle\int_{B_{r/2}(x)}f^{\frac{n}{2}+1}dv_{t}$ $\displaystyle\leq$
$\displaystyle\int_{B_{r}(x)}\phi^{m+4}f^{\frac{n}{2}+1}dv_{t}$
$\displaystyle\leq$ $\displaystyle
Ct^{-1}\int_{0}^{t}\int\phi^{m+2}f^{\frac{n}{2}+1}dv_{t}dt$
$\displaystyle\leq$ $\displaystyle
Ct^{-1}\int_{0}^{t}\left(\int_{B_{r}(x)}f^{\frac{n}{2}}dv_{t}\right)^{\frac{2}{n}}\left(\int(\phi^{m+2}f^{\frac{n}{2}})^{\frac{n}{n-2}}dv_{t}\right)^{\frac{n-2}{n}}dt$
$\displaystyle\leq$ $\displaystyle
Ct^{-1}N^{\frac{2}{n}}e_{0}(t)A\int_{0}^{t}\int|\nabla(\phi^{\frac{m}{2}+1}f^{\frac{n}{4}})|^{2}dv_{t}dt$
$\displaystyle\leq$ $\displaystyle
Ct^{-1}N^{\frac{2}{n}}e_{0}(t)A(N(e_{0}(t))^{\frac{n}{2}}+N(e_{0}(t))^{\frac{n}{2}}t)$
$\displaystyle\leq$ $\displaystyle
CN^{1+\frac{2}{n}}A(e_{0}(t))^{1+\frac{n}{2}}t^{-1},$

or equivalently

$\left(\int_{B_{r/2}(x)}f^{\frac{n+2}{2}}dv_{t}\right)^{\frac{2}{n+2}}\leq
CNA^{\frac{2}{n+2}}e_{0}(t)t^{-\frac{2}{n+2}},$ (2.19)

where $C$ is a constant depending only on $n$, here and in the sequel, we
often denote various constants by the same $C$. Setting $q=n+2$,
$p=\frac{n}{2}$ and $\mu=CNA^{\frac{2}{n+2}}e_{0}(T^{\prime})$, we obtain by
employing Theorem 2.1

$\displaystyle f(x,t)$ $\displaystyle\leq$ $\displaystyle
CA\left(\frac{1+A^{\frac{n}{2}}\mu^{\frac{n+2}{2}}}{t}+\frac{1}{r^{2}}\right)^{\frac{n+2}{n}}\left(\int_{0}^{t}\int_{B_{r}(x)}f^{\frac{n}{2}}dv_{t}dt\right)^{\frac{2}{n}}$
$\displaystyle\leq$ $\displaystyle
CAe_{0}(T^{\prime}){t}^{\frac{2}{n}}\left(\frac{1+A^{\frac{n}{2}}\mu^{\frac{n+2}{2}}}{t}+\frac{1}{r^{2}}\right)^{\frac{n+2}{n}}$

for $t\in[0,T^{\prime}]$. Recalling the definition of $e_{0}(T^{\prime})$ (see
(2.14) above), we can see that $Ae_{0}(T^{\prime})$ is bounded and

$A^{\frac{n}{2}}\mu^{\frac{n+2}{2}}=(CNAe_{0}(T^{\prime}))^{\frac{n+2}{2}}$
(2.20)

is also bounded. Therefore when $0<t<\min(T,C_{2}N^{-1}r^{2})$, $f(x,t)\leq
C_{1}t^{-1}$ for some constants $C_{1}$ and $C_{2}$ depending only on $n$,
$C_{0}$.

Using $u\leq c(n)f$ and $\partial_{t}dv_{t}\leq c(n)fdv_{t}$ and mimicking the
method of proving (2.15), we obtain

$\frac{\partial}{\partial
t}\int\phi^{m+2}u^{p}dv_{t}+\int|\nabla(\phi^{\frac{m}{2}+1}u^{\frac{p}{2}})|^{2}dv_{t}\leq\frac{C}{r^{2}}\int\phi^{m}u^{p}dv_{t}.$
(2.21)

Taking $m=0$, $p=n/2$ and integrating this inequality, we have by using
(2.12))

$\int_{0}^{t}\int|\nabla(\phi
u^{\frac{n}{4}})|^{2}dv_{t}dt\leq\int_{B_{r}(x)}u_{0}^{\frac{n}{2}}dv_{0}+\frac{C}{r^{2}}N(e_{0}(t))^{\frac{n}{2}}t.$
(2.22)

Integrating (2.21) with $m=2$, $p={(n+2)}/{2}$, and using the Sobolev
inequality (2.1), we obtain

$\displaystyle\int_{B_{r/2}(x)}u^{\frac{n+2}{2}}dv_{t}$ $\displaystyle\leq$
$\displaystyle\int_{B_{r}(x)}u_{0}^{\frac{n+2}{2}}dv_{0}+\frac{C}{r^{2}}\int_{0}^{t}\int\phi^{2}u^{\frac{n+2}{2}}dv_{t}dt$
$\displaystyle\leq$
$\displaystyle\int_{B_{r}(x)}u_{0}^{\frac{n+2}{2}}dv_{0}+\frac{C}{r^{2}}e_{0}(t)A\int_{0}^{t}\int|\nabla(\phi
u^{\frac{n}{4}})|^{2}dv_{t}dt,$

which together with (2.22) and (2.12) gives

$\displaystyle\int_{B_{r/2}(x)}u^{\frac{n+2}{2}}dv_{t}$ $\displaystyle\leq$
$\displaystyle\int_{B_{r}(x)}u_{0}^{\frac{n+2}{2}}dv_{0}+\frac{C}{r^{2}}e_{0}(t)A\left(\int_{B_{r}(x)}u_{0}^{\frac{n}{2}}dv_{0}+\frac{C}{r^{2}}Ne_{0}(t)^{\frac{n}{2}}t\right)$
(2.23) $\displaystyle\leq$
$\displaystyle\int_{B_{r}(x)}u_{0}^{\frac{n+2}{2}}dv_{0}+\frac{C}{r^{2}}NA(e_{0}(t))^{1+\frac{n}{2}}\left(1+\frac{1}{r^{2}}t\right).$

