6 problems

Yang–Yau sharpness conjecture. The Yang–Yau inequality is sharp for a family of metrics on surfaces of genus two.

Let $M$ be a closed orientable Riemannian surface of genus $\gamma$, and let $\Lambda_1(M)$ denote the supremum of the normalized first Laplace eigenvalue over all Riemannian metri…

Piecewise smooth immersion conjecture. There exists a piecewise smooth isometric immersion

Reznikov's Lindelöf-type conjecture. The geodesic period integrals on compact hyperbolic surfaces satisfy the estimate above. This would improve the known uniform bound and is pres…

Let $\Lambda^t_k(\gamma)$ denote the supremum of the normalized $k$-th Laplace–Beltrami eigenvalue over closed orientable Riemannian surfaces of genus $\gamma$, with $k\geq 1$. Let…

Let $(\Omega,g_1)$ and $(\Omega,g_2)$ be Riemannian surfaces. Two such surfaces are $\sigma$-isometric when their metrics are conformally equivalent, $g_2=\rho g_1$, with…