Geometric analysis on manifolds with ends

TL;DR

This survey reviews recent advances in geometric analysis on manifolds with ends, highlighting new constructions with oscillating volume functions that may influence heat kernel estimates.

Contribution

It introduces novel manifold constructions with oscillating volume functions, potentially affecting heat kernel behavior, and summarizes recent progress in the field.

Findings

Manifolds with oscillating volume functions can have different heat kernel estimates.

Recent progress includes new geometric analysis techniques on manifolds with ends.

Constructed examples demonstrate varied heat kernel behaviors.

Abstract

In this survey article, we discuss some recent progress on geometric analysis on manifold with ends. In the final section, we construct manifolds with ends with oscillating volume functions which may turn out to have a different heat kernel estimates from those provided by known results.