Two stupid question

The story of the infinite dimensional space of $\Delta$ is following, we eliminate ourself with compact smooth non-boundary manifold $M$ with metric $g$, then we have Betrami-Laplace operator $\Delta_g$. We could instead $\Delta_g$ by hodge laplace $dd^*+d^*d$, but let we consider $\Delta_g$ the eigenvalue problem:
$$\Delta_g u=\lambda u$$
A classical way to investigate the eigenvalue problem is according to consider variational principle and max-min principle. We equip the path integral on the function space $C^{\infty}(M)$:
$$E(f)=\frac{\int_M |\nabla u|^2}{\int_M |u|^2 }$$
Then it have a sequences of eigenvalue, negative of course: $$0<-\lambda_1<-\lambda_2<…<\lambda_k<…$$

Then things become interesting, the morse theory of infinite space involve, called the infinite space as $X$, so at least, shrink the far place of $X$ as a point, in physics, this mean, cut off at fix scale. And we can take the scale to infinite small, we use the cutoff one to approximation the real one. What I can do is the following, I can proof the eigenvalue function is uniformly distributed in $L^2(M_g)$ (after rescaling of course) and the classical weyl law(although can not give a good error term estimate), but thing become more complicated when I try to consider the infinite space $X_{M_g}$’s topology, at finite scale at least, i.e. $X_{M_g}^{h}$ which is the cut off at scale $h$. Among the other thing, I believe the following issue is true, but without ability to proof it:

>**Problem**
for every manifold $M$ and metric $g$ on $M$, the topology of infinite space $X_{M_g}$ is the same, beside this, the inverse could be true, i.e. If $X_{M_1},X_{M_2}$ is not homomorphism for some scale $h$ then $M_1,M_2$ is not homomorphism.

By intuition, I think it is depend by the underling manifold’s topology. But I do not have a rigorous proof, I definitely have a non-rigorous one, if ignore the coverage…

As I find this problem when I try to give a proof of weyl law, I do not check the reference, may be this problem is a classical one? As always, I will appreciate to any interesting comments and answers, thanks a lots!A

2.

We begin with our favorite situation, the Dirchlet problem on bounded simple-connected domain $\Omega$ in $\mathbb R^n$. Let $\lambda_1$ be the first eigenvalue of $$\Delta u=\lambda u \ in\ \Omega$$
$$u=0\ \ on\ \partial\Omega$$
Rescaling $u$ such that $\sup_{\Omega} u=1$, I think the following property of the first eigenvalue is true.
>**Problem**
We have, the Minkowski functional of $\Omega$, called $M_{\Omega}$ and the Minkowski functional with the ball $B$ such that $vol(B)=vol(\Omega)$, then along the level set of $u$, i.e. the fiber: $$\Omega=\cup_{t\in [0,1]}l_t, l_t:=\{t|x\in \Omega, u(x)=t\}$$
We pretend for the isolate point $l_1$ to be a ball with radius 0, so equipped it with the uniformly density at every direction in $S^1$, i.e. the mass distribution given by $M_B$ and the total mass coincide with the total mass induce by $M_{\Omega}$ in $l_0$, i.e.
$$\int_{e\in S_1}M_{\Omega}(e)d\mu=\int_{e\in S_1}M_{B}(e)d\mu$$
The measure $d\mu$ equipped on $S^1$ is the natural Haar measure. And the cost function is given by $c(x,y)=\|x-y\|^2$. Then, among this setting,
I wish the following property to be true:
Along the direction $1\to 0$, the transport of density $\partial_{t_0} M_{\cup_{t=t_0}^1l_t}$ given the unique optimal transport of the natural measure induce by $M(\Omega)$ and $M(B)$.

**Remark 1** As point out by SebastianGoette, the multiplicity of the first eigenvalue must be one, thanks to the eigenfunction never change the symbol, so we are in the best case.

**Remark 2**:I am not very sure this property could always true, there may be a center example when $\Omega$ is not convex, but I tend to believe it is true at least when $\Omega$ is convex.

**Remark 3**: As point out by Dirk, when you try to consider the optimal transport problem, you always need to point out the cost function $c(x,y)$ defined on $\Omega \times \Omega$, for there, I think the naive choice is $c(x,y)=\|x-y\|^2$

The thing I can proof is the following, the level set of $u$ should be convex by brunn-minkowski inequality, and some type of monotonically property, i.e. more and more like a ball when the level set is more and more shirking smaller form $\partial \Omega$ to the point $f$ arrive maximum.

Some interesting problems

There are some interesting problem, I post them at there in case I forget them. Excuse me if they are trivial, I have not took enough time to consider them about I think they are valuable to be consider.

Problem 1:

This problem is stated by graph coloring. there are two prat of it, in fact the first part I heard from someone else and I try to generate it to high dimension.

1. there are finite lines $\{l_i\}_{i\in I}, l_i\subset \mathbb R^2$, crossing each other and the is a set $J$ of crossing point. for technique reason, assume the position of lines are generic, i.e. no three of them intersect at one point. Then we could use 3 different colors to color  $J$ make Neighbor points have different color. And to proof 3 is smallest.
2. generate it to high dimension, to prove $\mathbb R^n$ case, $n+1$ is the number.

This seems to be a graph problem, but the underlying structure is linear structure and some topological obstacle. I am not very sure. But it seems we can use an energy decrement argument with the obesevation:

The existence of a reasonable definition of “energy of correlation”.

the simplex arrive with the maximum of “correlation energy” in a very symmetric way, and this situation is easy to handle (coloring).

If make sense, this argument could also generate to high dimension.

Problem 2:

Let us consider some example of map between two metric space, a toy model is a line and two parallel lines, I called two parallel lines by $X_1\cup X_2$, the single line by $X_3$. The problem is try to find a tuple $(d,f)$, where $d$ is a metric define on $X_1\cup X_2$ and $f: X_1\cup X_2\to X_3$. such that the distortion of $f^* d$ and the standard metric on $X_3$ arrive at a infimum, this of course could not be the case, such like the situation of Yamabe problem on manifold with conners. So, let us ask a more general problem, could we describe the behavior of $f$ in some sense? what could we say with this kind of $f$?

