1. Some Examples and Observations

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

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

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

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

The given “smooth” initial:

$latex \exists T>0$, the solution exists on $latex [0,T]$

Deturk Trick

Threshold Type Theorem

Ricci Flow:

Mean Curvature Flow:

HMF:

Calabi Flow:

Proof of Observation 4:
If the threshold condition holds for $latex [0,T]$, then we can bound the $latex C^k$ norm of the solution.

Geometry

2. Smooth Manifold with Conical Singularities

On a surface, we can define a conical singularity.

Definition 1 (Conical Singularity)
Let $latex M^2, p_i, \eta_i$, where $latex \eta > -1$. The angle of conical singularity $latex p_i$ is $latex 2\pi(1+\eta)$, if the conical background metric $latex g_0$ near $latex p$ is $latex g_0 = r^{2\eta}(dr^2 + r^2d\theta^2)$, where $latex (r,\theta)$ is the interpolation coordinate chart.

3. Rough Line of Proof

Initial $latex u_0$,

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

Step 1 (Short Time Existence):
State and prove the “magic theorem”.

Step 2 (Threshold Type Theorem, Long Time Existence):
Threshold as long as $latex ||u||_{L^\infty}$ is bounded.

Step 3 (More Regularity):

1. The question does not exist for smooth manifold (why).
2. Singular space.

Conical Kachler Ricci Flow [Chen.Wang]
Donaldson setting $latex C^{2,\alpha,\eta}$

4. More Seriously Treat with the Problem

In 07 years, consider the problem

$latex \frac{\partial u}{\partial t} = \Delta u$

on $latex M-{p}$

$latex u|_{t=0} = f$

where $latex f$ is a function with nice regularity.

Functional Analysis:
$latex \Delta: C_c^\infty(M-{p}) \longrightarrow C_c^\infty(M-{p})$

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

which is a self-adjoint extension. And then use the theory of operator semi-group, the problem can be solved.

Elementary Treat:
Consider the simplest case, smooth manifold with only one singularity.

5. Construct the Suitable Banach Space

Call the space constructed following the Mixed-Holder-Sobolev space for the simple case, consider smooth manifold with only one conical singularity.

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

Definition 3 ($latex |\cdot|_w$):
$latex |u|_w = \left(\int_S |\tilde{\nabla} u|^2 d\tilde{V}\right)^{\frac{1}{2}}$

Definition 4 ($latex W^{k,\alpha}$):
The set of all $latex f$ in $latex \varepsilon^{k,\alpha}$ with finite $latex |f|_w$.

Definition 5 ($latex ||\cdot||{\mathcal{H}^{l,\alpha,[0,T]}}$):
$latex ||f|| {\mathcal{H}^{l,\alpha,[0,T]}} = \sup_{k=0,1,2,…,\infty} ||f(2^{-k}h,\theta,4^{-k}t)||{C^{l,\alpha}((B_1-B{\frac{1}{2}})\times [0,4^{-k}T])} + ||f||_{C^{l,\alpha}(U\times[0,T])}$

Definition 6 ($latex |\cdot|v$):
$latex |f|^2_v = \max {t\in [0,T]}\int_S |\tilde{\nabla} f|^2 d\tilde{V} + \int_0^T\int_M |\frac{\partial f}{\partial t}|^2 d\tilde{V}dt$

Definition 7 ($latex u^{k,\alpha,[0,T]}$):
$latex u^{k,\alpha,[0,T]}$ is the set of $latex f$ in $latex \mathcal{H}^{l,\alpha,[0,T]}$ with finite $latex |f|_v$

6. What is a Solution of Equation

Trivial sense:
Satisfied equation point-wise on $latex S-{p}$.

Weak sense:

1. Trivial case
2. $latex |u|_v < +\infty$