On an alternate proof of Hamilton's matrix Harnack inequality for the Ricci flow

Details On an alternate proof of Hamilton's matrix Harnack inequality for the Ricci flow On an alternate proof of Hamilton's matrix Harnack inequality for the Ricci flow

2001-10-24

arxiv.org/abs/math/0110261v1

Mathematics

Differential Geometry

9 pages

Scientific paper

Based on a suggestion of Richard Hamilton, we give an alternate proof of his matrix Harnack inequality for solutions of the Ricci flow with positive curvature operator. This Harnack inequality says that a certain endomorphism, consisting of an expression in the curvature and its first two covariant derivatives, of the bundle of 2-forms Whitney sum 1-forms is nonnegative. The idea is to consider the 2-form which minimizes the associated quadratic form to obtain a symmetric 2-tensor. A long but straightforward computation implies this 2-tensor is a subsolution to heat-type equation. A standard application of the maximum principle implies the result.

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