论文标题
随着RICCI流的向后热方程的波浪近似
Wave Approximation of Backward Heat Equation with Ricci Flow
论文作者
论文摘要
在本文中，我们将带有RICCI流动的向后热方程式的溶液视为在较大的歧管上使用波前集分析的波浪方程溶液的无限尺寸极限。具体而言，Wave方程$ \左（\ frac {2t} {n} {n} \ cdot \ frac {\ partial^{2}} {\ partial t^{2}}}} + \ frac { δ_ {\ tilde {g}^{（n）}（t）} \ right）u = r（t，x）$之外的波前将其设置在$ \ mathbb {r} _ {t} _ {t} \ times m_ {x，x，g（t），g（t）} $求解后向后热方程$ \ e} = -r（t，x）$在某些适当的时间间隔内。我们从欧几里得案开始讨论这种近似，然后扩展到开放的Riemannian歧管情况。这个想法部分来自Perelman的原始论文，证明了Poincaré猜想以及Terence Tao在UCLA中的笔记。
In this paper, we consider solutions of the backward heat equation with Ricci flow on manifolds as a type of infinite dimensional limit of solutions of a wave equation on a larger manifold with an analysis of wavefront set. Specifically, the projection of the solution of the wave equation $ \left(\frac{2t}{N} \cdot \frac{\partial^{2}}{\partial t^{2}} + \frac{tR(t, x)}{N} \frac{\partial}{\partial t} - Δ_{\tilde{g}^{(N)}(t)} \right) u = R(t, x) $ outside its wavefront set onto $ \mathbb{R}_{t} \times M_{x, g(t)} $ solves the backward heat equation $ \partial_{t} u + Δ_{x, g(t)} u = -R(t, x) $ within some appropriate time interval. We discuss this approximation starting from Euclidean case, and then extend to the open Riemannian manifold situation. This idea partially comes from Perelman's original papers in proving Poincaré conjecture as well as Terence Tao's Notes in UCLA.