Notice that when $0\leq t\leq\min(C_{2}r^{2}/N,T)$, (2.19) implies

$\int_{B_{r/2}(x)}f^{\frac{n+2}{2}}dv_{t}\leq\mu t^{-1}.$

Without loss of generality we can assume $A>1$ (otherwise we can substitute
$A$ for $A+1$). In view of (2.20) and (2.23), we obtain by using Theorem 2.1
in the case $q=n+2$ and $p=(n+2)/2$

$\displaystyle u(x,t)$ $\displaystyle\leq$ $\displaystyle
CA^{\frac{n}{n+2}}\left(\frac{1}{t}+\frac{1}{r^{2}}\right)\left(\int_{0}^{t}\int_{B_{r/2}(x)}u^{\frac{n+2}{2}}dv_{t}dt\right)^{\frac{2}{n+2}}$
$\displaystyle\leq$ $\displaystyle
CA^{\frac{n}{n+2}}t^{-\frac{n}{n+2}}\left[\left(\int_{B_{r}(x)}u_{0}^{\frac{n+2}{2}}dv_{0}\right)^{\frac{2}{n+2}}+{r^{-\frac{4}{n+2}}}e_{0}(t)\right],$

provided that $0\leq t\leq\min(C_{2}r^{2}/N,T)$. $\hfill\Box$

Remark 2.4. We remark that Theorem 2.1 and Proposition 2.3 are very similar to
Theorem A.1 and Corollary A.10 of Dean Yang’s paper [17] respectively. The
differences are that we have heat flow type inequalities, but Dean Yang has
heat flow type inequalities with cut-off function. It seems that Dean Yang’s
Corollary A.10 is stronger than our Proposition 2.3, which is enough for our
use here. Also we should compare Theorem 2.1 with ([6, 7], Theorem 2.1), where
Dai-Wei-Ye obtained a similar result by using a similar method. Here the
constant $C$ of (2.6) depends only on $n$, $q$, $p$, but not on the Sobolev
constant $A$. While in [6, 7], since the Sobolev constants $C_{S}(t)$ along
the flow are bounded, they need not care how the constant $C$ exactly depends
on $C_{S}$.

##  3 Short time existence of the Ricci flow

In this section we focus on closed Riemannian manifolds. Precisely, following
the lines of [13, 18], we study the short time existence of the Ricci flow and
give the proof of Theorem 1.1. Assume $(M,g_{0})$ is a closed Riemannian
manifold of dimension $n(\geq 3)$ with $|{\rm Ric}(g_{0})|\leq K$. Consider
the Ricci flow

$\left\\{\begin{array}[]{lll}\displaystyle\frac{\partial g}{\partial
t}&=-2Ric(g),\\\\[5.16663pt] g(0)&=g_{0}.\end{array}\right.$ (3.1)

It is well known [10] that the Riemannian curvature tensor and the Ricci
curvature tensor satisfy the following evolution equations

$\displaystyle\frac{\partial{\rm Rm}}{\partial t}$ $\displaystyle=$
$\displaystyle\Delta{\rm Rm}+{\rm Rm}*{\rm Rm},$ (3.2)
$\displaystyle\frac{\partial{\rm Ric}}{\partial t}$ $\displaystyle=$
$\displaystyle\Delta{\rm Ric}+{\rm Rm}*{\rm Ric},$ (3.3)

where ${\rm Rm}*{\rm Rm}$ is a tensor that is quadratic in ${\rm Rm}$, ${\rm
Ric}*{\rm Rm}$ can be understood in a similar way. It follows that

$\displaystyle\frac{\partial|{\rm Rm}|}{\partial t}$ $\displaystyle\leq$
$\displaystyle\Delta|{\rm Rm}|+c(n)|{\rm Rm}|^{2},$ (3.4)
$\displaystyle{}\frac{\partial|{\rm Ric}|}{\partial t}$ $\displaystyle\leq$
$\displaystyle\Delta|{\rm Ric}|+c(n)|{\rm Rm}||{\rm Ric}|.$ (3.5)

To prove Theorem 1.1, it suffices to prove the following:

Proposition 3.1. Let $(M,g_{0})$ be a closed Riemannian manifold of dimension
$n(\geq 3)$ with $|{\rm Ric}(g_{0})|\leq K$. Suppose there exists a constant
$A_{0}>0$ such that the following local Sobolev inequalities hold for all
$x\in M$

$\|u\|_{{2n}/{(n-2)}}^{2}\leq A_{0}\|\nabla u\|_{2}^{2},\quad\forall u\in
C_{0}^{\infty}(B_{r}(x)).$

Then there exist constants $C_{1}$, $C_{3}$ depending only on $n$ and $K$, and
$C_{2}$ depending only on $n$ such that for $r\leq 1$, if

$\left(\int_{B_{r/2}(x)}|{\rm
Rm}(g_{0})|^{\frac{n}{2}}dv_{g_{0}}\right)^{2/n}\leq(C_{1}A_{0})^{-1}$

for all $x\in M$, then the Ricci flow (3.1) has a smooth solution for $0\leq
t\leq T$, where $T\geq C_{2}\min(r^{2}/N,K^{-1})$, such that for all $x\in M$

$\displaystyle\frac{1}{2}g_{0}\,\leq\,g(t)\,\leq 2g_{0},$ (3.8)
$\displaystyle\|u\|_{{2n}/{(n-2)}}^{2}\leq 4A_{0}\|\nabla
u\|_{2}^{2},\quad\forall u\in C_{0}^{\infty}(B_{r}(x)),$
$\displaystyle\left(\int_{B_{r/2}(x)}|{\rm
Rm}(g(t))|^{\frac{n}{2}}dv_{t}\right)^{2/n}\leq 2N(C_{1}A_{0})^{-1}.$

Proof. It is well known (see for example [8, 10]) that a smooth solution
$g(t)$ of the Ricci flow (3.1) exists for a short time interval and is unique.
Let $[0,T_{\rm max})$ be a maximum time interval on which $g(t)$ exists and
(3.8)-(3.8) hold. Clearly $T_{\rm max}>0$ since the strict inequalities in
(3.8)-(3.8) hold at $t=0$. Suppose $T_{\rm
max}<T_{0}=C_{2}\min(r^{2}/N,K^{-1})$ for some constant $C_{2}$ to be
determined later. Since the Ricci curvature satisfies (3.5), it follows from
Proposition 2.3 that for $0\leq t\leq T^{\prime}$,

