3 problems

Let $(\mathbb{B}^2,g)$ be the Euclidean unit disk, let $\eta=-y\,dx+x\,dy$ be the rotational $1$-form, and let $\sigma_1^{t\eta}(\mathbb{B}^2)$ denote the first eigenvalue of the m…

Let $(H^b)_{d_B}$ be the graph defined in the paper, with $b$ boundary vertices and boundary diameter $d_B$. For a graph with $b$ boundary vertices and boundary diameter $d_B$, let…

Let the cuboid be $[-a,a]\times[-b,b]\times[-1,1]$, with $a,b\leq 1$, and consider the surface plotting its first Steklov invariant in the first octant. Differentiability and origi…