{\L}ojasiewicz inequalities near simple bubble trees

TL;DR

This paper establishes a gap phenomenon for critical points of the H-functional on closed non-spherical surfaces with constant H, and proves Łojasiewicz inequalities for sequences approaching bubble trees, advancing understanding of geometric analysis.

Contribution

It introduces conditions under which Łojasiewicz inequalities hold near almost-critical points, specifically in the context of bubble trees on surfaces.

Findings

Proves a gap phenomenon for critical points of the H-functional.

Establishes Łojasiewicz inequalities near bubble trees.

Provides conditions for Łojasiewicz inequalities in a finite-dimensional setting.

Abstract

In this paper we prove a gap phenomenon for critical points of the $H$-functional on closed non-spherical surfaces when $H$ is constant, and in this setting furthermore prove that sequences of almost critical points satisfy {\L}ojasiewicz inequalities as they approach the first non-trivial bubble tree. To prove these results we derive sufficient conditions for {\L}ojasiewicz inequalities to hold near a finite-dimensional submanifold of almost-critical points for suitable functionals on a Hilbert space.