$\displaystyle{}|{\rm Ric}(g(t))|$ $\displaystyle\leq$ $\displaystyle
CA_{0}^{\frac{n}{n+2}}t^{-\frac{n}{n+2}}\left[\left(\int_{B_{r}(x)}|{\rm
Ric}(g_{0})|^{\frac{n+2}{2}}dv_{0}\right)^{\frac{2}{n+2}}+{r^{-\frac{4}{n+2}}}e_{0}(T^{\prime})\right]$
$\displaystyle\leq$ $\displaystyle
CA_{0}^{\frac{n}{n+2}}t^{-\frac{n}{n+2}}\left(K^{\frac{2}{n+2}}(e_{0}(T^{\prime}))^{\frac{n}{n+2}}+{r^{-\frac{4}{n+2}}}e_{0}(T^{\prime})\right)$
$\displaystyle\leq$ $\displaystyle
C(K^{\frac{2}{n+2}}+r^{-\frac{4}{n+2}})t^{-\frac{n}{n+2}},$ (3.9)

where $T^{\prime}$ and $e_{0}(T^{\prime})$ are defined by (2.14) in the case
$f$ is replaced by $|{\rm Rm}|$. It follows that for all $x\in M$, $u\in
C_{0}^{\infty}(B_{r}(x))$ and $0\leq t\leq T^{\prime}$,

$\displaystyle\left|\frac{d}{dt}\int_{B_{r}(x)}|u|^{\frac{2n}{n-2}}dv_{t}\right|$
$\displaystyle\leq$ $\displaystyle 2|{\rm
Ric}(g(t))|_{\infty}\int_{B_{r}(x)}|u|^{\frac{2n}{n-2}}dv_{t}$
$\displaystyle\leq$ $\displaystyle
Ct^{-\frac{n}{n+2}}\int_{B_{r}(x)}|u|^{\frac{2n}{n-2}}dv_{t}.$

This implies

$e^{-C\,t^{\frac{2}{n+2}}}\int_{B_{r}(x)}|u|^{\frac{2n}{n-2}}dv_{0}\leq\int_{B_{r}(x)}|u|^{\frac{2n}{n-2}}dv_{t}\leq
e^{C\,t^{\frac{2}{n+2}}}\int_{B_{r}(x)}|u|^{\frac{2n}{n-2}}dv_{0}.$

Similarly we have

$\left|\frac{d}{dt}\int_{B_{r}(x)}|\nabla u|^{2}dv_{t}\right|\leq
Ct^{-\frac{n}{n+2}}\int_{B_{r}(x)}|\nabla u|^{2}dv_{t},$

and

$e^{-C\,t^{\frac{2}{n+2}}}\int_{B_{r}(x)}|\nabla
u|^{2}dv_{0}\leq\int_{B_{r}(x)}|\nabla u|^{2}dv_{t}\leq
e^{C\,t^{\frac{2}{n+2}}}\int_{B_{r}(x)}|\nabla u|^{2}dv_{0}.$

Hence if $T_{\rm max}<T_{0}=C_{2}\min(r^{2}/N,K^{-1})$ for sufficiently small
$C_{2}$ depending only on $n$ and $K$, then (3.8) holds with strict
inequality.

To show (3.8) holds with strict inequality, we fix a tangent vector $v$ and
calculate

$\displaystyle\frac{d}{dt}|v|_{g(t)}^{2}=\frac{d}{dt}(g_{ij}(t)v^{i}v^{j})=-2{\rm
Ric}_{ij}v^{i}v^{j},$

which together with (3.9) gives

$\left|\frac{d}{dt}\log|v|_{g(t)}^{2}\right|\leq
C(K^{\frac{2}{n+2}}+r^{-\frac{4}{n+2}})t^{-\frac{n}{n+2}}.$

Therefore we obtain for $0\leq t<C_{2}\min(r^{2},K^{-1})$,

$\frac{1}{2}|v|_{g(0)}^{2}<|v|_{g(t)}^{2}<2|v|_{g(0)}^{2}.$

Using the same method of deriving (2.16), one can see that the strict
inequality in (3.8) holds when $0\leq t<C_{2}\min(r^{2},K^{-1})$ for
sufficiently small $C_{2}$. By Proposition 2.3, $|{\rm Rm}(g(t))|_{\infty}\leq
Ct^{-1}$ for all $t\in[0,T_{\rm max}]$. Hence one can extend $g(t)$ smoothly
beyond $T_{\rm max}$ with (3.8)-(3.8) still holding. This contradicts the
assumed maximality of $T_{\rm max}$. Therefore $T_{\rm max}\geq T_{0}$.
$\hfill\Box$

Proof of Theorem 1.1. By Proposition 3.1, there exists a unique solution
$g(t)$ of the Ricci flow (3.1) such that (3.8)-(3.8) hold. Then by Proposition
2.3, one concludes

$|{\rm Rm}(g(t))|\leq Ct^{-1},\quad|{\rm Ric}(g(t))|\leq Ct^{-\frac{n}{n+2}}$

for $t\in[0,T_{0}]$. This completes the proof of Theorem 1.1. $\hfill\Box$

##  4 Applications

In this section, we will prove Theorem 1.2 by applying Theorem 1.1. It follows
from (1.4)-(1.6) that the deformed metric $g(t)$ has uniform sectional
curvature bounds away from $t=0$ and $g(t)$ is close to $g(0)$ when $t$ is
close to $0$. We first show that diameters of the flow are under control,
namely

Lemma 4.1. Let $g(t)$ be the Ricci flow in Theorem 1.1. Then for $0\leq t\leq
c_{1}\min(r^{2},K^{-1})$, there exists a constant $c$ depending only on $n$
and $K$ such that

$e^{-ct^{\frac{2}{n+2}}}{\rm diam}(g_{0})\leq{\rm diam}(g(t))\leq
e^{ct^{\frac{2}{n+2}}}{\rm diam}(g_{0}).$ (4.1)

where ${\rm diam}(g(t))$ means the diameter of the manifold $(M,g(t))$.

