Morse Index Stability for Critical Points to Conformally invariant Lagrangians

TL;DR

This paper establishes the upper-semi-continuity of the combined Morse index and nullity for critical points of conformally invariant Lagrangians in 2D, including bubble analysis and degenerating domain cases.

Contribution

It proves the upper-semi-continuity of Morse index plus nullity under weak convergence for conformally invariant Lagrangians, extending to degenerating domains with collar length constraints.

Findings

Morse index plus nullity is upper semi-continuous under weak convergence.

The result accounts for bubbles and degenerating domains.

The analysis applies to maps from closed surfaces into smooth manifolds.

Abstract

We prove the upper-semi-continuity of the Morse index plus nullity of critical points to general conformally invariant Lagrangians in dimension 2 under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of an arbitrary sequence of weakly converging critical points to a general conformally invariant Lagrangians of maps from an arbitrary closed surface into an arbitrary closed smooth manifold passes to the limit in the following sense : it is asymptotically bounded from above by the sum of the Morse indices plus the nullity of the weak limit and the bubbles, while it was well known that the sum of the Morse index of the weak limit with the Morse indices of the bubbles is asymptotically bounded from above by the Morse indices of the weakly converging sequence. The main result is then extended to the case of sequences of maps from sequences of domains degenerating to a punctured Riemann surface assuming that the lengths of the images by the maps of the collars associated to this degeneration stay below some critical length.