A lower bound for the scalar curvature of noncompact nonflat Ricci shrinkers

TL;DR

This paper establishes a sharp quadratic decay bound for scalar curvature in noncompact nonflat Ricci shrinkers, building on recent work and providing examples that demonstrate the bound's optimality.

Contribution

It proves a sharp quadratic decay rate for scalar curvature in noncompact nonflat Ricci shrinkers, extending recent results and confirming their optimality with explicit examples.

Findings

Noncompact nonflat Ricci shrinkers have at most quadratic scalar curvature decay.

Examples by Feldman, Ilmanen, and Knopf show this decay rate is sharp.

The result builds on recent work by Ni and Wilking.

Abstract

We show that recent work of Ni and Wilking yields the result that a noncompact nonflat Ricci shrinker has at most quadratic scalar curvature decay. The examples of noncompact K\"{a}hler--Ricci shrinkers by Feldman, Ilmanen, and Knopf exhibit that this result is sharp.