Proof. Let $\gamma:[0,1]\rightarrow M$ be any smooth curve. Denote the length
of $\gamma$ by

$l_{\gamma}(t)=\int_{0}^{1}|\dot{\gamma}(s)|_{g(t)}^{2}ds.$

We calculate by using the Ricci bound in Theorem 1.2

$\displaystyle\left|\frac{d}{dt}l_{\gamma}(t)\right|=\left|\int_{0}^{1}-2{\rm
Ric}_{g(t)}(\dot{\gamma}(s),\dot{\gamma}(s))ds\right|\leq
ct^{-\frac{n}{n+2}}l_{\gamma}(t).$

This implies

$l_{\gamma}(0)e^{-ct^{\frac{2}{n+2}}}\leq l_{\gamma}(t)\leq
l_{\gamma}(0)e^{ct^{\frac{2}{n+2}}}.$

It follows that

$e^{-ct^{\frac{2}{n+2}}}{\rm dist}_{g_{0}}(p,q)\leq{\rm dist}_{g(t)}(p,q)\leq
e^{ct^{\frac{2}{n+2}}}{\rm dist}_{g_{0}}(p,q),$

where ${\rm dist}_{g(t)}(p,q)$ denote the distance between $p$ and $q$ in the
metric $g(t)$. This gives the desired result. $\hfill\Box$

The following proposition is a corollary of Gromov’s almost flat manifold
theorem [9]:

Proposition 4.2 (Gromov). Let $(M,g)$ be a compact Riemannian manifold of
dimension $n$. Assume the sectional curvature is bounded, i.e., $|{\rm
Sec}(g)|\leq\Lambda$. Then there exists a constant $\epsilon_{0}$ depending
only on $n$ such that if

$\Lambda({\rm diam}(g))^{2}\leq\epsilon_{0},$ (4.2)

then the universal covering of $(M,g)$ is diffeomorphic to $\mathbb{R}^{n}$.
If in addition the fundamental group $\pi(M)$ is commutative, then $(M,g)$ is
diffeomorphic to a torus.

Proof of Theorem 1.2. Let $g(t)$ be a unique solution to the Ricci flow (1.3).
By (1.5), for $0\leq t\leq c_{1}\min(r^{2},K^{-1})$,

$|{\rm Sec}(g(t))|\leq ct^{-1},$

where ${\rm Sec}(g(t))$ denotes the sectional curvature of $(M,g(t))$. Let
$\epsilon_{0}$ be given by Proposition 4.2. Take
$t_{0}=c_{1}\min(r^{2},K^{-1})$ and

$\delta=\left(\epsilon_{0}t_{0}c^{-1}e^{-2ct_{0}^{\frac{2}{n+2}}}\right)^{1/2}.$

If ${\rm diam}(g_{0})\leq\delta$, then we obtain by Lemma 4.1

$|{\rm Sec}(g(t_{0}))({\rm diam}(g(t_{0})))^{2}|\leq
ct_{0}^{-1}e^{2ct_{0}^{\frac{2}{n+2}}}({\rm
diam}(g_{0}))^{2}\leq\epsilon_{0}.$

Applying Proposition 4.2 to $g(t_{0})$, we conclude Theorem 1.2. $\hfill\Box$

Acknowledgements. The author is partly supported by the program for NCET. He
thanks Ye Li for introducing this interesting topic to him. Also he thanks the
referee for valuable comments and suggestions, which improve this paper.

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# Nanomechanical Photothermal Near Infrared Spectromicroscopy of Individual
Nanorods

Kostas Kanellopulos  Robert G. West  Silvan Schmid<EMAIL_ADDRESS>[

###### Abstract

Understanding light-matter interaction at the nanoscale requires probing the
optical properties of matter at the individual nano-absorber level. To this
end, we have developed a nanomechanical photothermal sensing platform that can
be used as a full spectromicroscopy tool for single molecule and single
particle analysis. As a demonstration, the absorption cross-section of
individual gold nanorods is resolved from the spectroscopic and polarization
standpoint. By exploiting the capabilities of nanomechanical photothermal
spectromicroscopy, the longitudinal localized surface plasmon resonance (LSPR)
in the NIR range is unravelled and quantitatively characterized. The
polarization features of the transversal surface plasmon resonance (TSPR) in
the VIS range are also analyzed. The measurements are compared with the finite
element method (FEM), elucidating the role played by electron-surface and bulk
scattering in these plasmonic nanostructures, as well as the interaction
between the nano-absorber and the nanoresonator, ultimately resulting in
absorption strength modulation. Finally, a comprehensive comparison is
conducted, evaluating the signal-to-noise ratio of nanomechanical photothermal
spectromicroscopy against other cutting-edge single molecule and particle
spectroscopy techniques. This analysis highlights the remarkable potential of
nanomechanical photothermal spectromicroscopy due to its exceptional
sensitivity.

###### keywords:

nanomechanics, photothermal microscopy, absorption spectroscopy, polarization-
resolved absorption, single molecule spectroscopy

TU Wien] Institute of Sensor and Actuator Systems, TU Wien, Gusshausstrasse
27-29, 1040 Vienna, Austria. NEMS, NIR, FEM, SPR

