On the Morse Index with Constraints I: An Abstract Formulation
Differential Geometry
2026-01-23 v1 Functional Analysis
Optimization and Control
Abstract
In this sequence, we first prove an abstract Morse index theorem in a Hilbert space modeling a variational problem with constraints. Then, our abstract formulation is applied to study several optimization setups including closed CMC hypersurfaces, capillary surfaces in a ball, and critical points of type-II partitioning. In this paper, we study the index and nullity of a symmetric bounded bilinear form in a Hilbert space. The main results determine precisely how these notions change when restricting to a subspace of a finite codimension.
Cite
@article{arxiv.2010.05952,
title = {On the Morse Index with Constraints I: An Abstract Formulation},
author = {Hung Tran and Detang Zhou},
journal= {arXiv preprint arXiv:2010.05952},
year = {2026}
}
Comments
First in a sequence of two papers; comments welcome