Minimizer of an isoperimetric ratio on a metric on R2\R^2 with finite total area

Let $g=(g_{ij})$ be a complete Riemmanian metric on $\R^2$ with finite total area and $I_g=\inf_{\gamma}I(\gamma)$ with $I(\gamma)=L(\gamma)(A_{in}(\gamma)^{-1}+A_{out}(\gamma)^{-1})$ where $\gamma$ is any closed simple curve in $\R^2$, $L(\gamma)$ is the length of $\gamma$, $A_{in}(\gamma)$ and $A_{out}(\gamma)$ are the area of the regions inside and outside $\gamma$ respectively, with respect to the metric $g$. We prove the existence of a minimizer for $I_g$. As a corollary we obtain a new proof for the existence of a minimizer for $I_{g(t)}$ for any $0<t<T$ when the metric $g(t)=g_{ij}(\cdot,t)=u\delta_{ij}$ is the maximal solution of the Ricci flow equation \1 g_{ij}/\1 t=-2R_{ij} on $\R^2\times (0,T)$ \cite{DH} where $T>0$ is the extinction time of the solution.Comment: 14 pages, some typos are corrected and some proofs are written in more detai

Another proof of Ricci flow on incomplete surfaces with bounded above Gauss curvature

We give a simple proof of an extension of the existence results of Ricci flow of G.Giesen and P.M.Topping [GiT1],[GiT2], on incomplete surfaces with bounded above Gauss curvature without using the difficult Shi's existence theorem of Ricci flow on complete non-compact surfaces and the pseudolocality theorem of G.Perelman [P1] on Ricci flow. We will also give a simple proof of a special case of the existence theorem of P.M.Topping [T] without using the existence theorem of W.X.Shi [S1].Comment: 13 page

A harmonic map flow associated with the standard solution of Ricci flow

Let $(\Bbb{R}^n,g(t))$, $0\le t\le T$, $n\ge 3$, be a standard solution of the Ricci flow with radially symmetric initial data $g_0$. We will extend a recent existence result of P. Lu and G. Tian and prove that for any $t_0\in [0,T)$ there exists a solution of the corresponding harmonic map flow $\phi_t:(\Bbb{R}^n,g(t))\to (\Bbb{R}^n,g(t_0))$ satisfying $\partial \phi_t/\partial t=\Delta_{g(t),g(t_0)}\phi_t$ of the form $\phi_t(r,\theta) =(\rho (r,t),\theta)$ in polar coordinates in $\Bbb{R}^n\times (t_0,T_0)$, $\phi_{t_0}(r,\theta)=(r,\theta)$, where $r=r(t)$ is the radial co-ordinate with respect to $g(t)$ and $T_0=\sup\{t_1\in (t_0,T]: \|\widetilde{\rho}(\cdot ,t)\|_{L^{\infty}(\Bbb{R}^+)} +\|\partial\widetilde{\rho}/\partial r(\cdot ,t)\|_{L^{\infty}(\Bbb{R}^+)} <\infty\quad\forall t_0<t\le t_1\}$ with $\widetilde{\rho}(r,t) =\log (\rho(r,t)/r)$. We will also prove the uniqueness of solution of the harmonic map flow. We will also use the same technique to prove that the solution $u$ of the heat equation in $(\Omega\setminus\{0\})\times (0,T)$ has removable singularities at $\{0\}\times (0,T)$, $\Omega\subset\Bbb{R}^m$, $m\ge 3$, if and only if $|u(x,t)|=O(|x|^{2-m})$ locally uniformly on every compact subset of $(0,T)$.Comment: 21 page

A pseudolocality theorem for Ricci flow

In this paper we will give a simple proof of a modification of a result on pseudolocality for the Ricci flow by P.Lu without using the pseudolocality theorem 10.1 of Perelman [P1]. We also obtain an extension of a result of Hamilton on the compactness of a sequence of complete pointed Riemannian manifolds $\{(M_k,g_k(t),x_k)\}_{k=1}^{\infty}$ evolving under Ricci flow with uniform bounded sectional curvatures on $[0,T]$ and uniform positive lower bound on the injectivity radii at $x_k$ with respect to the metric $g_k(0)$.Comment: 11 pages, I have add one mild assumption on the theorem and completely rewrites the proof of the theorem which avoids the use of the logarithmic Sobolev inequality completely. I also obtain an extension of the compactness result of Hamilton on a sequence of complete pointed Riemannian manifolds evolving under Ricci flo