##  1 Introduction

The advent and development of optical single-molecule and single-particle
measurement techniques have had a tremendous impact on the scientific research
for the past 30 years 1. The detection of single objects at the nanoscale
level affords us a unique perspective on the interactions occurring between
these tiny entities and their local environment, revealing their heterogeneity
without relying on ensemble average information. Due to its high signal-to-
noise ratio (SNR), optical fluorescence-based detection approaches have
rapidly evolved, being nowadays routinely employed in a huge variety of
scientific fields, from biology to condensed matter, to the design and
engineering of novel materials. However, the fluorescent label can photobleach
or quench as well as alter the system under study. For this reason, the
scientific community has pushed toward the development of label-free single-
molecule detection schemes 2, such as iSCAT 3, 4, ground-state depletion
microscopy 5 and photothermal microscopy 6, 7, among others. All these
approaches rely on the absorption rather than scattering of the nano-object
upon illumination of a probing light. The rationale behind this choice lies in
the fact that, while the optical scattering cross-section scales quadratically
with the target volume ($\sigma_{scatt}$ $\propto$ $V^{2}$), the optical
absorption cross-section scales linearly with it ($\sigma_{abs}$ $\propto$
$V$) 8, 9. In other words, the smaller the target size is, the more
effectively its absorption properties can be interrogated. More specifically,
this central aspect shows also the advantage of nanomechanical absorption
spectroscopy over other fully-optical single-molecule spectroscopic methods,
such as surface-enhanced Raman scattering (SERS) 10 or tip-enhanced Raman
spectroscopy (TERS) 11. The former approach uses the strong near-field
enhancement at the nanoscale on the surface of plasmonic nanoparticles or
nanostructures (so-called hotspots) to increase the SNR of the Stokes-shift
Raman scattering. However, the plasmonic nanostructure fabrication and control
of the placement of the particles/molecules in the sites of interest increase
the overall complexity of the measurement procedure. The latter approach uses
a nanoscopic probe to scan an area where the molecules of interest are fixed,
which requires a very good control in the tip fabrication. In contrast,
nanomechanical absorption spectroscopy overcomes all this complexity as it
measures directly the non-radiative energy losses of the illuminated molecule,
not limited to any specific sample preparation. In other words, the molecule
itself becomes part of the detector, due to the interplay between its
absorption properties and the light excitation used. The reduced analyte-
detector distance results also in a reduction of noise and unwanted external
interference. Based on this consideration, it has been possible to detect and
image single molecules by nanomechanical photothermal sensing.7 This work has
been made possible by the previous research, which showed the ability of
nanomechanical resonators to detect and quantify the absorption of single
plasmonic 12, 13, 14 and polymer 15 nanoparticles via photothermal heating. It
is worth noting that this photothermal spectromicroscopy approach is not based
on the thermo-optical effect as in photothermal contrast microscopy 16, 17,
18, 19, 6, where the absorber is detected due to the temperature dependence of
the surrounding embedding medium refractive index (glycerol, thermotropic
liquid crystal, near-critical $Xe$ or $CO_{2}$) 6, 20 via modulation of the
scattering of a second probing laser. In nanomechanical photothermal
spectromicroscopy, this thermal effect consists instead in a stress reduction
in the nanomechanical resonator, detuning its resonance frequency upon
illumination of the nano-absorber.

Here, this work pushes further the boundaries of single-molecule
nanomechanical photothermal sensing toward a full NIR spectro- and
polarization-microscopy technique. With a silicon nitride nano-optomechanical
drum resonator as a sensitive thermometer, individual gold nanorods are
localized and their spectra and polarization features fully characterized,
additionally shedding light on their interaction with the nanoresonator
itself. Among the huge variety of nanoparticle shapes and materials, gold
nanorods occupy a relevant position in gas- and liquid-phase chemical
detection, so as sensing platform for biomolecules 21, 22, or as photothermal
heating sources 23. In this work, the plasmonic properties of such nano-
objects are analyzed and their corresponding plasmonic damping mechanisms
unravelled, showing also a good agreement with finite element method (FEM)
simulation results. The performance of our approach is then compared with the
other state-of-the-art single molecule and particle techniques in terms of
normalized SNR, showing the capabilities offered by nanomechanical
photothermal spectromicroscopy with its superior SNR.

##  2 Experimental methods

Figure 1: Schematics of the operating set-up. The drum resonator is actuated
in vacuum ($p$ $<$ $10^{-5}$ $mbar$) by a piezoshaker. The displacement is
read out by a Fabry-Perot interferometer (blue laser). The interference signal
is processed and sent to the lock-in amplifier which records the frequency.
The scanning lasers (red and green) are used to generate the photothermal
signal by plasmonic excitation of the nanorod. BE: beam expander. BS: beam
splitter. CCD: charge-coupled device camera. DM: dichroic mirror. I: iris. L:
lens. LP: linear polarizer. HL: halogen lamp. M: mirror. WP: waveplate.

The 50 nm thick nano-optomechanical drum resonator is operated at room
temperature under high vacuum conditions ($p$ < $10^{-5}$ mbar) in order to
reduce air damping and eliminate heat dissipation by convection 7. It is
actuated by a piezo-electric element placed underneath the sample holder. The
nanoresonator’s displacement is measured with a Fabry-Perot interferometer
(Attocube IDS3010) 24, 25 (see Fig. 1). The mechanical resonance frequency is
recorded with a phase-locked loop (PLL) tracking scheme (HF2LI, Zurich
Instrument).

The operating set-up is equipped with a green laser at 513 nm (Toptica
TopMode) and a Ti:Sapphire laser (M Square SolsTiS) with a tunable wavelength
in the range of 700-1000 nm, used as probe beams to photothermally excite both
the longitudinal (LSPR) and transverse plasmonic resonances (TSPR) of each
nanorod. In fact, every time one of the two lasers is scanned across the
central area of the drum and hits the sample, the corresponding light
absorption results in a local heating, reducing the stress of the
nanoresonator and ultimately resulting in a detectable resonance frequency
detuning 7, 12, 15.

For the scanning laser probes, long working distance 50x objectives are used
(N.A. = 0.42, M Plan Apo NIR, Mitutoyo in the NIR range, N.A. = 0.55, M Plan
Apo, Mitutoyo in the VIS range). The lasers’ polarization angle is controlled
by means of a linear polarizer in the optical beam path. Raster scanning is
made possible by a closed-loop piezoelectric nanopositioning stage (PiMars,
Physikinstrumente). The analyte sampling is performed by spin-casting onto the
resonator a drop of diluted solution containing the nanorods, after being
filtered with 200 nm pore size PTFE membrane syringe filters (Acrodisc, Sigma-
Aldrich) to avoid particle aggregation. As already shown 7, NEMS-based
photothermal sensing enables the measurement of pure optical absorption of the
sample, resulting in the full characterization of its absorption cross-section

$\sigma_{abs}(\lambda)=\frac{P_{abs}}{I_{0}},$ (1)

with $I_{0}$ being the peak irradiance of a gaussian laser beam and $P_{abs}$
the absorbed power by the sample. The former is a function of the input laser
power $P_{0}$

$I_{0}=\frac{2P_{0}}{\pi r^{2}},$ (2)

with $r$ being the beam radius, which is always characterized by knife-edge
method before each measurement 26, 27. The latter can be calculated from the
measured resonance frequency shifts, assuming a full thermalization, as

$P_{abs}=\frac{\Delta f}{f_{0}\ R_{P}},$ (3)

$f_{0}$ being the resonance frequency, $\Delta f$ the frequency shift, and
$R_{P}$ the relative power responsivity ($W^{-1}$).

