Flows with surgery revisited

TL;DR

This paper presents a novel method for establishing the existence of geometric flows with surgery that avoids the need for a priori estimates, using a hybrid compactness theorem to handle limits near surgery regions.

Contribution

Introduces a new approach to flows with surgery that relies on a hybrid compactness theorem, applicable without prior smooth estimates, demonstrated in mean-convex surfaces in A3^3.

Findings

Provides a new proof of existing existence results for mean-convex surfaces with surgery.

Develops a hybrid compactness theorem combining smooth and weak limits.

Lays groundwork for extending the method to other geometric settings.

Abstract

In this paper, we introduce a new method to establish existence of geometric flows with surgery. In contrast to all prior constructions of flows with surgery in the literature our new approach does not require any a priori estimates in the smooth setting. Instead, our approach is based on a hybrid compactness theorem, which takes smooth limits near the surgery regions but weak limits in all other regions. For concreteness, here we develop our new method in the classical setting of mean-convex surfaces in $\mathbb{R}^3$, thus giving a new proof of the existence results due to Brendle-Huisken and Haslhofer-Kleiner. Other settings, including in particular free boundary surfaces, will be addressed in subsequent work.