# Minimal surfaces near short geodesics in hyperbolic $3$-manifolds

## TL;DR

This paper investigates how the minimal surface area functional in hyperbolic 3-manifolds behaves under geometric convergence, focusing on the interaction with short geodesics and establishing continuity properties.

## Contribution

It proves the lower semi-continuity and conditions for continuity of the minimal surface area functional in hyperbolic 3-manifolds, highlighting the role of short geodesics.

## Key findings

- The functional is lower semi-continuous under geometric convergence.
- Continuity holds when the minimal surface satisfies certain conditions.
- Interaction between minimal surfaces and short geodesics is characterized.

## Abstract

If $M$ is a finite volume complete hyperbolic $3$-manifold, the quantity $\mathcal A_1(M)$ is defined as the infimum of the areas of closed minimal surfaces in $M$. In this paper we study the continuity property of the functional $\mathcal A_1$ with respect to the geometric convergence of hyperbolic manifolds. We prove that it is lower semi-continuous and even continuous if $\mathcal A_1(M)$ is realized by a minimal surface satisfying some hypotheses. Understanding the interaction between minimal surfaces and short geodesics in $M$ is the main theme of this paper