##  3 Results and discussion

The nanorods analyzed here have lengths in the range of ca. 38-48 nm, radial
diameters in the range of 9.5-11.5 nm and silica coating with a thickness in
the range of 18-22 nm (Sigma-Aldrich silica coated gold nanorods) (Fig. 2a,
inset - the majority of the nanorods have been SEM imaged by deposition of 10
nm thick gold layer on top of them to reduce any possible charging effect).
Their optical properties in the visible and near-infrared range are
characterized by surface plasmon resonances (SPR), i.e. the electromagnetic
coupling between an impinging light and the collective motion of the
conduction band electrons 8, 28. The main interest in gold nanorods lies in
their large SPR amplitudes and broad spectral tunability 29, 30, 31, 32, 33,
34, 35, 36, 37, 38. In the specific case where a coating is present, depending
on its thickness, SPR features will be more or less affected by the
environment. Indeed, the plasmonic response is sensitive to the surrounding on
the spatial range of the order of the nanorod diameter, the region where the
field enhancement takes place 33, 39, 40, 41 (Supporting Information Fig. S1).
In the present study, a silica coating of roughly 20 nm is thin enough for the
SPR to remain sensitive to both the coating and the surrounding medium, but
within a reduced magnitude 40.

(a)

(b)

Figure 2: (a) Measured absorption cross-sections spectra of single nanorods
(Au-SNR, red curves) and small nanorods aggregations (Au-ANR, blue curves),
showing the heterogeneity characterizing these samples, mainly caused by the
particle size dispersion. Inset: SEM micrograph of a single silica-coated gold
nanorod landing on the drum resonator. (b) Red curve and dot: renormalized sum
of the measured absorption cross-section spectra of single nanorods (fig. 2a);
blue curve: renormalized sum of the absorption cross-sections of the nanorods
aggregation; black curve: ensemble average absorption spectrum given by the
datasheet.

Fig. 2a shows the measured absorption spectra of different, individual silica-
coated gold nanorods (Au-SNR) indicated by the red curves, together with some
aggregations of few units (Au-ANR) indicated by the blue curves.
Differentiating these two photothermal responses is aided by FEM
electromagnetic simulations (for an overview on how aggregations can be
differentiate from individual nanorods, see Supporting Information). For each
spectrum, the polarization of the probe beam (here Ti:Sapphire laser) was set
to maximize the absorption in the wavelength range of 700-900 nm. In fact, the
nanorods analyzed here present maximum absorption in the range of ca. 790-830
nm (see Fig.2b, black solid curve), due to LSPR excitation whenever the laser
polarization is parallel to their long axis. With the nanomechanical
photothermal technique, nanoparticle heterogeneity can be investigated,
revealing more than the information obtained in ensemble average measurements.
Here, the heterogeneity is mainly due to the size dispersion of the particles,
as stated by the vendor 42, and it relates both to the LSPR spectral position
$\lambda_{LSPR}$ and the absorption cross-section amplitude
$\sigma_{abs}(\lambda_{LSPR})$. Moreover, the latter has a strong dependence
on the substrate, specifically on its thickness and optical properties (see
Fig. 3; further details in the Supporting Information).

The red solid line in Fig. 2b indicates the corresponding renormalized sum of
the responses from individual nanorods (Au-SNR in Fig. 2a), showing a very
good match with the reference spectrum given in the datasheet (black solid
line), recovering a typical ensemble average absorption spectrum 29, 31. The
ensemble Au-SNR wavelength $\lambda_{LSPR}$ is measured to be 809 nm, which is
close to the nominal value of 808 nm 42. Fig. 2b also shows the renormalized
spectrum considering only responses from nanorod aggregations (Au-ANR, blue
solid line). For this spectrum, an Au-ANR wavelength of 786 nm is extracted,
corresponding to a blue-shift of the 2.8$\%$ from the Au-SNR wavelength. As
shown by Jain et al. 43, such a blue-shift occurs in nanorod aggregations of
two or more units assembled in a side-by-side orientation, for a polarization
parallel to their long axis. As the authors reported, the strength of this
shift depends on the interdistance between the nanorods involved, on their
aspect ratios, the relative orientational angle and on the number of units
considered. For the spectral distributions seen in Fig. 2 (blue curves), we
expect these signals to originate from side-by-side assembled nanorod
aggregations.

The individual nanorods have also been measured with a wavelength of 513 nm,
to excite the TSPR (red dot in Fig. 2b). As expected, the absorption at this
wavelength is roughly one order of magnitude smaller than LSPR, due to the
overlap between the transverse localized plasmonic resonance and the
electronic interband transitions, which start to arise at 2.4 eV ca. in gold
44, 45, 46, ultimately resulting in enhanced plasmonic damping.

(a)

(b)

Figure 3: (a) Measured absorption cross-section of a individual nanorod (red
dots), compared to FEM simulated absorption spectrum in the presence of the
silicon nitride substrate (black dot) and in free space (blue dots), obtained
for nanorod dimensions of L = 48 nm, r = 6 nm, with the silica coating
thickness of 20 nm. (b) FEM absorption cross-section at 800 nm wavelength for
the same nanorod (black empty dot and solid line), and 1D averaged FEM
electromagnetic intensity in the vicinity of the gold core in the presence of
the substrate only (red crosses and solid line), for different silicon nitride
slab thicknesses.