Sarnak conjecture, understand with standard model

Sarnak conjecture is a conjecture lie in the overlap of dynamic system and number theory. It is mainly focus on understanding the behavior of entropy zero dynamic system by look at the correlation of an observable and the Mobius function .

We state it in a rigorous way:

let $(X,T)$ be a entropy zero topological dynamic system. Let Mobius function be defined as $\mu(n)=(-1)^t$, where $latex$ is the number of different primes occur in the decomposition of $n$.

Then for any continuous function $f:X\to R$ and $x\in X$, observable $\xi(n)=f(T^n(x))$ is orthogonal to the Mobius function; i.e. ,

$\lim_{N\to \infty}\frac{1}{N}\sum_{n=0}^{N-1}\mu(n)\xi(n)=o(N).$

I mainly focus on the special cases when dynamic system $X$ is the skew product on $T^2$ and when the dynamic system which is a interval exchange in $[0,1]$.

Skew product

For the first one, $\Theta=(T,T^2)$,$T:T^2\longrightarrow T^2$:
$T(x)=x+\alpha,T(y)=cx+y+h(x)$
$y_1(n)=T^{n}(x)=x+n\alpha,y_2(n)=T^n(y)=nx+\frac{n(n-1)}{2}\alpha+y+\sum_{n=1}^{N-1}h(x+i\alpha)$, where $c=1,-1$.

by Bourgain-Ziegelar-Sarnak theorem we know the difficulties is focus on deal with the exponent

$S_{p,q}(N)=\sum_{n=1}^N\mu(n)e^{\phi(n)+\sum_{m\in Z}e(mx)\hat H(m)(\frac{e(npm\alpha)-1}{e(m\alpha)-1}- \frac{e(nqm\alpha)-1}{e(m\alpha)-1})}$

for all $p,q$ is suffice large primes pair.

and a much simper case is the affine map:$T:(x,y)\to (x+\alpha,cx+y+\beta)$ on $\mathbb T^2$ and the general case $T:(x_1,...,x_n)\to A(x_1,...,x_n)$ where A is a upper-triangle matrix with diagonal 1; i.e. $A=I+B$, B is nilpotent. So the sarnak conjecture in this case is reduce to the Davenport estimate on exponent by B-Z-S theorem:

$|\sum_{n=0}^{N}e^{2\pi if(n)}|\leq c_A\frac{N}{(log N)^A}$, $\forall A>0$.

Interval exchange map

For the interval exchange map, we can explain it by a composition of rotation of some part of $S_1$ step by step and with a renormalization process to glue the neighbor rotations.

Now let us explain a little with this interesting dynamic system. We focus in the simplest nontrivial case, which is the 3-interval exchange map. In this case, just consider the permutation of intervals $I_1,I_2,I_3$, and it is easy to see there is only one case is nontrivial that is permutation: $I_1\to I_3,I_2\to I_2,I_3\to I_1$. We explain a little more with other trivial case:

When  $I_1\to I_2,I_2\to I_3,I_3\to I_1$, the interval exchange map is just a rotation and for which the sarnak conjecture is just come from:

$|\sum_{n=0}^{N}e^{2\pi in\alpha}\mu(n)|=o(N)$, $\forall \alpha\in R$.

Which is trivial because $\sum_{n=0}^{N}e^{2\pi in\alpha}\mu(n)=\frac{1-e^{2\pi iN\alpha}}{1-e^{2\pi i\alpha}}$.

For the case $I_1\to I_2, I_2\to I_1, I_3\to i_3$ the map $T$ is a rotation on $I_1\cap I_2$ but it is a identity map on $I_3$ and the orbits of point only lying one of $I_1\cap I_2, I_3$, lying in which one depend on the original point $x$ we take is lying in which one.

Now we focus on the most difficult situation. It is annoying but it is the obstacle we must get over to go far. Fortunately it could be explained as in the following picture.

Now we explain what happen in the picture, it is mainly say one identity, which explain how to look 3-interval exchange map as a composition of rotation map with a renormalization map to glue them. Rotation is a kind of map we have good understanding but we do not understand very well with the renormalization map which is glue the two endpoints of $I_2,I_3$ which are not the common endpoint of them. Then you get two circle glue like a “8” , and $T_2$ is just rotate one of it and make the other one to be invariance.

Now we roughly could think about what is the thing we need to charge with, it is just:

$\sum_{n=0}^{N}f((T_1\circ R\circ T_1)^n(x))\mu(n)=o(N)$.

Now we do some calculate with this geometric explain of interval exchange map.

Let $A=I_1, B=I_2\cap I_3$, then $A\cap B=\emptyset, A\cup B=[0,1]$. And $|A|=\alpha$, $0<\beta<|B|$. the rotation $T_1:x\to x-\alpha$, $T_2:x\to x+\beta$.

Standard model

Is there a standard model of entropy zero dynamic system?

This problem seems to be too ambitious. But it occur naturally when I an trying to have a global understand of the Sarnak conjecture.

Hilbert 16th problem

Introduction

the statement of Hilbert’s 16th problem:

$H(n)<\infty?$

definition of $H(n)=max$

Limit cycle:

Try beginning with Bendixon-Poincaré theorem, which is classical stuff and belongs to a lot of textbooks on vector fields.

Affine invariance

The number of limit cycle is invariant under affine map.

