In 1966, Mark Kac asked the famous question “Can one hear the shape of a drum?”
In his article with this question as the title, he translated it into eigenvalue problems for planar domains.
This question highlighted the relationship between eigenvalues and geometry.
One can then ask how eigenvalues are related to the geometry of the boundary.
In this talk, we consider a special type of eigenvalues, called Steklov eigenvalues, that are closely tied to boundary geometry.
We will introduce Steklov eigenvalues and explain their basic background and applications.
Then we will discuss our recent results on inequalities relating Steklov eigenvalues to the boundary area of compact manifolds.