陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「285G, Lecture 13: Li-Yau-Hamilton Harnack inequalities and κ-solutions」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

We now turn to the theory of parabolic Harnack inequalities , which control the variation over space and time of solutions to the scalar heat equation

which are bounded and non-negative, and (more pertinently to our applications) of the curvature of Ricci flows

已知结果和反例

whose Riemann curvature or Ricci curvature is bounded and non-negative. For instance, the classical parabolic Harnack inequality of Moser asserts, among other things, that one has a bound of the form

whenever is a bounded non-negative solution to (1) on a complete static Riemannian manifold M of bounded curvature, are spacetime points with , and is a constant which is uniformly bounded for fixed when range over a compact set. (The even more classical elliptic Harnack inequality gives (1) in the steady state case, i.e. for bounded non-negative harmonic functions.) In terms of heat kernels , one can view (1) as an assertion that the heat kernel associated to dominates (up t

证明或构造的主线

The classical proofs of the parabolic Harnack inequality do not give particularly sharp bounds on the constant . Such sharp bounds were obtained by Li and Yau , especially in the case of the scalar heat equation (1) in the case of static manifolds of non-negative Ricci curvature, using Bochner-type identities and the scalar maximum principle. In fact, a stronger differential version of (3) was obtained which implied (3) by an integration along spacetime curves (closely analog

In this current lecture, we shall discuss all of these inequalities (although 下面会 not give the full details for the proof of Hamilton’s Harnack inequality, as the computations are quite involved), and derive several important consequences of that inequality for -solutions. The material here is based on several sources, including Evans’ PDE book , Müller’s book , Morgan-Tian’s book , the paper of Cao-Zhu , and of course the primary source papers mentioned in this article.

阅读时建议盯住的点

Before we turn to the inequalities for Ricci flows (which are our main interest), we first consider the simpler case of scalar non-negative bounded solutions to the heat equation (1) on a static complete smooth Riemannian manifold . This case will not actually be used in our applications but serve as an important motivating example of the method. Our basic tools will be the scalar maximum principle and the following identity.

Exercise 1. Let be a smooth function. Establish the Bochner formula

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

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