Classification of singular point

Bezout theorem

Example

1.$\frac{dx}{dt}=y,\frac{dy}{dt}=x$.The graph is just like:

2.$\frac{dx}{dt}=x^2+y^2,\frac{dy}{dt}=x-y$.The graph is just like:

3.$\frac{dx}{dt}=x^2-y^2,\frac{dy}{dt}=5-y$.The graph is just like:

4.$\frac{dx}{dt}=x^2-y^2,\frac{dy}{dt}=-y$.The graph is just like:

5.$\frac{dx}{dt}=x^3-y^3,\frac{dy}{dt}=5-y$.The graph is just like:

6.$\frac{dx}{dt}=y^2-x^2+1,\frac{dy}{dt}=y$. The graph of it is just like:

Now we try to explain the phenomenon we see. At first we can see there is no limit cycle in the picture. The bifurcation place is just the place $\frac{dx}{dt}=0\ or \frac{dy}{dt}=0$ and is just like 3 lines. And there are two singularity $(-1,0),(1,0)$.

7.$\frac{dx}{dt}=y,\frac{dy}{dt}=y-x^2-x^2y$

8.$\frac{dx}{dt}=y,\frac{dy}{dt}=y-x-x^2y$:

9.$\frac{dx}{dt}=y(y-3),\frac{dy}{dt}=(y-3-x-x^2y)(y-x-x^2y)$:

exist 2 limits cycles.

10.$\frac{dx}{dt}=y(y-3)(y-5),\frac{dy}{dt}(y-5-x-x^2y)(y-3-x-x^2y)(y-x-x^2y)$

Tree structure

In general, the lower bound of $H(n)$ is established. first $H(n)=O(n^2)$ by Otrokov,and later proved to be $H(x)=O(ln(n)n^2)$.

If right, this upper bound estimate is combine of two things:

1.every limit cycle do not intersect.

2.every unclear limit cycle contain a singularity.

Rough strategy attack Hilbert 16th problem

Step1:

The first step is to do some simplify, we know the number of limit cycles do not change under a affine map $(x,y)\to (\hat x,\hat y)=(x,y)A$, where A is a nonsingular 2*2 matrix. So we could classify the topological graph of the dynamic system.

Step2:

Classification the singularity under affine map, and investigate the topological graph of the topological graph of the singularity by floer cohomology. There is only finite type and we could focus on them one by one.

Every color in the graph is an area $P,Q$ do not have same component in it. And the boundary of areas is just the same component of $P,Q$ if it exists.

Step3:

Bezout theorem tell us if two  polynomials $P(x,y), Q(x,y)$ do not have same connected component then the intersection $I_{P,Q}\leq deg(P)deg(Q)$. And we can divide the space $R^2$ into finite parts, $P(x,y),Q(x,y)$ restrict to every part do not have the same component. And we have a upper bound control on the number of part when $\max\{deg(P(x,y)),deg(Q(x,y))\} \leq n$. A easy arrive bound could be $4^n$.

Step4:

Now we focus ourself on 1 part where $P(x,y),Q(x,y)$ do not have same component on it. Now we begin to proof there will be a relationship between the limit cycle, very like a tree structure, it will combine with the following two thing:

1. Every limit cycle contains at least one singularity or one smaller limit cycle inside it.

2. Every pair of limit cycle $(A,B)$, $A,B$ are limit cycle and $A$ is inside of $B$, then there is at least one singularity or another limit cycle contain in $\Omega$.

This kind of topological result will lead to a upper bound of number of limit cycle and end the proof of Hilbert 16th problem.

Planar polynomial vector field for a harmonic pair of polynomials

In this case you can consider the heat equation $\partial_t u(z,t)=\Delta u(z,t)$. If the number of the limit cycle change, it must be the time to pass a singularity. and take $t=\infty$, the dynamic system coverage to a very simlpe one and in particular it do not have limit cycle. So we need only look at the moments passing singularity.

Has the system of ODEs:

$\frac{dx}{dt}=P(x,y)\\ \frac{dy}{dt}=Q(x,y)$

been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of a holomorphic polynomial $F=F(z)$, $z=x+iy$?

I am looking to learn a bit about (complex) ODEs and their interplay with algebraic geometry by some examples, but I couldn’t find anything on this special case in Ilyashenko’s survey on Hilbert 16 (I guess this case is too special and/or not very interesting as far as Hilbert 16 is concerned).

Nontheless, it seems very natural. If we set $\gamma(t)=x(t)+iy(t)$, this amounts to the equation $latex \int_{\gamma_t}\frac{dz}{F(z)}=t$
where $\gamma_t$ is the curve $\gamma$ “truncated” at $t$ and the RHS is in particular **real**. This can be taken further, for example by assuming $\gamma$ is closed and using the residue theorem to obtain constraints on (the coefficients of) $F$.

First, this case is totally uninteresting regarding Hilbert XVI. Indeed, there are no limit cycles in such systems. The $\alpha / \omega$-limit of a trajectory is either a point or a non-isolated cycle (center case).

A singularity at $a\in \mathbb C$ (*i.e.* a root of $F$) can only be of three types, according to the value of $F'(a)$:

1. Source/focus: $F'(a)\notin i\mathbb R$.
2. Center: $F'(a)\in i\mathbb R_{\neq 0}$.
3. Flower with $2k$ petals: $F'(a)=0$ with multiplicity $k$.

In addition there is a pole at infinity (if $\deg(F)>0$) with exactly $2\deg(F)$ separatrices, reaching the singularity in finite time. The bassins of attraction / center regions attached to the above singularities are delimited by the separatrices.

[![enter image description here][2]][2]

S. Smale began to get interested in the question in the early 80’s while laying the foundations for BSS computational model (*The fundamental theorem of algebra and complexity theory*, 1981). He proposed a numerical root solver for polynomials by following the flow of $\frac{F}{F'}$. This started some works on the topic, for instance by Schub, Tischler, William (*The Newtonian graph of a complex polynomial*, 1988) or Benzinger (*Plane autonomous systems with rational vector fields*, 1991)…

In the case of these vector fields, the topological class is entirely encoded by their Newtonian graph (or the «dual» spinal graph) given by the incidence graph of the $\alpha / \omega$-limits of trajectories (in red on the picture). The main result for polynomials is that it is a tree. See *e.g.* Sverdlove (*Inverse problems for dynamical systems*,1981) and Schecter, Singer (*A class of vectorfields on $\mathbb S^2$ that are topologically equivalent to polynomial vectorfields*,1985) and Jongen, Jonker, Twilt (*On the classification of plane graphs representing structurally stable rational Newton flows*,1991).