Super fast vanishing solutions of the fast diffusion equation

We will extend a recent result of B.Choi, P.Daskalopoulos and J.King. For any $n\ge 3$, $00$, we will construct subsolutions and supersolutions of the fast diffusion equation $u_t=\frac{n-1}{m}\Delta u^m$ in $\mathbb{R}^n\times (t_0,T)$, $t_0<T$, which decay at the rate $(T-t)^{\frac{1+\gamma}{1-m}}$ as $t\nearrow T$. As a consequence we obtain the existence of unique solution of the Cauchy problem $u_t=\frac{n-1}{m}\Delta u^m$ in $\mathbb{R}^n\times (t_0,T)$, $u(x,t_0)=u_0(x)$ in $\mathbb{R}^n$, which decay at the rate $(T-t)^{\frac{1+\gamma}{1-m}}$ as $t\nearrow T$ when $u_0$ satisfies appropriate decay condition.Comment: 37 pages, typos corrected, reference update

Uniqueness of solutions of Ricci flow on complete noncompact manifolds

We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow on complete noncompact manifolds with bounded curvatures.Comment: A simple example of a manifold with bounded curvature and injectivity radius going to zero as the point tends to infinity is given. A proof and argument why the crucial lemma Lemma 2.2 of the Chen-Zhu's paper \cite{CZ} cannot hold is give

Generalized \Cal{L}-geodesic and monotonicity of the generalized reduced volume in the Ricci flow

Suppose $M$ is a complete n-dimensional manifold, $n\ge 2$, with a metric $\bar{g}_{ij}(x,t)$ that evolves by the Ricci flow $\partial_t \bar{g}_{ij}=-2\bar{R}_{ij}$ in $M\times (0,T)$. For any $0<p<1$, $(p_0,t_0)\in M\times (0,T)$, $q\in M$, we define the \Cal{L}_p-length between $p_0$ and $q$, \Cal{L}_p-geodesic, the generalized reduced distance $l_p$ and the generalized reduced volume $\widetilde{V}_p(\tau)$, $\tau=t_0-t$, corresponding to the \Cal{L}_p-geodesic at the point $p_0$ at time $t_0$. Under the condition $\bar{R}_{ij}\ge -c_1\bar{g}_{ij}$ on $M\times (0,t_0)$ for some constant $c_1>0$, we will prove the existence of a \Cal{L}_p-geodesic which minimize the \Cal{L}_p(q,\bar{\tau})-length between $p_0$ and $q$ for any $\bar{\tau}>0$. This result for the case $p=1/2$ is conjectured and used many times but no proof of it was given in Perelman's papers on Ricci flow. My result is new and answers in affirmative the existence of such \Cal{L}-geodesic minimizer for the $L_p(q,\tau)$-length which is crucial to the proof of many results in Perelman's papers on Ricci flow. We also obtain many other properties of the generalized \Cal{L}_p-geodesic and generalized reduced volume.Comment: 64 page

A simple proof on the non-existence of shrinking breathers for the Ricci flow

Suppose $M$ is a compact n-dimensional manifold, $n\ge 2$, with a metric $g_{ij}(x,t)$ that evolves by the Ricci flow $\partial_tg_{ij}=-2R_{ij}$ in $M\times (0,T)$. We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality.Comment: 15 page

Maximum principle and convergence of fundamental solutions for the Ricci flow

In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in $B_k\times (0,T)$ to the minimal fundamental solution of the conjugate heat equation as $k\to\infty$. We will prove the uniqueness of the fundamental solution under some exponential decay assumption on the fundamental solution. We will also give a detail proof of the convergence of the fundamental solutions of the conjugate heat equation for a sequence of pointed Ricci flow $(M_k\times (-\alpha,0],x_k,g_k)$ to the fundamental solution of the limit manifold as $k\to\infty$ which was used without proof by Perelman in his proof of the pseudolocality theorem for Ricci flow.Comment: 15 page

Removable singularity of the polyharmonic equation

Let $x_0\in\Omega\subset\Bbb{R}^n$, $n\ge 2$, be a domain and let $m\ge 2$. We will prove that a solution $u$ of the polyharmonic equation $\Delta^mu=0$ in $\Omega\setminus\{x_0\}$ has a removable singularity at $x_0$ if and only if $|\Delta^ku(x)|=o(|x-x_0|^{2-n})\quad\forall k=0,1,2,...,m-1$ as $|x-x_0|\to 0$ for $n\ge 3$ and $=o(\log (|x-x_0|^{-1}))\quad\forall k=0,1,2,...,m-1$ as $|x-x_0|\to 0$ for $n=2$. For $m\ge 2$ we will also prove that $u$ has a removable singularity at $x_0$ if $|u(x)|=o(|x-x_0|^{2m-n})$ as $|x-x_0|\to 0$ for $n\ge 3$ and $|u(x)| =o(|x-x_0|^{2m-2}\log (|x-x_0|^{-1}))$ as $|x-x_0|\to 0$ for $n=2$.Comment: 6 page