The classification of locally conformally flat Yamabe solitons

We provide the classification of locally conformally flat gradient Yamabe solitons with positive sectional curvature. We first show that locally conformally flat gradient Yamabe solitons with positive sectional curvature have to be rotationally symmetric and then give the classification and asymptotic behavior of all radially symmetric gradient Yamabe solitons. We also show that any eternal solutions to the Yamabe flow with positive Ricci curvature and with the scalar curvature attaining an interior space-time maximum must be a steady Yamabe soliton

Classification of singularities in the complete conformally flat Yamabe flow

We show that an eternal solution to a complete, locally conformally flat Yamabe flow, $\frac{\partial}{\partial t} g = -Rg$, with uniformly bounded scalar curvature and positive Ricci curvature at $t = 0$, where the scalar curvature assumes its maximum is a gradient steady soliton. As an application of that, we study the blow up behavior of $g(t)$ at the maximal time of existence, $T < \infty$. We assume that $(M,g(\cdot, t))$ satisfies (i) the injectivity radius bound {\bf or} (ii) the Schouten tensor is positive at time $t = 0$ and the scalar curvature bounded at each time-slice. We show that the singularity the flow develops at time $T$ is always of type I.Comment: The paper has been withdrawn due to a crucial error in the argumen

On the extinction profile of solutions to fast-diffusion

We study the extinction behavior of solutions to the fast diffusion equation $u_t = \Delta u^m$ on $\R^N\times (0,T)$, in the range of exponents $m \in (0, \frac{N-2}{N})$, $N > 2$. We show that if the initial data $u_0$ is trapped in between two Barenblatt solutions vanishing at time $T$, then the vanishing behaviour of $u$ at $T$ is given by a Barenblatt solution. We also give an example showing that for such a behavior the bound from above by a Barenblatt solution $B$ (vanishing at $T$) is crucial: we construct a class of solutions $u$ with initial data $u_0 = B (1 + o(1))$, near $|x| >> 1$, which live longer than $B$ and change behaviour at $T$. The behavior of such solutions is governed by $B(\cdot,t)$ up to $T$, while for $t >T$ the solutions become integrable and exhibit a different vanishing profile. For the Yamabe flow ($m = \frac{N-2}{N+2}$) the above means that these solutions $u$ develop a singularity at time $T$, when the Barenblatt solution disappears, and at $t >T$ they immediately smoothen up and exhibit the vanishing profile of a sphere. In the appendix we show how to remove the assumption on the bound on $u_0$ from below by a Barenblatt

Evolution of non-compact hypersurfaces by inverse mean curvature

We study the evolution of complete non-compact convex hypersurfaces in $\mathbb{R}^{n+1}$ by the inverse mean curvature flow. We establish the long time existence of solutions and provide the characterization of the maximal time of existence in terms of the tangent cone at infinity of the initial hypersurface. Our proof is based on an a'priori pointwise estimate on the mean curvature of the solution from below in terms of the aperture of a supporting cone at infinity. The strict convexity of convex solutions is shown by means of viscosity solutions. Our methods also give an alternative proof of the result by Huisken and Ilmanen on compact start-shaped solutions, based on maximum principle argument.Comment: 24 pages, 4 figure

C^{1,\al} regularity of solutions to parabolic Monge-Amp\'ere equations

We study interior C^{1, \al} regularity of viscosity solutions of the parabolic Monge-Amp\'ere equation u_t = b(x,t) \ddua, with exponent $p >0$ and with coefficients $b$ which are bounded and measurable. We show that when $p$ is less than the critical power $\frac{1}{n-2}$ then solutions become instantly C^{1, \al} in the interior. Also, we prove the same result for any power $p>0$ at those points where either the solution separates from the initial data, or where the initial data is $C^{1, \beta}$

Fully Degenerate Monge Amp\'ere Equations

In this paper, we consider the following nonlinear eigenvalue problem for the Monge-Amp\'ere equation: find a non-negative weakly convex classical solution $f$ satisfying {equation*} {cases} \det D^2 f=f^p \quad &\text{in $\Omega$} f=\vp \quad &text{on $\partial\Omega$} {cases} {equation*} for a strictly convex smooth domain $\Omega\subset\R^2$ and $0<p<2$. When $\{f=0\}$ contains a convex domain, we find a classical solution which is smooth on $\bar{\{f>0\}}$ and whose free boundary $\partial\{f=0\}$ is also smooth

Type II extinction profile of maximal solutions to the Ricci flow in R2\R^2

We consider the initial value problem $u_t = \Delta \log u$, $u(x,0) = u_0(x)\ge 0$ in $\R^2$, corresponding to the Ricci flow, namely conformal evolution of the metric $u (dx_1^2 + dx_2^2)$ by Ricci curvature. It is well known that the maximal (complete) solution $u$ vanishes identically after time $T= \frac 1{4\pi} \int_{\R^2} u_0$. Assuming that $u_0$ is compactly supported we describe precisely the Type II vanishing behavior of $u$ at time $T$: we show the existence of an inner region with exponentially fast vanishing profile, which is, up to proper scaling, a {\em soliton cigar solution}, and the existence of an outer region of persistence of a logarithmic cusp. This is the only Type II singularity which has been shown to exist, so far, in the Ricci Flow in any dimension. It recovers rigorously formal asymptotics derived by J.R. King \cite{K}

Eternal Solutions to the Ricci Flow on R2\R^2

We provide the classification of eternal (or ancient) solutions of the two-dimensional Ricci flow, which is equivalent to the fast diffusion equation $\frac{\partial u}{\partial t} = \Delta \log u$ on $\R^2 \times \R.$ We show that, under the necessary assumption that for every $t \in \R$, the solution $u(\cdot, t)$ defines a complete metric of bounded curvature and bounded width, $u$ is a gradient soliton of the form $U(x,t) = \frac{2}{\beta (|x-x_0|^2 + \delta e^{2\beta t})}$, for some $x_0 \in \R^2$ and some constants $\beta >0$ and $\delta >0$

Extinction profile of complete non-compact solutions to the Yamabe flow

This work addresses the {\em singularity formation} of complete non-compact solutions to the conformally flat Yamabe flow whose conformal factors have {\em cylindrical behavior at infinity}. Their singularity profiles happen to be {\em Yamabe solitons}, which are {\em self-similar solutions} to the fast diffusion equation satisfied by the conformal factor of the evolving metric. The self-similar profile is determined by the second order asymptotics at infinity of the initial data which is matched with that of the corresponding self-similar solution. Solutions may become extinct at the extinction time $T$ of the cylindrical tail or may live longer than $T$. In the first case the singularity profile is described by a {\em Yamabe shrinker} that becomes extinct at time $T$. In the second case, the singularity profile is described by a {\em singular} Yamabe shrinker slightly before $T$ and by a matching {\em Yamabe expander} slightly after $T$

Yamabe Flow: steady solitons and type II singularities

We study the convergence of complete non-compact conformally flat solutions to the Yamabe flow to Yamabe steady solitons. We also prove the existence of Type II singularities which develop at either a finite time $T$ or as $t \to +\infty$