# Some sharp bounds for Steklov eigenvalues

## TL;DR

This paper establishes sharp lower bounds for Steklov eigenvalues on star-shaped domains in hypersurfaces of revolution and paraboloids, extending classical results to more general geometries.

## Contribution

It provides new sharp lower bounds for Steklov eigenvalues on star-shaped domains in curved spaces, generalizing previous Euclidean results.

## Key findings

- Derived sharp lower bounds for Steklov eigenvalues.
- Extended bounds to domains in hypersurfaces of revolution.
- Applied bounds to domains in paraboloids.

## Abstract

This work is an extension of a result given by Kuttler and Sigillito (SIAM Rev $10$:$368-370$, $1968$) on a star-shaped bounded domain in $\mathbb{R}^2$. Let $\Omega$ be a star-shaped bounded domain in a hypersurface of revolution, having smooth boundary. In this article, we obtain a sharp lower bound for all Steklov eigenvalues on $\Omega$ in terms of the Steklov eigenvalues of the largest geodesic ball contained in $\Omega$ with the same center as $\Omega$. We also obtain similar bounds for all Steklov eigenvalues on star-shaped bounded domain in paraboloid, $P = \left\lbrace (x, y, z) \in \mathbb{R}^{3} : z = x^2 + y^2\right\rbrace$.