The conformal classification has been initiated by Douady, Estrada and Sentenac (unpublished monograph, 2005) for the generic case (only focus/source singularities) and completed by Branner and Dias (*Classification of complex polynomial vector fields in one complex variable*, 2010). In addition to the combinatorial (topological) invariant, a complex «time-shift» (related to the integrals $\int_\gamma\frac{1}{F(z)} dz$) is associated to the separatrices, providing a complete conformal invariant.

In that latter context, the function $\int\frac{1}{F(z)} dz$ is called a Fatou coordinates. It is a rectifying chart for the vector field, and has many interesting dynamical properties.

Notice also the deep and beautiful relationship between spinal graph and *Dessins d’enfants*, as established by Pilgrim (*Polynomial vector fields, dessins d’enfants, and circle packings*,2006), related to [this question](https://mathoverflow.net/questions/118527).

Reference

Classification of Singularities and Bifurcations of Critical Points of Even Functions

E.A.Kudryavtseva, E.Lakshtanov

Classification of the singularity in even degree case.

Adjoint harmonic case have been studied. Look into the recent paper
Langley, J. K. Trajectories escaping to infinity in finite time. Proc. Amer. Math. Soc. 145 (2017), no. 5, 2107–2117, and the reference list in this paper.

They were also studied by physicists:

Bender, Carl M.; Hook, Daniel W.
Complex classical motion in potentials with poles and turning points.
Stud. Appl. Math. 133 (2014), no. 3, 318–336.

EDIT. I forgot to mention this:

B. Branner, K. Dias, Classification of complex polynomial vector fields in
one complex variable, Journal
Journal of Difference Equations and Applications
Volume 16, 2010 – Issue 5-6:

1. some example and observations

${(M^2,g)}$,${g(t)=e^{2u(t)}g_0}$,

$\displaystyle \frac{\partial u}{\partial t}=e^{-2u}\tilde\Delta u+\frac{r}{2}-e^{-2u}K_0$

${(M^n,g_{ij}(t))}$

$\displaystyle \frac{\partial g_{ij}(t)}{\partial t}=-2Ric(g_{ij})$

The given “smooth” initial :
${\exists }$ T small ,${T>0}$,the solution exists on ${[0,T]}$

equation is possible system.

Deturk Trick

“Threshold type theorem”
Ricci flow:
Mean curvature flow:
HMF:
Calabi flow:
pf of observation 4:if threshold condition hold for ${[0,T]}$,then we can bound ang ${C^k}$ norm of solution.

“geometry”

2. smooth manifold with conical singularities

on surface we can define conical singularity.

Definition 1 (conical singularity) ${M^2,p_i}$,${\beta_i}$,where ${\beta > -1}$,the angle of conical singularity ${p_i}$ is ${2\pi(1+\beta)}$.
iff conical background metric ${g_0}$: ${g_0}$ near p,${g_0=r^{2\beta}(dr^2+r^2d\theta^2)}$,${(r,\theta)}$ is the interpolation coordinate chart.

it is easy to chake the form ${g_0=r^{2\beta}(dr^2+r^2d\theta^2)}$ is independent with the coordinate chart ,so the definition is well defined.

3. rough line of proof

initial ${u_0}$,

$\displaystyle \frac{\partial u}{\partial t}=e^{-2u}\tilde\Delta u+\frac{r}{2}-e^{-2u}K_0$

Step 1:(Short time existence)
state and proof the “magic theorem”:
1.we need to explain what is smooth,to define a Banach space ${A}$,maybe ${W^{k,p},C^{k,p}}$ type.
2.to proof the “magic theorem” under the setting.maybe use shauder fix point theorem or contraction map theorem or else.
3.so the problem reduce to get this type estimate,
if ${\frac{\partial u_{i+1}}{\partial t}=e^{-2u_i}\tilde\Delta u_{i+1}+\frac{r}{2}-e^{-2u_i}K_0}$.
define ${T_{[0,T]}:A(\Omega \times [0,T]) \longrightarrow A(\Omega \times [0,T])}$. ${T_{[0,T]}}$ is continuous,${Dom(T_{[0,T]})}$ is convex,${Im(T_{[0,T]})}$ is a pre-compact set.for ${T_{[0,T]}}$ suffice small
the difficult is to set up the continuous of operator ${T_{[0,T]}}$
Schauder estimate tell us:
$\displaystyle ||u_{i+1}||_{C^{2,\alpha}} \leq ||u_{i+1}||_{L^{\infty}}+||\frac{r}{2}-e^{-2u_i}K_0||_{C^{\alpha}}$

this give us some useful information to construct space ${A}$ .
Step 2:(Threshold type theorem,long time existence)
Threshold as long as ${||u||_{L^{\infty}}}$ is bounded.
This type theorem is relate to the maximal internal in which solution existence is closed.
Basically is based on Alzalo-Ascoli theorem.
Step3:(More regularity) 1.the question does not existence for smooth manifold.(why)
2.singular space.
${u_0\in }$ small space ${\Longrightarrow}$ ${u(t)\in}$ small space.
what is the optimal regularity?
the problem naturally come from both “pure PDE” and “application for geometry problem”.

