# A Convex Surface with Fractal Curvature

## TL;DR

This paper constructs a fractal-curved convex surface derived from an octahedron, analyzes its spectral properties using finite element methods, and explores the spectrum's behavior.

## Contribution

It introduces a novel convex surface with fractal curvature based on the Sierpinski gasket and computes its Laplacian spectrum.

## Key findings

- Computed the bottom of the spectrum of the Laplacian.
- Used finite element method on polyhedral approximations.
- Speculated on the spectrum's overall behavior.

## Abstract

We construct a surface that is obtained from the octahedron by pushing out 4 of the faces so that the curvature is supported in a copy of the Sierpinski gasket in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the finite element method on polyhedral approximations of our surface, and speculate on the behavior of the entire spectrum.