# Stability for a second type partitioning problem

## TL;DR

This paper investigates the stability properties of type-II partitioning problems, classifying stable hypersurfaces in space forms, providing topological restrictions in convex domains, and establishing Morse index bounds based on topology.

## Contribution

It offers a complete classification of stable type-II stationary hypersurfaces in space forms and topological restrictions for surfaces in convex domains, along with Morse index bounds.

## Key findings

- Stable hypersurfaces are totally geodesic n-balls in space forms
- Topological restrictions for stable surfaces in convex domains
- Lower bounds for Morse index based on topology

## Abstract

In this paper, we study stability and instability problem for type-II partitioning problem. First, we make a complete classification of stable type-II stationary hypersurfaces in a ball in a space form as totally geodesic $n$-balls. Second, for general ambient spaces and convex domains, we give some topological restriction for type-II stable stationary immersed surfaces in two dimension. Third, we give a lower bound for the Morse index for type-II stationary hypersurfaces in terms of their topology.