conical Kachler Ricci flow[Chen.Wang]
Donaldson setting ${C^{2,\alpha,\beta}}$

4. More seriously treat with the problem

in 07 years,consider the problem
$\displaystyle \frac{\partial u}{\partial t}=\Delta u$

on ${M-\{p\}}$
$\displaystyle u|_{t=0}=f$

where f is a function with nice regularity.
Functional analysis:
$\displaystyle \Delta: C_c^{\infty}(M-\{p\}) \longrightarrow C_c^{\infty}(M-\{p\})$

extension to:
$\displaystyle \Delta: L^{2}(M-\{p\}) \longrightarrow L^2(M-\{p\})$

which is a self-adjoint extension.and then use the theory of operator semi-group.the problem can be solved.
remark:the extension is not unique so the information we know for the solution is very little.and because the really true extension which is suit for our geometry setting is just one extension.so the treat of Functional analysis is not enough for us.
Elementary treat:
consider the simplest case,smooth manifold with only one singularity.

we set ${M_i}$ is the manifold cut off form ${M}$ with a boundary more and more near the singularity. consider the equation on each ${M_i}$,i.e.:
$\displaystyle \frac{\partial u}{\partial t}=\Delta u$

on ${M_i}$
$\displaystyle u|_{t=0}=f$

with boundary condition:
Drichlet condition
$\displaystyle u|_{\partial M_i=0}$

or Neumann condition
$\displaystyle \frac{\partial u}{\partial v}=0$

we choose Neumann condition there and at last we will see the solution come from Dirichlet condition is the same with the solution come from Neumann condition.
Under the general setting this become:

$\displaystyle \frac{\partial u_k}{\partial t}=a(k,t)\Delta u_k+b(k,t)\partial^i u_k +c(x,t)$

$\displaystyle \frac{\partial u_k}{\partial v}|_{\partial M_k}=0$

when ${k \longrightarrow \infty }$, do we have ${u_k \longrightarrow u}$?
we need priori estimate: Schauder estimate for serious parabolic equation tell us:
for equation ${\frac{\partial u}{\partial t}=\Delta u}$ on ${M}$ with priori estimate ${||u||_{L^{\infty}}\leq C}$, we have:

$\displaystyle |\nabla^k u(p)|\leq \frac{C}{r^k}$

wher ${r}$ is the maximal such that geodesic ball ${B(r,p) \subset\subset M_i}$.
For general setting :

$\displaystyle \frac{\partial u_k}{\partial t}=\Delta u_k+f$

$\displaystyle u_k|_{t=0}=u_0$

$\displaystyle \frac{u_k}{\partial t}|_{\partial M_k}=0$

we know

$\displaystyle ||u(t)||_{C^0(M_k)}\leq ||u_0||_{C^0(M_k)}+t||f||_{C^0(M_k)}$

this is what Schauder estimate tell us.
1.the uniform estimate with k:
${||u_k||_{***}\leq C}$ independent of k.(now we do not know what the norm ${||\cdot||_{***}}$ need to be)
we have ${C^0}$ estimate and the energy estimate as follows:
from maximal principle,easy to get ${C^0}$ norm estimate.

the point is the equation ${\frac{\partial }{\partial t}u_k= \Delta u_k +f}$ is strict parabolic so we have strong maximal principle and to construct suit bump function we can estimate ${C^0}$ norm of ${u_k}$.

from energy method we can estimate ${\int_{M_k} ||\nabla u_k||^2}$.

the point is:
${\frac{\partial}{\partial t}\int_{M_k} |\nabla u_k|^2=2\int_{M_k} \nabla u_k \cdot \frac{\partial}{\partial t}(\nabla u_k) }$
${=2\int_{M_k} \Delta u_k \cdot \frac{\partial}{\partial t} u_k }$
${=-2\int_{M_k}(\frac{\partial}{\partial t} u_k -f)\cdot \frac{\partial}{\partial t}u_k}$
${=-2[\int_{M_k}|\frac{\partial}{\partial t}u_k|^2-\int_{M_k} f\cdot \frac{\partial}{\partial t}u_k]}$
${=-2\int_{M_k}|\frac{\partial}{\partial t}u_k|^2-\int_{M_k}f \cdot (\Delta u_k +f)}$
${\leq 2\int_{M_k} \nabla u_k\cdot \nabla f}$
${\leq\int_{M_k} |\nabla f|^2 +\int_{M_k} |\nabla u_k|^2}$.
so we get:

$\displaystyle \frac{\partial}{\partial t}\int_{M_k}|u_k|^2\leq \int_{M_k}|\nabla f|^2+\int_{M_k} |\nabla u_k|^2$

so we can bounded ${\int_{M_k}|u_k|^2}$.
for the general case:the equation becomes:
$\displaystyle \frac{\partial u}{\partial t}=a(x,t)\Delta u+b(x,t) \partial^i u+c(x,t)$

$\displaystyle \frac{\partial u}{\partial v}|_{\partial M_k}=0$

$\displaystyle u(0)=u_0$

but there is a hide Dragon,we need the condition ${\frac{\partial u(0)}{\partial v}|_{M_k}=0}$.
otherwise we will get solution ${u \notin W^{1,2}(M_k)\cap C^{2}(M_k)}$.
but in this case we still have the two necessary estimate(esay to see the above argument still make sense).
in this case to prove the short time existence we need follow four claims is ture.
$\displaystyle ||u_{i+1,k}(t)||_C^0\longrightarrow ||u||_{i,k}{C^0}$

as ${t \longrightarrow 0}$
$\displaystyle \int_{M_k}|u_{i+1,k}|^2 \longrightarrow \int_{M_{k}}|u_{i,k}|^2$

as ${t \longrightarrow 0}$
$\displaystyle ||u_{i,k+1}(t)||_C^0\longrightarrow ||u||_{i,k}{C^0}$

as ${t \longrightarrow 0}$
$\displaystyle \int_{M_{k+1}}|u_{i,k+1}|^2 \longrightarrow \int_{M_{k}}|u_{i,k}|^2$

as ${t \longrightarrow 0}$
5. Construct the suitable Banach space

call the space construct follow the Mixed-Holder-Sobolev space for simply case,consider smooth manifold with only one conical singularity.
first cover the whole manifold by a open set have positive distance t=with the conical singularity and a countable group of set ${A_n=B(\frac{d}{2^n},p)-B(\frac{d}{2^{n+1}},p)}$,which is balls center at singularity ${p}$ with radius ${\frac{d}{2^n}}$.(where ${M=B(d,p)\cup U}$)
i.e. ${M-\{p\}=U \cup (\cup_{i=1}^{\infty}A_i)}$