Fig. 3a focuses the attention on the nanomechanical photothermal spectrum of a
individual nanorod (red dots). This specific sample has maximum absorption at
$\lambda\approx 840$ nm with a cross-section of $\sigma_{abs}\approx 2.5\cdot
10^{-15}$ $m^{2}$, close to what it is reported in literature 34, 32, 33. The
measurements are compared with FEM simulations (black and blue dots, Fig. 3a),
showing a good agreement with data (black dots). Indeed, FEM approaches offer
the possibility to evaluate the absorption and scattering cross-sections of an
arbitrarily shaped particle 47, 48, 49, 50, 51, 52, 53. For this individual
nanorod, a linewidth of ca. 132 meV is measured in terms of full-width-at-
half-maximum (FWHM), two time higher than expected for electron-bulk
scattering alone (72 meV, see Supporting Information). Intrinsic size effects
for the gold core have to be taken into account, such as electron-surface
scattering and radiative damping (Supporting Information Eq. S3). The former
affects the gold nanorods’ LSPR linewidth stronger than the latter due to
their reduced volume, in contrast to what has been observed in spherical gold
nanoparticles 46. Similar FWHM values are found for the other nanorods,
corroborating the evidence that electron-surface scattering is a major source
of damping in these plasmonic-assisted nano-absorbers (Supporting Information
Fig. S3b).

Fig. 3a also clearly shows how the presence of the silicon nitride resonator
affects ultimately the absorption cross-section of the nanorod under study.
The FEM analysis in the presence of the substrate (black dots) reproduces well
the measured absorption spectrum, where the FEM analysis conducted in free
space in the absence of the slab does not (blue dots). More precisely, at the
LSPR wavelength (840 nm), the absorption cross-section results are roughly
half of the free space case ($\approx 2\cdot 10^{-15}$ $m^{2}$ in the presence
of the slab and $\approx 4.2\cdot 10^{-15}$ $m^{2}$ in air). In contrast, the
LSPR wavelength and FWHM are weakly affected by the presence of the substrate
($\leq$ 1$\%$ difference for both quantities). In general, a dielectric
substrate underneath a metal nanoparticle screens the electromagnetic
restoring force acting on the plasmon oscillations. This screening can be
modelled qualitatively as a nanoparticle image with a reduced number of
charges, whose electromagnetic strength is determined by the nanoparticle-
substrate interdistance and the slab dielectric permittivity 52. The
interdistance of 20 nm (silica coating thickness) and the relatively small
refractive index of low-stress silicon nitride (whose spectral distribution
has been taken from Ref. 54) give reason for this weak effect 40, 52. To
better understand the role played by the silicon nitride slab, FEM simulations
have been performed at a single wavelength (800 nm) for different thicknesses
$h_{subs}$.

In Fig. 3b the absorption cross-section follows a period pattern for an
increasing substrate thickness. This modulation follows perfectly the
variation in intensity at the interface air-silicon nitride in the vicinity of
the nanorod. In fact, the electromagnetic losses $Q_{h}$ due to absorption are
directly proportional to the intensity of the electric field, $Q_{h}\ \propto\
|\bf{E}(\bf{r})|^{2}$ (Supporting Information Eq. S1). This intensity
modulation is due to the interference occurring between the input electric
field and the reflected light from the slab, whose magnitude depends on the
thickness and refractive index. The intensity value calculated here is an
average over its spatial distribution in the proximity of the nanorod, but
without it, in the presence of the substrate only (Supporting Information Fig.
S4). For the thickness used in this study (50 nm) along with the considered
wavelengths (730-900 nm) and the refractive index of silicon nitride 54, no
interference inside the slab is present as it is the case in Ref. 55. There, 1
$\mu m$ thick silicon cantilevers worked as optical cavity for specific
wavelengths in the VIS range, modulating the scattering of deposited 100 nm
gold nanoparticles. In this study, however, the most relevant interference
occurs at the interface between the free space and the substrate. Therefore,
by controlling the substrate refractive index and thickness, it is possible to
tailor the absorption spectrum of individual nano-absorbers.

Figure 4: Polar plot of the absorption cross-section of an individual nanorod
measured at $\lambda_{exc}=808$ nm as a function of the polarization angle
$\theta_{pol}$ (red dots). The ratio between the absorption cross-section for
a polarization parallel to the nanorod long-axis ($\theta_{pol}\approx
157.5$°) and perpendicular to it ($\theta_{pol}\approx 90$°) is roughly
$\sigma_{abs,\parallel}$(808 nm)/$\sigma_{abs,\perp}$(808 nm) $\approx 100$.
FEM simulations show good agreement with the measurement (black dots). Both
the red and black solid curves represent the $cos^{2}(\theta)$ pattern. Blue
dots represent nanomechanical photothermal measurements at $\lambda_{exc}$ =
513 nm.

For these non-spherical nanoparticles, absorption is strongly polarization
dependent, as clearly seen in Fig. 4&5. Fig. 4 shows how the absorption cross-
section varies with the laser polarization angle for a individual nanorod,
with the red dots representing the nanomechanical photothermal measurements
close to its LSPR ($\lambda_{exc}=808$ nm). Each point is acquired by changing
the polarization of the scanning laser probe with steps of 22.5° by means of a
half-waveplate (HWP) and a linear polarizer, while maintaining the same laser
input power. The ratio between the absorption cross-section for a polarization
parallel to the nanorod long-axis ($\theta_{pol}\approx 157.5$°) and
perpendicular to it ($\theta_{pol}\approx 90$°) is roughly:

$\frac{\sigma_{abs,\parallel}(808\ nm)}{\sigma_{abs,\perp}(808\ nm)}\approx
100.$ (4)

This high polarization contrast gives us therefore an insight into the
absorption efficiency achievable in this nano-absorber upon control of the
incident laser polarization. The absorption efficiency for the parallel case
is $Q_{abs,\parallel}\approx 3.64$, while for the perpendicular case
$Q_{abs,\perp}\approx 0.03$ (the area of the nanorod being $A_{nr}\approx
3.29\cdot 10^{-16}\ m^{2}$, whose calculation is based on the sizes extracted
from FEM simulations). The measurements have been compared also to the FEM
simulations (black dots), showing a good match. It is worth noting how the
measurements follow the typical pattern $\sigma_{abs}(\lambda,\theta_{pol})\
=\ \sigma_{abs,\parallel}(\lambda)cos^{2}(\theta_{pol})$ expected for perfect
dipoles 56, 29, 36.

