arXiv subjects
Publications and source records attributed to Richard Hamilton.
We consider an ancient solution $g(\cdot,t)$ of the Ricci flow on a compact surface that exists for $t\in (-\infty,T)$ and becomes spherical at time $t=T$. We prove that the metric $g(\cdot,t)$ is either a family of contracting spheres, which is a type I ancient solution, or a Rosenau solution, which is a type II ancient solution.
We consider an embedded convex ancient solution $Γ_t$ to the curve shortening flow in $\mathbb{R}^2$. We prove that there are only two possibilities: the family $Γ_t$ is either the family of contracting circles, which is a type I ancient solution, or the family of evolving Angenent ovals, which correspond to a type II ancient solution to the curve shortening flow. We also give a necessary and sufficient curvature condition for an embedded, closed ancient solution to the curve shortening flow to be convex.
In this paper we consider the class $\mathcal{A}$ of those solutions $u(x,t)$ to the conjugate heat equation $\frac{d}{dt}u = -Δu + Ru$ on compact Kähler manifolds $M$ with $c_1 > 0$ (where $g(t)$ changes by the unnormalized Kähler Ricci flow, blowing up at $T < \infty$), which satisfy Perelman's differential Harnack inequality on $[0,T)$. We show $\mathcal{A}$ is nonempty. If $|\ric(g(t))| \le \frac{C}{T-t}$, which is alaways true if we have type I singularity, we prove the solution $u(x,t)$ satisfies the elliptic type Harnack inequlity, with the constants that are uniform in time. If the flow $g(t)$ has a type I singularity at $T$, then $\mathcal{A}$ has excatly one element.
We introduce a geometric evolution equation for 3-manifolds with sectional curvature of one sign which is in some sense dual to the Ricci flow. On a closed 3-manifold with negative sectional curvature, we establish short time existence and a pair of monotonicity formulas for solutions to the flow. One of these formulas shows that, provided the solution exists for all time, the metric approaches hyperbolic in an integral sense. Long time existence is still an open problem.