Definition 2 (${||\cdot||_{\varepsilon^{k,\alpha}(S)}}$)
$\displaystyle ||f||_{\varepsilon^{k,\alpha}(S)}=sup_{k=1,2,...,\infty}||f(2^{-k},\theta)||_{C^{k,\alpha}(B_1-B_{\frac{1}{2}})}+||f||_{C^{k,\alpha}(U)}$

easy to see the definition is independent with the cover and the local interpolation coordinate chart.
one thing is also trivial,is that we have the schauder estimate under the norm ${||\cdot||_{\varepsilon^{k,\alpha}(S)}}$.
that is
$\displaystyle \delta u=f$

on ${S-\{p\}}$. ${|u| on ${S}$. then
$\displaystyle ||u||_{\varepsilon^{k+2,\alpha}}\leq C(||u||_{L^{\infty}}+||f||_{\varepsilon^{k,\alpha}})\leq C(C_1+||f||_{\varepsilon^{k,\alpha}})$

in fact we only need to add each inequality come from each open set of the cover by Schauder estimate to proof this.
on the other hand we need a suitable Sobolev type norm.
Definition 3 (${|\cdot|_w}$)
$\displaystyle |u|_w=(\int_S |\tilde \nabla u|^2d \tilde V)^{\frac{1}{2}}$

Definition 4 (${W^{k,\alpha}}$) the set of all f in ${\varepsilon^{k,\alpha}}$ with finite ${|f|_w}$,
$\displaystyle ||f||_W^{k,\alpha}=||f||_{\varepsilon^{k,\alpha}}+|f|_w$

in Banach space.
Assume norm ${C^{k,\alpha}(B\times [o,T])}$ on ${B\times [0,T]}$

Definition 5 (${||\cdot||_{\rho^{l,\alpha,{0,T}}}}$]
${f: S\times [0,T] \longrightarrow R }$

$\displaystyle ||f||_{\rho^{l,\alpha,[0,T]}}=sup_{k=0,1,2,...,\infty}||f(2^{-k}\rho,\theta,4^{-k}t)||_{C^{l,\alpha}((B_1-B_{\frac{1}{2}})\times [0,4^{-k}T])}+||f||_{C^{l,\alpha}(U\times[0,T])}$

\end) from the definition,easy to see
$\displaystyle \frac{\partial u}{\partial t}=\Delta u+f$

om ${M}$
$\displaystyle u|_{t=0}=u_0$

${\Longrightarrow}$
$\displaystyle ||u||_{\rho^{l+2,\alpha,[0,T]}}\leq C(||u_0||_{\varepsilon^{l,\alpha}}+||f||_{\rho^{l,\alpha,[0,T]}}+||u||_{C^0(S\times [0,t])})$

easy from the classical schauder estimate.

Definition 6 (${|f|_v}$) ${f:S\times [0,T] \longrightarrow R}$
$\displaystyle |f|^2_v=max_{t\in [0,T]}\int_S|\tilde \nabla f|^2d\tilde V +\int_0^T\int_M |\frac{\partial f}{\partial t}|^2 d\tilde Vd t$

Key point:

Definition 7 (${\nu^{k,\alpha,{0,T}}}$] ${\nu^{k,\alpha,[0,T]}}$ is the set of ${f}$ in ${\rho^{l,\alpha,[0,T]}}$ with finite ${|f|_v}$
$\displaystyle ||\cdot||_{\nu^{k,\alpha,[0,T]}}=||\cdot||_{\rho^{l,\alpha,[0,T]}}+|\cdot|_v$

\end)
6. What is a solution of equation

trivial sense:
satisfied equation point-wise on ${S-\{p\}}$.
weak sense:
1.trivial case
2.${|u|_v<+\infty}$

Atiyah-Singer index theorem 2

1. rough outline of heat kernel proof of Atiyah-singer index theorem

1.1. proof strategy

Theorem 1 (Mckean-Singer formula.)
$\displaystyle ind(D^+)=Str(e^{-tD^2})=\int\limits_{x \in M} Str(K(x,y)).$

from this we know Fredholm operator deformation invariance,in the same time we need chern-weil theory.
Main challenge:
1. in the expansion on heat kernel ,we need to proof when ${ t \rightarrow 0}$, the limit exist and find a way to calculate it.
2. indentify the limit as ${t \rightarrow 0}$.
Proof: Our proof road锛� mckean-singer formula ${\rightarrow}$ local-index thm ${\rightarrow}$ A-S index thm ${\rightarrow}$ Riemann-roch-Hirzebunch theorem. $\Box$

1.2. preliminary work

superbundle: ${E=E^+ \oplus E^-}$.
on compact manifold ${M}$, ${D: \Gamma(M,E) \rightarrow \Gamma(M,E)}$ is a self-adjoint operator . ${ D = }$. ${D^+=D|_E^+,D^-=D|_E^-}$.
observe that:
${D}$ is symmetric ${\Longrightarrow}$ eigenvalue space of ${D^2}$ is finite dimention ${\Longrightarrow}$ in particular ${Ker D^2}$ is finite dimesion ${\Longrightarrow}$ ${Ker D}$ is finite dimension.
dimention of superspace ${E = E^+ \oplus E^-}$:

$\displaystyle dim E=dim E^+ - dim E^-.$

$\displaystyle kerD=kerD^+ \oplus ker D^-$

Def: ${ind D^+=dim ker D^+ -dim ker D^-}$.
Lemma:
1.Let ${D}$ be a self-adjoint Dirac operator on a clifford module ${E}$ over a compact manifold ${M}$,then