Figure 5: 2D maps of the same region at $\lambda_{exc}=808$ nm for three
different polarization angles $\theta_{pol}$: 90°, 135°, 180°. The two
responses are from two individual nanorods. For the perpendicular
polarizations, 90° and 180°, the absorbers behave in an opposite way, meaning
that they are almost perpendicular one to each other, while absorbing almost
the same amount of light for the central map ($\theta_{pol}=135$°).

For comparison, Fig. 4 shows also the nanomechanical photothermal signal at
$\lambda_{exc}$ = 513 nm (the blue dots). The TSPR shows almost no
polarization contrast since it starts to overlap with polarization-independent
electronic transitions in gold 44. For this reason, the plasmonic damping
increases, resulting in an transverse plasmonic resonant absorption strength
smaller than the longitudinal one.

Nanomechanical photothermal spectromicroscopy also allows the precise
determination of the orientation on the substrate of different nano-absorbers,
as seen in Fig. 5. 2D maps of the same region on the drum resonator for three
different polarization angles $\theta_{pol}$ (90°, 135°, 180°) are acquired at
$\lambda_{exc}=808$ nm. The two responses are from two individual nanorods,
whose absorption amplitude varies as a function of the laser polarization.
Focusing the attention on the two perpendicular polarizations, 90° and 180°,
the two absorbers behave in an opposite way, meaning that they are almost
perpendicular to one another, while absorbing almost the same amount of light
for the central scenario ($\theta_{pol}=135$°). Nanomechanical photothermal
spectromicroscopy can be therefore employed in the analysis of the more
complex optical features, for nanorods and more exotic structures, like
plamon-assisted optical chirality in metallic nanoparticles 57, 58, 37.

Finally, to further stress the advantages offered by nanomechanical
photothermal spectromicroscopy compared to other label-free single-particle
and molecule spectromicroscopy techniques, a comparison between SNR of
different approaches is carried out in the following way59:

$Norm.SNR=\frac{SNR_{exp}}{SNR_{0}}\frac{P_{heat,0}}{P_{heat,exp}}\sqrt{\frac{\tau_{m,0}}{\tau_{m,exp}}}.$
(5)

For the sake of completeness, Eq. 5 takes into account, together with the SNR
itself, also the power absorbed by the sample under study $P_{heat}$ and the
time constant of the experiment $\tau_{m}$. The reference values of the three
quantities used for normalization (labelled with the subscript $0$)
corresponds to the value obtained for the individual nanorod of Fig. 3&4. The
calculations are plotted in Fig. 6, for which the experimental values
extracted from the listed references are used (for the used values, see SI)
56, 60, 3, 31, 17, 18, 19, 36, 61, 7.

Figure 6: Signal-to-noise ratio (SNR) comparison between different single-
molecule absorption sensing techniques. Black: UV-Vis exctinction 56; blue:
spatial modulation spectroscopy 60; violet: extinction microscopy plus balance
detection 3; brown: ground-state depletion microscopy 5; dark green:
photothermal constrast microscopy (PCM) with Glycerol 16; green: PCM with
thermotropic liquid crystal 5CB 17; light green: PCM with near-critical Xe 18,
19; orange: optical microresonator 36, 61; red: nanomechanical photothermal
microscopy 7.

The different colors correspond to different techniques (see caption). It is
worth noting that the nanomechanical photothermal approach of Ref. 7 (first
red empty dot) presents the highest SNR, followed by photothermal contrast
microscopy with near-critical Xe 18, 19 and this work. More precisely, the
difference of two orders of magnitude between this work and Ref. 7 lies in the
different prestress of the nano-optomechanical resonator. There, oxygen plasma
treatment is exploited to reduce the stress of the resonator, with the aim to
improve its relative power responsivity $R_{P}$ to detect single Atto 633
molecules. Here, there has been no need to further reduce the stress due to
the already high sensitivity of the resonator used for the nanorods detection.
Still, this work shows the superior capabilities of nanomechanical
photothermal spectromicroscopy over a huge range of label-free absorption
techniques.

##  4 Conclusions

In conclusion, the optical absorption cross-section of individual silica-
coated gold nanorods in the NIR range has been measured and quantitatively
characterized using nanomechanical photothermal spectroscopy and microscopy,
likewise elucidating their polarization features. With this approach, where
the substrate acts as a temperature sensor, it is possible to shed light on
the variations in nano-absorbers’ properties to investigate concealed
heterogeneity, as expected for these complex systems, as well as their
reciprocal intercoupling, which opens up a wealth field of research by its
own. It has also been shown that these nanorods present, on one hand, a
pronounced plasmonic electron-surface scattering, broadening their LSPR in
conjunction with bulk scattering. On the other hand, a strong polarization
contrast of the order of few hundreds has been observed. The interaction
between the silicon nitride slab and the nanorod has been also investigated,
consisting in a modulation of its absorption strength over the whole
considered spectrum, while weakly affecting the plasmonic resonant energy and
its broadening. This result underlines the importance of taking into account
the interaction of the substrate in all the experiments where a support is
used for spectroscopic measurements.

Primarily, this work demonstrates the capabilities of nanomechanical
photothermal NIR spectromicroscopy for localizing individual nanoparticles,
obtaining their plasmon spectra, and resolving their polarization features,
pushing further our understanding of light-matter interaction at the nanoscale
level. A comparison conducted among the different label-free single-molecule
techniques shows that nanomechanical photothermal sensing presents a superior
signal-to-noise ratio within a less complex experimental setup and measurement
procedure.

The following files are available free of charge.

  * •

Supporting Information: FEM simulations details (computational approach, gold
dielectric function model), LSPR linewidth (measurements and modeling), FEM
substrate analysis, table with all the parameters used for the SNR comparison
(PDF)

The authors are thankful to Sophia Ewert and Patrick Meyer for the device
fabrication, and for the assistance of Sophia Ewert with scanning electron
microscopy. The authors thank also Hendrik Kähler and Andreas Kainz for the
fruitful discussions on finite element method. Furthermore, the authors thank
Johannes Hiesberger, Paolo Martini and Niklas Luhmann for useful discussions
on the set-up. Finally, K.K. thanks BioRender for their illustration software.
This work is supported by the European Research Council under the European
Unions Horizon 2020 research and innovation program (Grant Agreement No.
716087-PLASMECS).

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