$\displaystyle \Gamma(M,E^{\pm})=ker D^{\pm} \oplus im D^{\mp}$

in particular,
$\displaystyle ind D^+ = dim ker D^+ -dim coker D^+$

where coker ${D^+ :=\Gamma(M,E^-)/im D^+}$.
2.Let ${D}$ be a differential operator acting on a ${Z_2}$-graded vector bundle ${E}$,then ${Str[D,K]=0}$.
the proof of this two lemma is easy,leave sas exercise.
1.3. Mckean-Singer formula

the formula is:

$\displaystyle ind(D^+)=Str(e^{-tD^2})=\int\limits_{x \in M} Str(K(x,y)).$

the expression of heat operator by spectral measure is:
$\displaystyle e^{-tD^2}=\int\limits_{0}^\infty d^{\lambda t}dE_{\lambda}.$

proof 1:
we have first eigenvalue estimate on compact manifold:
$\displaystyle |Str(e^{-tD^2}-P_0)| \leq Cvol(M)e^{-t\lambda}.$

$\displaystyle \Longrightarrow$

$\displaystyle \lim\limits_{t \rightarrow \infty}Str(e^{-tD^2}) = Str p_0 =dim kerD^+ -dim ker D^- =ind D^+.$

on the other hand ,we need to show ${Str e^{-tD^2}}$ is independent with ${t}$,in fact:
$\displaystyle \frac{d}{dt} Str (e^{-tD^2})=-Str(D^2 e^{-tD^2}).$

${D}$ odd parity : ${\Longrightarrow \ D^2e^{-tD^2}=[D,D E^{-tD^2}]}$.
${[\ \ ,\ \ ]}$ supercommunater ${\Longrightarrow \ \frac{d}{dt}Str (e^{-tD^2})= -Str[D,D e^{-tD^2}]=0.}$
q.e.d
proof2:
by spectral decompositon of ${e^{-tD^2}}$:
$\displaystyle Str (e^{-tD^2})=\sum\limits_{\lambda \geq 0}(n_{\lambda}^+ - n_{\lambda}^-)e^{-t\lambda}$

observe that: ${n_{\lambda}^+=n_{\lambda}^-}$ for ${\lambda \not=0}$. ${\Longrightarrow ind D=n_0^+ - n_0^-}$. (detail in [BGV])
q.e.d
Corallary: the index of a smooth on-parameter family of Dirac operator is constant.
what we have proved is:
$\displaystyle ind D = Str (e^{-tD^2})=\int Str(k(x,y)).$

1.4. analytic formula of ind${D^+}$

from the discuss of heat kernel in section 2,we know following result(section 2 only discuss the case of function but use the similar way we can get similar result on bundle):

$\displaystyle K_t(x,y) \sim (4 \pi t)^{\frac{n}{2}} \sum\limits_{i=0}^{+\infty} t^i K_i(x), K_i \in \Gamma(M,End(E)).$

on the other hand: ${ D=\left[\begin{array}{ccc} 0 & D^- \\ D^+ & 0 \end{array}\right] \Longrightarrow D^2=\left[\begin{array}{ccc} D^-D^+ & 0\\ 0 & D^+D^- \end{array}\right] }$.
and use the Mckean-Singer formula,we get: ind (D^+)&=&Str(e^{-tD^2})
&=&Tr(e^{-tD^+D^-})-Tr(e^{-tD^=D^-})
&=&\int\limits_M Tr(K_t(x,y,D^-D^+)) – \int\limits_M Tr(k_t(x,y,D^+D^-))
&=&\sum\limits_{i=0}^{\infty} t^{i-\frac{n}{2}}a_i(D^+D^-) – \sum\limits_{i=0}^{\infty} t^{i – \frac{n}{2}}a_i(D^+D^-). where ${a_i}$ is the heat trace invariants.
take ${ t \rightarrow 0}$,the only thing make sense is the series of order ${\frac{n}{2}}$,and we want to proof:
$\displaystyle ind(D^+) =a_{\frac{n}{2}}(D^-D^+) -a_{\frac{n}{2}}(D^+D^-).$

But the difficult thing is that the high order series is very hard ro calculate….
our strategy is following:

Step1: proof ${Str(K_t(x,y))}$ has a limit as ${t \rightarrow 0}$ i.e ${Str(K_t(x,y))\stackrel{t \rightarrow 0}{\longrightarrow}}$ index density.
step2:use a rescaling of space,time,clifford bundles ,to find a way that make us only need to calculate the leader coefficient.

1.5. From the McKean鈥揝inger formula to the index theorem

Let ${M}$ be a compact oriented Riemannian manifold of even dimension ${n}$. We will write ${k_t(x, y)}$ for the heat kernel associated to ${D^2}$. The diagonal ${k_t(x, x)}$ is a section of ${End(E )}$ which is iso- morphic to ${Cl(M) \otimes End_{Cl(M)}(E )}$. Using this isomorphism, we define a filtration on ${End(E )}$, induced by the filtration on ${Cl(M)}$. Elements of ${End_{Cl(M)}(E )}$ are given 0-degree. Denote by ${Cl_i(M)}$ the subbundle of ${Cl(M)}$ consisting of all elements of degree less or equal to ${i}$.
the following theorem hold:
Theorem 1. The following statements hold:
1. The coefficients ${k_i}$ have degree less or equal to ${2i}$. In other words, ${k_i \in \Gamma (M, Cl_{2i}(M) \otimes End_{Cl(M)}(E))}$. 2. If ${ \sigma(k):= \sum \limits_{i=0}^{n/2}\sigma_{2i}(k_i) \in A(M,End_{Cl(M)}(E))}$,where ${\sigma_j :Cl_j(M)鈫扐_j(M)}$ denotes the ${i=0}$ restriction of the symbol map, then:

$\displaystyle \sigma (k) = A藛(M) exp(鈭扚E /S ).$

$\displaystyle ind (D^+)=Str(e^{-tD^2})=\sum\limits_{i=0}^{\infty} t^{i-\frac{n}{2}}a_i(D^+D^-) - \sum\limits_{i=0}^{\infty} t^{i - \frac{n}{2}}a_i(D^+D^-).$

the important observation is that the ${Str}$ of order less than n all vanish,in the other word:

lemma2:for any quadratic space ${v}$ of dimension ${n}$,${CL_{n-1}(V) =[CL(V),CL(V)]}$.

Proof: Let ${e_1,...,e_n}$ be a basis of ${V}$. For any multi-index ${I \subset \{1,...,n\}}$, denote by ${c_I}$ the Clifford product ${\prod_{i \in I} c}$.Then the set ${\{c_I\}}$ is a basis for ${Cl(V)}$.If${|I|,there is at least one ${j}$ such that ${j \notin I}$, and we have
$\displaystyle c(e_I)=-\frac{1}{2}[c_j,c_j c_I].$

q.e.d
so take ${t \rightarrow 0}$ , we get:
$\displaystyle ind (D) =(4 \pi)^{\frac{n}{2}} \sum\limits_{i = \frac {n}{2}}^{\infty} t^{i- \frac{n}{2}}Str(k_i(x)).$

now we want to identify the term ${Str(k_{\frac{n}{2}}(x))}$ as a characteristic form on ${M}$ :
Lemma 3. Let ${V}$ be a Euclidean space. There is, up to a constant factor, a unique supertrace on ${Cl(V)}$, equal to ${T \circ \sigma}$ where ${\sigma}$ denotes the symbol map and ${T}$ is the projection of ${\alpha \in \wedge V}$ onto the coefficient of ${e_1 \wedge, ... ,\wedge e_n}$ if ${e_1,...,e_n}$ form an oriented orthonormal basis of ${V}$. Furthermore, the supertrace defined above equals:
$\displaystyle Str(a) = (鈭�2i)^{\frac{n}{2}} (T \circ \sigma(a)).$

rmk:(The map ${T}$ is also called the canonical Berezin integral.)
proof:
the dimension of ${Str}$ space is one because of ${CL_{n-1}(V)=[CL(V),CL(V)]}$ and it never be empty because there is a natural defined supertrace on ${CL(V)}$.so the only thing we need to do is to determine the constant,in face we only need to calculate the supertarce on any non-zero element .for instance the chirality operator ${\Gamma}$. We have that ${ Str(\Gamma) = dim (\wedge P) = 2^{n/2}}$ and therefore ${Str(a) = (鈭�2i)^{n/2} T \circ \sigma (a)}$ for all ${a \in Cl(M)}$.
q.e.d
so we know:
${\forall \ }$ section ${a \otimes b \in \Gamma(M \otimes CL(M) \otimes End_{CL(M)}(E)}$:
$\displaystyle Str_E( a \otimes b)(x) = (-2i)^{\frac{n}{2}} \sigma_n(a(x))Str _{E/S}(b(x)).$

$\displaystyle Str_E(k_{\frac{n}{2}})(x) = (-2i)^{\frac{n}{2}} Str _{E/S}(\sigma_n(k_{\frac{n}{2}})).$

theorem1 implies then the following theorem for the index of a Dirac operator associated to a Clifford connection which is known as the local index theorem.

Theorem 2. (Local index theorem) Let ${M}$ be a compact, oriented even-dimensional manifold and let ${E}$ be a Clifford module with Clifford connection ${\nabla E}$ . Let ${D}$ be the associated Dirac operator. Then ${\lim\limits_{t \rightarrow 0} Str(k_t(x, x))|dx| }$ exists and is obtained by taking the ${n}$ -th form piece of
$\displaystyle (2\pi i)^{鈭抧/2} \widehat A(M)ch(E /S ).$

rmk:This theorem only holds for Dirac operators associated to Clifford connections, which are those which are compatible with the Clifford action. However, since the index of a Dirac operator is independent of the Clifford superconnection used to define it, we get Atiyah鈥揝inger index formula for any Dirac operator.

Theorem 3. (Atiyah鈥揝inger Index Theorem) Let ${M}$ be a compact, oriented, even-dimensional manifold and let ${D}$ be a Dirac operator on a Clifford module ${E}$ . Then the index of ${D}$ is given by :
$\displaystyle ind D = (2 \pi i)^{-\frac{n}{2}}\int\limits_{M} \widehat A(M) ch(E/S) .$

hence theorem 3 is a consequence of theorem 1.
up to now,to prove index theorem ,we only need to prove theorem 1.
1.6. idea of the proof of theorem 1

we give the clear proof in appendix锛宼here we explain the idea:
To prove Theorem 4.11 we mainly follow Chapter 4 of [BGV] but rearrange the different steps in order to make the proof clearer, at least for us. Let us summarize the first part of the proof:

1. The idea of the proof is to work in normal coordinates ${x}$ around a point ${x_0 \in M}$. Near the diagonal, we use parallel transport to pull back the heat kernel ${k_t(x, x_0)}$ which is a section of the vector bundle ${E_x \otimes E_x^*}$ and define a new kernel ${k(t, x) := \iota(x_0, x)k_t(x, x_0)}$, a section of ${End(E_{x_0} ) \cong Cl(V^*) \otimes End(W)}$ for some twisting space ${W}$. Using the symbol isomorphism ${蟽}$, we can look at ${k(t, x)}$ as a section of ${\wedge(V^*) \otimes E (W)}$.

2. We use Lichnerowicz鈥� formula to get the explicit form of the operator ${L}$ such that the kernel ${k(t, x)}$ satisfies the heat equation ${(\partial t + L)k(t, x) = 0}$.

3. In a third step, we define a rescaling of space, time and the Clifford algebra, introduced by Getzler. This rescaling has the effect that the leading coefficient of the asymptotic expansion of the rescaled kernel is exactly the differential form ${\sigma(k)}$ of theorem 1 which leads to a reformulation of Theorem 1.

\appendix

2. Chern-Weil theory

some text in Appendix A

3. Complete proof of theorem 1

some text in Appendix B